2.1 K-theoretic symmetric functions

A is a filling of a Young diagram $\lambda / \mu $ with positive integers such that the rows are weakly increasing left-to-right and strictly increasing top-to-bottom. The of a semistandard tableau T is

$$\begin{align*}\operatorname{\mathrm{wt}}(T) = \prod_{i=1}^{\infty} x_i^{m_i}, \end{align*}$$

where $m_i$ is the number of i’s that appear in T. This is a finite product since there are finitely many boxes in $\lambda / \mu $ .

The is the generating function

$$\begin{align*}s_{\lambda / \mu}({\mathbf{x}}) = \sum_T \operatorname{\mathrm{wt}}(T), \end{align*}$$

where we sum over all semistandard tableaux T of shape $\lambda / \mu $ . The Schur functions $\{ s_{\lambda } \}_{\lambda \in \mathcal {P}}$ form a basis for the ring of symmetric functions, and so we can define the by declaring the Schur functions are an orthonormal basis

$$\begin{align*}\left\langle s_{\lambda}, s_{\mu} \right\rangle = \delta_{\lambda\mu}. \end{align*}$$

Furthermore, we have a natural grading on symmetric functions by having the degree of $s_{\lambda }$ be $\left \lvert \lambda \right \rvert $ . There is also an algebra involution defined by $\omega s_{\lambda / \mu } = s_{\lambda ' / \mu '}$ . We refer the reader to [Reference StanleySta99, Ch. 7] and [Reference MacdonaldMac15, Ch. I] for more details on symmetric functions.

A of skew shape $\lambda / \mu $ is a filling of the Young diagram by hook shaped tableau satisfying the local conditions

(provided the requisite box exists). Note that this is a generalization of the semistandard conditions as the conditions are equivalent to the usual standard ones when $\mathsf {a},\mathsf {b},\mathsf {c}$ all consist of a single entry.

For $\mu \subseteq \lambda $ , the Footnote ² is the generating function

$$\begin{align*}G_{\lambda / \mu}({\mathbf{x}}; \boldsymbol{\alpha}, \boldsymbol{\beta}) = \sum_T \prod_{\mathsf{b} \in T} (-\alpha_i)^{a(\mathsf{b})} (-\beta_j)^{b(\mathsf{b})} \operatorname{\mathrm{wt}}(\mathsf{b}), \end{align*}$$

where we sum over all hook-valued tableaux T of shape $\lambda / \mu $ , product over all entries $\mathsf {b}$ in T with $a(\mathsf {b})$ (resp. $b(\mathsf {b})$ ) the arm (resp. leg) of the shape of $\mathsf {b}$ and i (resp. j) the row (resp. column) of the entry. We indicate various specializations and relation with the literature in Table 1. We note that technically the canonical Grothendieck functions lie in the completion of the ring of symmetric functions given by the grading; so we are allowed to take infinite sums with finite sums in each graded component. However, this does not affect our computations or results, and so we suppress this distinction in this paper. A basis for (the completion of) symmetric functions is given by $\{G_{\lambda }({\mathbf {x}}; \boldsymbol {\alpha }, \boldsymbol {\beta })\}_{\lambda \in \mathcal {P}}$ since

$$\begin{align*}G_{\lambda}({\mathbf{x}}; \boldsymbol{\alpha}, \boldsymbol{\beta}) = s_{\lambda}({\mathbf{x}}) + \sum_{\lambda \subsetneq \mu} (-1)^{\left\lvert \mu \right\rvert - \left\lvert \lambda \right\rvert} E_{\lambda}^{\mu}(\boldsymbol{\alpha}, \boldsymbol{\beta}) s_{\mu}({\mathbf{x}}), \end{align*}$$

where $E_{\lambda }^{\mu }(\boldsymbol {\alpha }, \boldsymbol {\beta }) \in \mathbb {Z}_{\geq 0}[\boldsymbol {\alpha }, \boldsymbol {\beta }]$ [Reference Hawkes and ScrimshawHS20, Reference Hwang, Jang, Soo Kim, Song and SongHJK⁺24, Reference Hwang, Jang, Soo Kim, Song and SongHJK⁺25].

Table 1 The relationship between our sign choices and some other papers in the literature.

We note that if $\boldsymbol {\alpha } = 0$ , then all entries of the hook-valued tableau must be column shapes, which we can equate with sets and recovers the set-valued tableau description of [Reference Chan and PfluegerCP21], which refines [Reference Skovsted BuchBuc02]. Likewise, if $\boldsymbol {\beta } = 0$ , then we have row shapes that we equate with multisets, refining [Reference Lam and PylyavskyyLP07]. Similarly, we call $G_{\lambda }({\mathbf {x}}; \boldsymbol {\beta }) := G_{\lambda }({\mathbf {x}}; 0, \boldsymbol {\beta })$ a Footnote ³ and $J_{\lambda }({\mathbf {x}}; \boldsymbol {\alpha }) := G_{\lambda }({\mathbf {x}}; \boldsymbol {\alpha }, 0)$ a .

The are defined as the dual basis to the canonical Grothendieck functions under the Hall inner product. A combinatorial definition was given in [Reference Hwang, Jang, Soo Kim, Song and SongHJK⁺25], which can be seen as the obvious refinement of the rim border tableaux description of [Reference YeliussizovYel17]. As such, we can extend the definition to skew shapes. For our purposes, we will only use the dual basis to the (weak) Grothendieck functions (that is, $\boldsymbol {\alpha } = 0$ or $\boldsymbol {\beta } = 0$ ), and thus we restrict to describing the combinatorics of the special cases for the dual basis to the (weak) Grothendieck functions following [Reference Lam and PylyavskyyLP07, Thm. 9.15]. A (resp. ) is a semistandard Young tableau where we are allowed to merge boxes in the same column (resp. row) with the entry considered to be aligned at the bottom (resp. right). The weight is the same as for semistandard tableaux. We then have the generating functions

$$ \begin{align*} g_{\lambda/\mu}({\mathbf{x}}; \boldsymbol{\beta}) &= g_{\lambda/\mu}({\mathbf{x}}; 0, \boldsymbol{\beta}) = \sum_T \prod_{j=1}^{\ell-1} \beta_j^{r_j} \operatorname{\mathrm{wt}}(T), \\ j_{\lambda/\mu}({\mathbf{x}}; \boldsymbol{\beta}) &= g_{\lambda/\mu}({\mathbf{x}}; \boldsymbol{\alpha}, 0) = \sum_T \prod_{j=1}^{\lambda_1-1} \alpha_j^{c_j} \operatorname{\mathrm{wt}}(T), \end{align*} $$

where $r_j$ (resp. $c_j$ ) is the number of boxes in row (resp. column) j that have been merged with the box below (resp. to the right) and we sum over all reverse plane partitions (resp. valued-set tableaux) of shape $\lambda / \mu $ . The definition of valued-set tableau follows [Reference Hawkes and ScrimshawHS20], which is conjugate to that of [Reference Lam and PylyavskyyLP07].

We note that our description of reverse plane partitions matches the classical definition by simply filling in the merged boxes with the entry of the merged box. The inverse map is merging duplicated entries in the same column. Thus, the description of $g_{\lambda / \mu }({\mathbf {x}}; \boldsymbol {\beta })$ matches that introduced in [Reference Galashin, Grinberg and LiuGGL16]. We have described reverse plane partitions as above to better demonstrate the following symmetry that motivated the definition of canonical Grothendieck polynomials, even though we will write our reverse plane partitions below using the classical description.

Theorem 2.1 [Reference Hwang, Jang, Soo Kim, Song and SongHJK⁺24, Thm. 1.7].

We have

$$\begin{align*}\omega G_{\lambda/\mu}({\mathbf{x}}; \boldsymbol{\alpha}, \boldsymbol{\beta}) = G_{\lambda'/\mu'}({\mathbf{x}}; \boldsymbol{\beta}, \boldsymbol{\alpha}), \qquad\qquad \omega g_{\lambda/\mu}({\mathbf{x}}; \boldsymbol{\alpha}, \boldsymbol{\beta}) = g_{\lambda'/\mu'}({\mathbf{x}}; \boldsymbol{\beta}, \boldsymbol{\alpha}). \end{align*}$$

As a consequence of Theorem 2.1, we have the relationship

$$\begin{align*}\omega G_{\lambda/\mu}({\mathbf{x}}; \boldsymbol{\alpha}) = J_{\lambda/\mu}({\mathbf{x}}; \boldsymbol{\alpha}), \qquad\qquad \omega g_{\lambda/\mu}({\mathbf{x}}; \boldsymbol{\alpha}) = j_{\lambda/\mu}({\mathbf{x}}; \boldsymbol{\alpha}), \end{align*}$$

between the (dual) weak Grothendieck functions and the (dual) Grothendieck functions. These are a refinement of [Reference Lam and PylyavskyyLP07, Prop. 9.22].

For the canonical Grothendieck polynomials, we note that the skew shape description is not natural from the branching rules (a precise description is given below), the skewing operator [Reference IwaoIwa22, Sec. 4], nor coproduct formula [Reference Skovsted BuchBuc02, Sec. 5] perspective. Refining [Reference Skovsted BuchBuc02, Eq. (6.4)] and [Reference YeliussizovYel17, Prop. 8.8], we define [Reference Iwao, Motegi and ScrimshawIMS24, Sec. 4.1]

(2.1) $$ \begin{align} G_{\lambda/\!\!/\mu}({\mathbf{x}}; \boldsymbol{\alpha}, \boldsymbol{\beta}) := \sum_{\nu \subseteq \mu} \prod_{(i,j) \in \mu/\nu} -(\alpha_i + \beta_j) G_{\lambda / \nu}({\mathbf{x}}; \boldsymbol{\alpha}, \boldsymbol{\beta}), \end{align} $$

where $\nu $ is formed by removing some of the corners of $\mu $ (that is, boxes $(i, \mu _i)$ such that $\mu _i> \mu _{i+1}$ ). We remark that we have the following identity by the same proof as [Reference Iwao, Motegi and ScrimshawIMS24, Thm. 4.4].

Proposition 2.2. We have

$$\begin{align*}\omega G_{\lambda/\!\!/\mu}({\mathbf{x}}; \boldsymbol{\alpha}, \boldsymbol{\beta}) = G_{\lambda'/\!\!/\mu'}({\mathbf{x}}; \boldsymbol{\beta}, \boldsymbol{\alpha}). \end{align*}$$

Equation (2.1) allows us to give the following (cf. [Reference YeliussizovYel17, Prop. 8.7, 8.8]).

Proposition 2.3 (Branching rules [Reference Iwao, Motegi and ScrimshawIMS24, Prop. 4.5]).

We have

$$ \begin{align*} G_{\lambda/\mu}({\mathbf{x}}, {\mathbf{y}}; \boldsymbol{\alpha}, \boldsymbol{\beta}) & = \sum_{\nu \subseteq \lambda} G_{\lambda/\!\!/\nu}({\mathbf{y}}; \boldsymbol{\alpha}, \boldsymbol{\beta}) G_{\nu/\mu}({\mathbf{x}}; \boldsymbol{\alpha}, \boldsymbol{\beta}), \\ G_{\lambda/\!\!/\mu}({\mathbf{x}}, {\mathbf{y}}; \boldsymbol{\alpha}, \boldsymbol{\beta}) & = \sum_{\mu \subseteq \nu \subseteq \lambda} G_{\lambda/\!\!/\nu}({\mathbf{y}}; \boldsymbol{\alpha}, \boldsymbol{\beta}) G_{\nu/\!\!/\mu}({\mathbf{x}}; \boldsymbol{\alpha}, \boldsymbol{\beta}), \\ g_{\lambda/\mu}({\mathbf{x}}, {\mathbf{y}}; \boldsymbol{\alpha}, \boldsymbol{\beta}) & = \sum_{\mu \subseteq \nu \subseteq \lambda} g_{\lambda / \nu}({\mathbf{y}}; \boldsymbol{\alpha}, \boldsymbol{\beta}) g_{\nu/\mu}({\mathbf{x}}; \boldsymbol{\alpha}, \boldsymbol{\beta}). \end{align*} $$

As was shown in [Reference Iwao, Motegi and ScrimshawIMS24, Eq. (4.3)], we have

$$\begin{align*}G_{\lambda/\mu}({\mathbf{x}}; \boldsymbol{\alpha}, \boldsymbol{\beta})=\prod_{(i,j)\in \mu/\lambda}(\alpha_i+\beta_j)\cdot G_{\lambda/(\lambda\cap \mu)}({\mathbf{x}}; \boldsymbol{\alpha}, \boldsymbol{\beta}), \end{align*}$$

and in particular, we do not have the vanishing property for $G_{\lambda /\mu }({\mathbf {x}}; \boldsymbol {\alpha }, \boldsymbol {\beta })$ that $G_{\lambda /\!\!/\mu }({\mathbf {x}}; \boldsymbol {\alpha }, \boldsymbol {\beta })$ exhibits: that $G_{\lambda /\!\!/\mu }({\mathbf {x}}; \boldsymbol {\alpha }, \boldsymbol {\beta }) = 0$ whenever $\mu \subseteq \lambda $ .

We will also need the skew Cauchy formula from [Reference Iwao, Motegi and ScrimshawIMS24, Thm. 4.6] (nonskew versions can be found in [Reference Hwang, Jang, Soo Kim, Song and SongHJK⁺24, Reference Hwang, Jang, Soo Kim, Song and SongHJK⁺25] or as a consequence of [Reference Chan and PfluegerCP21, Rem. 3.9]). This is a refined version of [Reference YeliussizovYel19, Thm. 1.1].

Theorem 2.4 (Skew Cauchy formula).

We have

$$\begin{align*}\sum_{\lambda} G_{\lambda /\!\!/ \mu}({\mathbf{x}}; \boldsymbol{\alpha}, \boldsymbol{\beta}) g_{\lambda / \nu}({\mathbf{y}}; \boldsymbol{\alpha}, \boldsymbol{\beta}) = \prod_{i,j} \frac{1}{1 - x_i y_j} \sum_{\eta} G_{\nu /\!\!/ \eta}({\mathbf{x}}; \boldsymbol{\alpha}, \boldsymbol{\beta})g_{\mu / \eta}({\mathbf{y}}; \boldsymbol{\alpha}, \boldsymbol{\beta}). \end{align*}$$

2.2 Supersymmetric functions

We set some additional standard notation from symmetric function theory. We (again) refer the reader to the standard textbooks [Reference MacdonaldMac15, Reference StanleySta99] for more information. Let

$$ \begin{align*} e_m({\mathbf{x}}) & = \sum_{i_1> \cdots > i_m} x_{i_1} \dotsm x_{i_m}, & e_{\lambda}({\mathbf{x}}) & = e_{\lambda_1} \cdots e_{\lambda_{\ell}}, \\ h_m({\mathbf{x}}) & = \sum_{i_1 \leq \cdots \leq i_m} x_{i_1} \dotsm x_{i_m}, & h_{\lambda}({\mathbf{x}}) & = h_{\lambda_1} \cdots h_{\lambda_{\ell}}, \\ p_m({\mathbf{x}}) & = \sum_{i=1}^{\infty} x_i^m, & p_{\lambda}({\mathbf{x}}) & = p_{\lambda_1} \cdots p_{\lambda_{\ell}}, \end{align*} $$

denote the elementary, homogeneous, and power sum symmetric functions, respectively. We consider $e_0({\mathbf {x}}) = h_0({\mathbf {x}}) = p_0({\mathbf {x}}) = 1$ . We have $\omega p_m({\mathbf {x}}) = (-1)^{m-1} p_m({\mathbf {x}}) = -p_m(-{\mathbf {x}})$ for $m> 0$ . Since, the powersum symmetric functions generate the ring of symmetric functions as a polynomial ring (over $\mathbb {Q}$ ) and the monomials (in the variables $p_m := p_m({\mathbf {x}})$ ) form a basis. Thus, we can define polynomials by the equations

$$\begin{align*}E_{\mu}(p_1, p_2, \ldots) = e_{\mu}({\mathbf{x}}), \qquad\qquad H_{\mu}(p_1, p_2, \ldots) = h_{\mu}({\mathbf{x}}), \qquad\qquad S_{\mu}(p_1, p_2, \ldots) = s_{\mu}({\mathbf{x}}). \end{align*}$$

For example, $E_{21}(p_1, p_2, \ldots ) = \frac {1}{2} p_1^3 - \frac {1}{2} p_2 p_1$ .

Now we recall some particular ; we refer the reader to [Reference MacdonaldMac15, Ch. I] for more details. We define the supersymmetric elementary, homogeneous, powersum, and Schur functions as

$$ \begin{align*} e_m({\mathbf{x}}/{\mathbf{y}}) & = \sum_{k=0}^m (-1)^{m-k} e_k({\mathbf{x}}) h_{m-k}({\mathbf{y}}), \\ h_m({\mathbf{x}}/{\mathbf{y}}) & = \sum_{k=0}^m (-1)^{m-k} h_k({\mathbf{x}}) e_{m-k}({\mathbf{y}}), \\ p_m({\mathbf{x}}/{\mathbf{y}}) & = p_m({\mathbf{x}}) - p_m({\mathbf{y}}), \\ s_{\lambda}({\mathbf{x}}/{\mathbf{y}}) & = \sum_{\mu} (-1)^{\left\lvert \lambda \right\rvert - \left\lvert \mu \right\rvert} s_{\mu}({\mathbf{x}}) s_{\lambda' / \mu'}({\mathbf{y}}). \end{align*} $$

When ${\mathbf {y}} = \emptyset $ , that is we have set all of the ${\mathbf {y}}$ indeterminates to $0$ , we have $f({\mathbf {x}} / {\mathbf {y}}) = f({\mathbf {x}})$ for any supersymmetric function f. The involution $\omega $ also extends to supersymmetric functions by $ \omega s_{\lambda / \mu }({\mathbf {x}} / {\mathbf {y}}) = s_{\lambda '/\mu '}({\mathbf {x}} / {\mathbf {y}}). $

The supersymmetric functions can also be described in terms of plethystic substitution. While we will not give a detailed account, we will briefly review the relevant descriptions for understanding the results in [Reference Hwang, Jang, Soo Kim, Song and SongHJK⁺24, Reference Hwang, Jang, Soo Kim, Song and SongHJK⁺25] and refer the reader to [Reference Loehr and RemmelLR11] and [Reference MacdonaldMac15, Ch. I] for a more detailed description. Let $X = x_1 + x_2 + \cdots $ and $Y = y_1 + y_2 + \cdots $ . For a symmetric function f, we define $f[X] = f(x_1, x_2, \ldots )$ , and if $Z = z_1 + z_2 + \cdots + z_n$ , then we have $f[Z] = f(z_1, z_2, \dotsc , z_n, 0, 0, \ldots )$ . We also can define

$$ \begin{align*} h_m[X - Y] = h_m({\mathbf{x}}/{\mathbf{y}}), \qquad\qquad e_m[X - Y] = e_m({\mathbf{x}}/{\mathbf{y}}), \qquad\qquad p_m[X - Y] = p_m({\mathbf{x}}/{\mathbf{y}}). \end{align*} $$

As a consequence, we have that $h_m[-Y] = (-1)^m e_m({\mathbf {y}})$ and $e_m[-Y] = (-1)^m h_m({\mathbf {y}})$ . Furthermore, we have the well-known plethystic identities

$$ \begin{align*} h_m\big( ({\mathbf{x}} \sqcup {\mathbf{x}}^{\prime})/({\mathbf{y}} \sqcup {\mathbf{y}}^{\prime}) \big) & = h_m[X + X^{\prime} - Y - Y^{\prime}] = \sum_{a+b=m} h_a({\mathbf{x}}/{\mathbf{y}}) h_b({\mathbf{x}}^{\prime}/{\mathbf{y}}^{\prime}), \\ e_m\big( ({\mathbf{x}} \sqcup {\mathbf{x}}^{\prime})/({\mathbf{y}} \sqcup {\mathbf{y}}^{\prime}) \big) & = e_m[X + X^{\prime} - Y - Y^{\prime}] = \sum_{a+b=m} e_a({\mathbf{x}}/{\mathbf{y}}) e_b({\mathbf{x}}^{\prime}/{\mathbf{y}}^{\prime}), \end{align*} $$

(see, e.g., [Reference Hwang, Jang, Soo Kim, Song and SongHJK⁺24, Prop. 2.1]). Next, we recall the notation given in [Reference Hwang, Jang, Soo Kim, Song and SongHJK⁺24, Def. 2.3]:

$$\begin{align*}h_m[X \ominus Y] := \sum_{a-b=m} h_a[X] h_b[Y], \qquad\qquad e_m[X \ominus Y] := \sum_{a-b=m} e_a[X] e_b[Y]. \end{align*}$$

We note that these can have infinite nonzero terms and be nonzero even when m is negative.

In order to avoid confusion with the plethystic negative and negating the variables, we will not use plethystic notation, and instead follow [Reference Iwao, Motegi and ScrimshawIMS24], where we write $h_m({\mathbf {x}} /\!\!/ {\mathbf {y}}) := h_m[X \ominus Y]$ and $e_m({\mathbf {x}} /\!\!/ {\mathbf {y}}) := e_m[X \ominus Y]$ .

Additionally, note the ordering of the variables for the elementary symmetric functions. We have chosen this so that the definitions extend to the introduced by Fomin and Greene [Reference Fomin and GreeneFG98]. We summarize the results that we need as follows. For a sequence of linear operators $\{\tau _i \colon {\mathbf {k}}[{\mathcal {P}}] \to \mathbf {k}[\mathcal {P}] \}_{i=1}^{\infty }$ , the are

(2.2a) $$ \begin{align} \tau_j \tau_i \tau_k & = \tau_j \tau_k \tau_i & &\text{for all } i \geq j> k, \quad i - k \geq 2, \end{align} $$

(2.2b) $$ \begin{align} \tau_i \tau_k \tau_j & = \tau_k \tau_i \tau_j & &\text{for all } i> j \geq k, \quad i - k \geq 2,\end{align} $$

(2.2c) $$ \begin{align} (\tau_i + \tau_{i+1}) \tau_{i+1} \tau_i & = \tau_{i+1} \tau_i (\tau_i + \tau_{i+1}) & & \text{for all } i. \end{align} $$

The (strong) Knuth relations are formed by removing the requirement $i - k \geq 2$ from (2.2).

2.3 Free fermions and Schur operators

We describe the free-fermion presentation of the (dual) canonical Grothendieck polynomials from [Reference Iwao, Motegi and ScrimshawIMS24]. For more details, we refer the reader to [Reference Alexandrov and ZabrodinAZ13, Reference KacKac90, Reference Miwa, Jimbo and DateMJD00]. Let $\mathbf {k}$ be a field of characteristic $0$ . We consider the unital associative $\mathbf {k}$ -algebra of generated by $\{\psi _n, \psi _n^* \mid n \in \mathbb {Z}\}$ with relations

$$\begin{align*}\psi_m \psi_n + \psi_n \psi_m = \psi_m^* \psi_n^* + \psi_n^* \psi_m^* = 0, \qquad\qquad \psi_m \psi_n^* + \psi_n^* \psi_m = \delta_{m,n}, \end{align*}$$

known as the canonical anticommuting relations. This is a Clifford algebra arising from the canonical bilinear form of an infinite dimensional vector space V with a basis $\{v_i\}_{i \in \mathbb {Z}}$ with its (restricted) dual space $V^*$ spanned by $\{v_i^*\}_{i \in \mathbb {Z}}$ . As such, there is an antialgebra involution on $\mathcal {A}$ defined by $\psi _n \leftrightarrow \psi _n^\ast $ ; that is $(xy)^\ast =y^\ast x^\ast $ for any $x,y\in \mathcal {A}$ . We will also define the fields

$$\begin{align*}\psi(z) = \sum_{n \in \mathbb{Z}} \psi_n z^n, \qquad\qquad \psi^*(w) = \sum_{n \in \mathbb{Z}} \psi_n w^{-n}. \end{align*}$$

The Footnote ⁴ are defined as

$$\begin{align*}a_k := \sum_{i \in \mathbb{Z}} \psi_i \psi_{i+k}^*, \end{align*}$$

and satisfy the Heisenberg algebra relations and duality

$$\begin{align*}[a_m, a_k] = m \delta_{m,-k}, \qquad\qquad a_k^* = a_{-k}. \end{align*}$$

We will use the

$$\begin{align*}H({\mathbf{x}} / {\mathbf{y}}) := \sum_{k> 0} \frac{p_k({\mathbf{x}}/{\mathbf{y}})}{k} a_k, \end{align*}$$

and the corresponding . These satisfy the relations (see, e.g., [Reference IwaoIwa23, Eq. (17), Eq. (18)])

(2.3a) $$ \begin{align} e^{H({\mathbf{x}}/{\mathbf{y}})} \psi_k e^{-H({\mathbf{x}}/{\mathbf{y}})}& = \sum_{i=0}^{\infty} h_i({\mathbf{x}}/{\mathbf{y}}) \psi_{k-i}, \end{align} $$

(2.3b) $$ \begin{align} e^{-H({\mathbf{x}}/{\mathbf{y}})} \psi^*_k e^{H({\mathbf{x}}/{\mathbf{y}})}& = \sum_{i=0}^{\infty} h_i({\mathbf{x}}/{\mathbf{y}}) \psi^*_{k+i}. \end{align} $$

Note that $-H({\mathbf {x}}/{\mathbf {y}}) = H({\mathbf {y}}/{\mathbf {x}})$ . Let $H^*({\mathbf {x}}/{\mathbf {y}}) = (H({\mathbf {x}}/{\mathbf {y}}))^*$ denote the . For $k> 0$ , we have the relations (see, e.g., [Reference Alexandrov and ZabrodinAZ13, Eq. (2.4)])

(2.4) $$ \begin{align} [a_{-k}, e^{H(t)}] = -t^k e^{H(t)}, \qquad\qquad [e^{H^*(t)}, a_k] = -t^k e^{H^*(t)}, \end{align} $$

and $[a_k, e^{H(t)}] = [e^{H^*(t)}, a_{-k}] = 0$ . We will also use the

$$\begin{align*}J({\mathbf{x}}/{\mathbf{y}}) := \omega H({\mathbf{x}} / {\mathbf{y}}) = -H(-{\mathbf{x}} / -{\mathbf{y}}). \end{align*}$$

Therefore, we can write

(2.5) $$ \begin{align} e^{H({\mathbf{x}} / {\mathbf{y}})} = \sum_{k=0}^{\infty} H_k(a_1, a_2, \ldots) p_k({\mathbf{x}} / {\mathbf{y}}), \qquad\qquad e^{J({\mathbf{x}} / {\mathbf{y}})} = \sum_{k=0}^{\infty} E_k(a_1, a_2, \ldots) p_k({\mathbf{x}} / {\mathbf{y}}). \end{align} $$

We will consider the spinor representation $\mathcal {F}$ of semi-infinite wedge products subject to a finiteness condition, but we will not present this here in detail (see, e.g., [Reference KacKac90, Reference Kac, Raina and RozhkovskayaKRR13, Reference Miwa, Jimbo and DateMJD00] for more information). This is sometimes referred to as . Instead, we will realize $\mathcal {F}$ as the cyclic $\mathcal {A}$ -representation generated by the that satisfies the relations

$$\begin{align*}\psi_n \lvert 0 \rangle = \psi_m^* \lvert 0 \rangle = 0, \qquad\qquad n < 0, \quad m \geq 0. \end{align*}$$

Therefore, we can describe the basis as the vectors

$$\begin{align*}\psi_{n_1} \psi_{n_2} \cdots \psi_{n_r} \psi_{m_1}^* \psi_{m_2}^* \cdots \psi_{m_s}^* \lvert 0 \rangle, \qquad (r,s \geq 0, n_1> \cdots n_r \geq 0 > m_s > \cdots > m_1). \end{align*}$$

We define the as

$$\begin{align*}\lvert m \rangle = \begin{cases} \psi_{m-1} \dotsm \psi_0 \lvert 0 \rangle & \text{if } m \geq 0, \\ \psi_m^* \dotsm \psi_{-1}^* \lvert 0 \rangle & \text{if } m < 0, \end{cases} \qquad\qquad \langle m \rvert = \begin{cases} \langle 0 \rvert \psi_0^* \dotsm \psi_{m-1}^* & \text{if } m \geq 0, \\ \langle 0 \rvert \psi_{-1} \dotsm \psi_m & \text{if } m < 0. \end{cases} \end{align*}$$

Note that

$$\begin{align*}e^{H({\mathbf{x}}/{\mathbf{y}})} \lvert m \rangle = \lvert m \rangle, \qquad\qquad \langle m \rvert e^{H^*({\mathbf{x}}/{\mathbf{y}})} = \langle m \rvert, \end{align*}$$

for all m. For finitely many (noncommutative) expressions $\Psi _1,\dots ,\Psi _\ell $ , we will use the notation

$$\begin{align*}\prod_{1\leq i\leq \ell}^{\rightarrow} \Psi_i=\Psi_1\Psi_2\cdots \Psi_\ell, \qquad\qquad \prod_{1\leq i\leq \ell}^{\leftarrow} \Psi_i=\Psi_\ell\cdots \Psi_2\Psi_1, \end{align*}$$

to indicate the order of multiplication. We will use the vectors

$$ \begin{align*} \lvert \lambda \rangle^{\sigma}_{[\boldsymbol{\alpha},\boldsymbol{\beta}]} & := \prod^{\rightarrow}_{1 \leq i \leq \ell} \left( e^{-H(A_{\sigma_i-1})} \psi_{\lambda_i-i} e^{H(\beta_i)} e^{H(A_{\sigma_i-1})} \right) \lvert -\ell \rangle, \\ \lvert \lambda \rangle_{\sigma}^{[\boldsymbol{\alpha},\boldsymbol{\beta}]} & := \prod^{\rightarrow}_{1 \leq i \leq \ell} \left( e^{H^*(A_{\sigma_i})} \psi_{\lambda_i-i} e^{-H^*(\beta_i)} e^{-H^*(A_{\sigma_i})} \right) { e^{H^\ast(A_{\sigma_{\ell}})}} \lvert -\ell \rangle, \end{align*} $$

where $\sigma = (\sigma _1, \sigma _2, \dotsc , \sigma _{\ell }) \in \mathbb {Z}_{\geq 0}^{\ell }$ and $A_k = -\boldsymbol {\alpha }_k = (-\alpha _1, \dotsc , - \alpha _k)$ . When $\sigma = \lambda $ , we simply write $\lvert \lambda \rangle _{[\boldsymbol {\alpha },\boldsymbol {\beta }]} := \lvert \lambda \rangle ^{\lambda }_{[\boldsymbol {\alpha },\boldsymbol {\beta }]}$ and $\lvert \lambda \rangle ^{[\boldsymbol {\alpha },\boldsymbol {\beta }]} := \lvert \lambda \rangle _{\lambda }^{[\boldsymbol {\alpha },\boldsymbol {\beta }]}$ . We use the notation

$$\begin{align*}\lvert \lambda \rangle_{[\boldsymbol{\beta}]} := \lvert \lambda \rangle^\sigma_{[0,\boldsymbol{\beta}]}, \qquad\qquad \lvert \lambda \rangle^{[\boldsymbol{\beta}]} := \lvert \lambda \rangle_\sigma^{[0,\boldsymbol{\beta}]}, \qquad\qquad \lvert \lambda \rangle := \lvert \lambda \rangle_{\sigma}^{[0,0]} = \lvert \lambda \rangle^{\sigma}_{[0,0]}. \end{align*}$$

(note these are independent of $\sigma $ ) for brevity. We will typically restrict ourselves to the subspace $\mathcal {F}^0$ , which we describe as the span of either of the bases [Reference Iwao, Motegi and ScrimshawIMS24, Thm. 3.10]

$$\begin{align*}\{\lvert \lambda \rangle_{[\boldsymbol{\alpha},\boldsymbol{\beta}]}\}_{\lambda \in \mathcal{P}}, \qquad\qquad \{\lvert \lambda \rangle^{[\boldsymbol{\alpha},\boldsymbol{\beta}]}\}_{\lambda \in \mathcal{P}}. \end{align*}$$

There is also the dual $\mathcal {A}$ -representation $\mathcal {F}^*$ , which has a canonical bilinear pairing called the that satisfies

$$\begin{align*}\langle k | m \rangle = \delta_{km}, \qquad\qquad (\langle w \rvert X) \lvert v \rangle = \langle w \rvert (X \lvert v \rangle), \end{align*}$$

for all $k,m \in \mathbb {Z}$ , $X \in \mathcal {A}$ , $\langle w \rvert \in \mathcal {F}^*$ , and $\lvert v \rangle \in \mathcal {F}$ . We abbreviate $\langle X \rangle = \langle 0 \rvert X \lvert 0 \rangle $ , and note that $\lvert k \rangle ^* = \langle k \rvert $ . Define

$$\begin{align*}{}_{[\boldsymbol{\alpha},\boldsymbol{\beta}]}\langle \lambda \rvert_{\sigma} = (\lvert \lambda \rangle_{\sigma}^{[\boldsymbol{\alpha},\boldsymbol{\beta}]})^*, \qquad\qquad {}^{[\boldsymbol{\alpha},\boldsymbol{\beta}]}\langle \lambda \rvert^{\sigma} = (\lvert \lambda \rangle^{\sigma}_{[\boldsymbol{\alpha},\boldsymbol{\beta}]})^*, \end{align*}$$

and similar abbreviations as above. With respect to this inner product, we have the orthonormal bases [Reference Iwao, Motegi and ScrimshawIMS24, Thm. 3.10]

(2.6) $$ \begin{align} {}_{[\boldsymbol{\alpha},\boldsymbol{\beta}]}\langle \lambda | \mu \rangle_{[\boldsymbol{\alpha},\boldsymbol{\beta}]} = {}^{[\boldsymbol{\alpha},\boldsymbol{\beta}]}\langle \lambda | \mu \rangle^{[\boldsymbol{\alpha},\boldsymbol{\beta}]} = \delta_{\lambda\mu}. \end{align} $$

Moreover, there is an isomorphism from $\mathcal {F}^0$ to symmetric functions defined by $\lvert v \rangle \mapsto \langle 0 \rvert e^{H({\mathbf {x}}/{\mathbf {y}})} \lvert v \rangle $ , which satisfies [Reference Iwao, Motegi and ScrimshawIMS24, Cor. 4.2, Eq. (4.1)]

(2.7) $$ \begin{align} G_{\lambda /\!\!/ \mu}({\mathbf{x}}; \boldsymbol{\alpha}, \boldsymbol{\beta}) = {}^{[\boldsymbol{\alpha},\boldsymbol{\beta}]} \langle \mu \rvert e^{H({\mathbf{x}})} \lvert \lambda \rangle^{[\boldsymbol{\alpha},\boldsymbol{\beta}]}, \qquad\qquad g_{\lambda / \mu}({\mathbf{x}}; \boldsymbol{\alpha}, \boldsymbol{\beta}) = {}_{[\boldsymbol{\alpha},\boldsymbol{\beta}]} \langle \mu \rvert e^{H({\mathbf{x}})} \lvert \lambda \rangle_{[\boldsymbol{\alpha},\boldsymbol{\beta}]}. \end{align} $$

From [Reference Iwao, Motegi and ScrimshawIMS24, Thm. 4.3], we also have

(2.8) $$ \begin{align} G_{\lambda' /\!\!/ \mu'}({\mathbf{x}}; \boldsymbol{\alpha}, \boldsymbol{\beta}) = {}^{[\boldsymbol{\alpha},\boldsymbol{\beta}]} \langle \mu \rvert e^{J({\mathbf{x}})} \lvert \lambda \rangle^{[\boldsymbol{\alpha},\boldsymbol{\beta}]}, \qquad\qquad g_{\lambda' / \mu'}({\mathbf{x}}; \boldsymbol{\alpha}, \boldsymbol{\beta}) = {}_{[\boldsymbol{\alpha},\boldsymbol{\beta}]} \langle \mu \rvert e^{J({\mathbf{x}})} \lvert \lambda \rangle_{[\boldsymbol{\alpha},\boldsymbol{\beta}]}. \end{align} $$

2.4 Particle processes

We describe the four different versions of the discrete (TASEP) given in [Reference Dieker and WarrenDW08]. All of these processes will be considered using a bosonic presentation following [Reference Dieker and WarrenDW08], where they are given by particles labeled $1, \dotsc , \ell $ that can occupy the same sites on the lattice $\mathbb {Z}_{\geq 0}$ . The position of these particles will remain in order and only move to the right (i.e., increase the value of their position), and so we can index states by partitions $\lambda $ , where $\lambda _i$ corresponds to the position of the i-th particle. To obtain a fermionic presentation and justifying calling this TASEP, simply move the i-th particle from position $\lambda _i$ to $\lambda _i - i$ , and hence, we can identify the shape $\lambda = \emptyset $ with the step initial condition:

Unless otherwise noted, we will consider our particle configurations using the bosonic description.

Example 2.6. We identify $\lambda =(3,3,1,0)$ with the (bosonic) particle distribution

with the first and second particles in position $3$ , the third particle is in position 1, and the fourth particle is in position $0$ . In terms of the fermionic presentation, we have

The j-th particle at time i will attempt to move $w_{ji}$ (which we take as a random variable) steps to the right according to either the geometric or Bernoulli distribution

$$ \begin{gather*} \mathsf{P}_{Ge}(w_{ji} = k) = (1 - \pi_j x_i) (\pi_j x_i)^k \quad (k \in \mathbb{Z}_{\geq 0}), \\ \mathsf{P}_{Be}(w_{ji} = 1) = \frac{\rho_i x_j}{1 + \rho_j x_i} \quad \text{and} \quad \mathsf{P}_{Be}(w_{ji} = 0) = (1 + \rho_j x_i)^{-1}, \end{gather*} $$

respectively, where $\pi _j x_i \in (0, 1)$ and $\rho _j x_i> 0$ . We can assemble all of these random variables, ranging over all particles and time, into a (random) matrix $W = [w_{ji}]_{i,j}$ . Since we are working in discrete time, we need a rule to determine the behavior when two particles decide to move simultaneously that results in a conflict. There are two natural ways to resolve this.

• The larger particle pushes the smaller particle.

• The smaller particle blocks the larger particle.

For the geometric (resp. Bernoulli) distribution, we will update the particles from largest-to-smallest (resp. smallest-to-largest). As such, we obtain four different variations of discrete TASEP, which we organize using the same indexing as [Reference Dieker and WarrenDW08]:

1. 1. Geometric distribution with pushing behavior.

2. 2. Bernoulli distribution with blocking behavior.

3. 3. Geometric distribution with blocking behavior.

4. 4. Bernoulli distribution with pushing behavior.

Table 2 provides a summary of these cases and Figure 1 gives an example of the blocking versus pushing behavior. We also give a recursive description of these processes following (1.2). Here we give the equivalent bosonic version. We introduce $ \Omega _\ell ^b:=\{ (z_1,z_2,\dots ,z_\ell ) \in \mathbb {Z}_{ \geq 0}^\ell : z_1 \geq z_2 \geq \cdots \geq z_\ell \} $ for $\ell \ge 2$ . We introduce four types of evolution of particles $Y_t^{\mathrm {A}}$ , $Y_t^{\mathrm {B}}$ , $Y_t^{\mathrm {C}}$ and $Y_t^{\mathrm {D}}$ on $\Omega _\ell ^b$ .

Table 2 Summary of the four cases of discrete TASEP that we consider in this paper.

Figure 1 Examples of the third particle making a jump of $6$ steps with the pushing (left) and blocking (right) behaviors.

Definition 2.7. The four particle processes whose positions at time $t+1$ are given recursively as

$$ \begin{align*} Y_{t+1}^{\mathrm{A}}(\ell)&=Y_t^{\mathrm{A}}(\ell)+\xi^{\mathrm{A}}(\ell,t+1), \ \ \ Y_{t+1}^{\mathrm{B}}(1)=Y_t^{\mathrm{B}}(1)+\xi^{\mathrm{B}}(1,t+1), \nonumber \\ Y_{t+1}^{\mathrm{C}}(1)&=Y_t^{\mathrm{C}}(1)+\xi^{\mathrm{C}}(1,t+1), \ \ \ Y_{t+1}^{\mathrm{D}}(\ell)=Y_t^{\mathrm{D}}(\ell)+\xi^{\mathrm{D}}(\ell,t+1),\end{align*} $$

and

(2.9a) $$ \begin{align} Y_{t+1}^{\mathrm{A}}(k) &= \max(Y_t^{\mathrm{A}}(k), Y_{t+1}^{\mathrm{A}}(k+1))+\xi^{\mathrm{A}}(k,t+1), \end{align} $$

(2.9b) $$ \begin{align} Y_{t+1}^{\mathrm{B}}(k) &= \min(Y_t^{\mathrm{B}}(k)+\xi^{\mathrm{B}}(k,t+1) ,Y_{t+1}^{\mathrm{B}}(k-1)),\end{align} $$

(2.9c) $$ \begin{align}Y_{t+1}^{\mathrm{C}}(k) &= \min(Y_t^{\mathrm{C}}(k)+\xi^{\mathrm{C}}(k,t+1),Y_{t}^{\mathrm{C}}(k-1)), \end{align} $$

(2.9d) $$ \begin{align} Y_{t+1}^{\mathrm{D}}(k) &= \max(Y_t^{\mathrm{D}}(k)+\xi^{\mathrm{D}}(k,t+1) ,Y_{t+1}^{\mathrm{D}}(k+1)),\end{align} $$

for $k=1,\dots ,\ell -1$ for Case (A), (D) and $k=2,\dots ,\ell $ for Case (B), (C), where $\xi ^{\mathrm {D}}(k,t+1)$ and $\xi ^{\mathrm {C}}(k,t+1)$ (resp/ $\xi ^{\mathrm {B}}(k,t+1)$ and $\xi ^{\mathrm {D}}(k,t+1)$ ) are random variables having distribution $\mathsf {P}_{Ge}$ (resp. $\mathsf {P}_{Be}$ ).

For a sequence of integers $\lambda =(\lambda _1,\lambda _2,\dots ,\lambda _\ell ) (\lambda _1 \ge \lambda _2 \ge \cdots \ge \lambda _\ell )$ , we write $Y=\lambda $ if $Y \in \Omega _\ell ^b$ satisfies $Y(j)=\lambda _j$ , $j=1,2,\dots ,\ell $ .

The bosonic version of Theorem 1.1 is that the transition probabilities of the four particle systems $Y=Y^{\mathrm {A}},Y^{\mathrm {B}}, Y^{\mathrm {C}}, Y^{\mathrm {D}}$ from the initial configuration $Y_0=\mu $ at time zero to the final configuration $Y_n=\lambda $ at time n are given by

$$ \begin{align*} \mathsf{P}(Y^{\mathrm{A}}_n=\lambda|Y^{\mathrm{A}}_0=\mu) & =\mathsf{P}_{A,n}(\lambda | \mu), \ \mathsf{P}(Y^{\mathrm{B}}_n=\lambda|Y^{\mathrm{B}}_0=\mu) =\mathsf{P}_{B,n}(\lambda | \mu), \\ \mathsf{P}(Y^{\mathrm{C}}_n=\lambda|Y^{\mathrm{C}}_0=\mu) & =\mathsf{P}_{C,n}(\lambda | \mu), \ \mathsf{P}(Y^{\mathrm{D}}_n=\lambda|Y^{\mathrm{D}}_0=\mu) =\mathsf{P}_{D,n}(\lambda | \mu). \end{align*} $$

Recall that $\mathsf {P}_{A,n}(\lambda | \mu )$ , $\mathsf {P}_{B,n}(\lambda | \mu )$ , $\mathsf {P}_{C,n}(\lambda | \mu )$ , $\mathsf {P}_{D,n}(\lambda | \mu )$ are Grothendieck polynomials with overall factors multiplied, explicitly given in Theorem 1.1. By abuse of notation, we also denote transition probabilities using these notations. In Section 4, where we give a proof, we regard these notations as transition probabilities.

Let us give an equivalent two-dimensional description. Let $G(j, i)$ denote the position of the j-th particle at time i in the bosonic formulation. For the pushing behavior, we can further realize it as a directed last-passage percolation model on W. To see the last-passage percolation, define

$$\begin{align*}G(k, n) = \max_{\Pi} \sum_{(j,i) \in \Pi} w_{ji}, \qquad\qquad \mathbf{G}(n) = \big(G(1, n), G(2, n), \dotsc, G(\ell, n) \big), \end{align*}$$

where the maximum is taken over a certain set of paths from $(\ell , 1)$ to $(k, n)$ . The paths are given in the natural matrix coordinates $(r, c)$ being the r-th row and c-th column. For the geometric distribution, the paths use unit steps to the right or up; that is, starting at position $(j_a, i_a)$ , then either $(j_{a+1}, i_{a+1}) = (j_a, i_a+1), (j_a-1, i_a)$ . Translating this into the update rule described in [Reference Dieker and WarrenDW08] (cf. (2.9a)), we have

$$\begin{align*}G(j,i) = \max(G(j,i-1), G(j+1,i)) + w_{ji}. \end{align*}$$

Note $G(j,i)$ corresponds to $Y_i(j)$ and $w_{ji}$ corresponds to $\xi (j,i)$ in the previous bosonic particle process description.

On the other hand, the Bernoulli distribution uses paths such that for every time $w_{j_a i_a} = 1$ , we must have the next step move right $(j_{a+1}, i_{a+1}) = (j_a, i_a+1)$ , and we also allow paths to end at $(k', n)$ for some $k \leq k' \leq \ell $ . Then from a simple argument using conditional probability, we see that the position of the particles at time n is given by $\mathbf {G}(n)$ (see, e.g., [Reference Dieker and WarrenDW08, Reference JohanssonJoh00, Reference Motegi and ScrimshawMS25]). To obtain the positions of the particles for the blocking behavior, this becomes a first-passage percolation model as we replace $\max $ by $\min $ , but we have to make some other changes. The first is that we shift the indices so that the $w_{ji}$ correspond to the weights on the horizontal edges $(i,j-1) \to (i, j)$ with the other edges being SE diagonal (resp. vertical) edges with weight $0$ for the geometric (resp. Bernoulli) case. The other is that we instead start at $(0, 0)$ .

Example 2.8. Consider $\ell = 3$ and $n = 4$ . Then the correspondence between a random matrix and the motion of particles in Case A, resp. Case C, is given by

First note that all of these transition probabilities $\mathsf {P}_X(\lambda | \mu ) := \mathsf {P}_X(\mathbf {G}(1) = \lambda | \mathbf {G}(0) = \mu )$ in Case X are independent of the value of $\ell $ provided $\ell (\lambda ) \leq \ell $ . Indeed, the j-th particle for $j> \ell (\lambda )$ must be fixed in place, which occurs with probability $1$ . Note that by taking $\ell \to \infty $ , we can effectively ignore the value of $\ell $ if desired and identify states with elements of the fermionic Fock space $\mathcal {F}$ (considered with the shifted positions $\lambda _j - j$ ). Furthermore, the step initial condition becomes $\lvert 0 \rangle $ , and this is sometimes known as the Dirac sea.

3.1 Noncommutative blocking operators

We denote by $\kappa _i \colon \mathbf {k}[\mathcal {P}] \to \mathbf {k}[\mathcal {P}]$ the i-th (row) that adds a box to the i-th row of a partition $\lambda $ if $\lambda _i < \lambda _{i-1}$ (that is, we can add the box and obtain a partition) and is $0$ otherwise. The Schur operators satisfy the Knuth relations.

We define the linear operator $U_i^{(\boldsymbol {\alpha }, \boldsymbol {\beta })}$ by

$$\begin{align*}U_i^{(\boldsymbol{\alpha}, \boldsymbol{\beta})} := \kappa_i + \Theta_i,\quad \text{where}\quad \Theta_i \cdot \lambda := \begin{cases} -\alpha_{\lambda_i} \lambda & \text{if } \lambda_i < \lambda_{i-1}, \\ \beta_{i-1} \lambda & \text{if } \lambda_i = \lambda_{i-1}, \end{cases} \end{align*}$$

for any $\lambda \in \mathcal {P}$ . We consider $\lambda _0 = \infty $ and $\alpha _0 = 0$ (although our proofs could have $\alpha _0$ be an arbitrary parameter). When there is no ambiguity in the parameters, we will simply write $U_i := U_i^{(\boldsymbol {\alpha },\boldsymbol {\beta })}$ . When $\boldsymbol {\alpha } = 0$ and $\boldsymbol {\beta } = \beta $ , the operators $\{U_i^{(0, \beta )}\}_{i=1}^{\infty }$ are the operators introduced in [Reference IwaoIwa20, Sec. 6].

Proof. Since $[\Theta _i,\Theta _j]=0$ for any $i,j$ and

(3.1) $$ \begin{align} [\kappa_i,\Theta_j]=0\quad \text{for } i\neq j-1,j, \end{align} $$

the “nonlocal commutativity” $U_iU_j=U_jU_i$ for $|i-j|\geq 2$ immediately follows from $\kappa _i\kappa _j=\kappa _j\kappa _i$ for $|i-j|\geq 2$ . This fact implies (2.2a) and (2.2b).

Define a linear operator $T_i \colon \mathbf {k}[\mathcal {P}] \to \mathbf {k}[\mathcal {P}]$ by

$$\begin{align*}T_i\cdot \lambda:= \begin{cases} 0 & \text{if } \lambda_{i+1} < \lambda_{i}, \\ \kappa_i\cdot \lambda & \text{if } \lambda_{i+1} = \lambda_{i} \end{cases}\qquad \text{for any } \lambda\in\mathcal{P}. \end{align*}$$

Note that, for any partition $\lambda $ , $T_i\cdot \lambda \neq 0$ if and only if $\lambda _{i+1}=\lambda _i<\lambda _{i-1}$ . We need the following commutation relations to prove (2.2c):

(3.2a) $$ \begin{align} T_i \kappa_i & = \kappa_{i+1}^2T_i=0, \end{align} $$

(3.2b) $$ \begin{align} \Theta_{i+1} \kappa_{i+1}T_i &= \beta_i \kappa_{i+1}T_i. \end{align} $$

(3.2c) $$ \begin{align} [\kappa_i, \kappa_{i+1}] & = -\kappa_{i+1}T_i, \end{align} $$

(3.2d) $$ \begin{align} \Theta_{i+1} T_i & = T_i\Theta_i, \end{align} $$

(3.2e) $$ \begin{align} [\kappa_i, \Theta_{i+1}] & = \beta_i T_i - \Theta_{i+1}T_i. \end{align} $$

Equation (3.2a) follows from the facts that (i) $\mu :=\kappa _i\cdot \lambda $ satisfies $\mu _{i+1}<\mu _{i}$ whenever $\mu \neq 0$ and that (ii) $\nu :=T_i\cdot \lambda $ satisfies $\nu _{i}=\nu _{i+1}+1$ whenever $\nu \neq 0$ for any $\lambda \in \mathcal {P}$ . Equation (3.2b) follows from the fact that $\zeta :=\kappa _{i+1}T_i\cdot \lambda $ satisfies $\zeta _{i+1}=\zeta _i$ whenever $\zeta \neq 0$ . Equation (3.2c) is proved by noting that $\lambda _{i+1}<\lambda _{i}\Rightarrow \zeta = [\kappa _i,\kappa _{i+1}]\cdot \lambda = 0$ and $\lambda _{i+1}=\lambda _{i}\Rightarrow \zeta =[\kappa _i,\kappa _{i+1}]\cdot \lambda =-\kappa _{i+1}\kappa _i\cdot \lambda $ . To prove (3.2d), it suffices to check that $\Theta _{i+1}T_i\cdot \lambda =T_i\Theta _{i}\cdot \lambda =\alpha _{\lambda _{i}}\kappa _i\cdot \lambda $ whenever $T_i\cdot \lambda \neq 0$ . To prove (3.2e), we consider the following three cases: (a) If $T_i\cdot \lambda \neq 0$ ( $\Leftrightarrow \lambda _{i+1}=\lambda _i<\lambda _{i-1} \Leftrightarrow T_i\cdot \lambda =\kappa _i\cdot \lambda \neq 0$ ), we have $\kappa _i\Theta _{i+1}\cdot \lambda =\beta _i\kappa _i\cdot \lambda =\beta _iT_i\cdot \lambda $ . (b) If $\lambda _{i+1}<\lambda _i$ , we have $\kappa _i\Theta _{i+1}\cdot \lambda =\Theta _{i+1}\kappa _i\cdot \lambda =\beta _i\kappa _i\cdot \lambda $ . (c) If $\lambda _{i}=\lambda _{i-1}$ , we have $\kappa _i\Theta _{i+1}\cdot \lambda =\Theta _{i+1}\kappa _i\cdot \lambda =0$ . In each case, (3.2e) holds.

Thus we have

$$ \begin{align*} [U_i,U_{i+1}]=[\kappa_i+\Theta_i,\kappa_{i+1}+\Theta_{i+1}] \stackrel{({\scriptstyle 3.1})}{=} [\kappa_i,\kappa_{i+1}]+[\kappa_{i},\Theta_{i+1}] \stackrel{({\scriptstyle 3.2c}),({\scriptstyle 3.2e})}{=}(\beta_i-U_{i+1})T_i, \end{align*} $$

$$ \begin{align*} [U_i,U_{i+1}]U_i =(\beta_i-U_{i+1})T_iU_i \stackrel{({\scriptstyle 3.2a})}{=}(\beta_i-U_{i+1})T_i\Theta_i \stackrel{({\scriptstyle 3.2d})}{=}(\beta_i-U_{i+1})\Theta_{i+1}T_i. \end{align*} $$

and

$$ \begin{align*} U_{i+1}[U_{i},U_{i+1}] =U_{i+1}(\beta_i-U_{i+1})T_i =(\beta_i-U_{i+1})U_{i+1}T_i =(\beta_i-U_{i+1})(\kappa_{i+1}+\Theta_{i+1})T_i. \end{align*} $$

Therefore, we obtain

$$ \begin{align*} [U_i+U_{i+1},U_{i+1}U_i] &=[U_i,U_{i+1}]U_i-U_{i+1}[U_{i},U_{i+1}] =(\beta_i-U_{i+1})\kappa_{i+1}T_i\\ & \stackrel{({\scriptstyle 3.2a})}{=}(\beta_i-\Theta_{i+1})\kappa_{i+1}T_i \stackrel{({\scriptstyle 3.2b})}{=}0, \end{align*} $$

which implies (2.2c).

Theorem 3.2. We have

(3.3) $$ \begin{align} {}^{[\boldsymbol{\alpha},\boldsymbol{\beta}]} \langle \lambda \rvert S_{\mu}(a_1, a_2, \ldots) = {}^{[\boldsymbol{\alpha},\boldsymbol{\beta}]} \langle s_{\mu}(\mathbf{U}/\boldsymbol{\beta}) \cdot \lambda \rvert. \end{align} $$

To prove Theorem 3.2, we need the following computations. For simplicity, let $v(\lambda; \sigma ) := {}^{[\boldsymbol {\alpha },\boldsymbol {\beta }]}\langle \lambda \rvert ^{\sigma }$ . Let $\epsilon _k$ be the k-th standard basis vector in $\mathbb {Z}^N$ for some $N \gg 1$ . For any subset $K = \{k_1. \dotsc , k_m \} \subseteq [N]$ , let $\epsilon _K = \epsilon _{k_1} + \cdots + \epsilon _{k_m}$ .

Lemma 3.3. For any sequences $\lambda $ and $\sigma $ (not necessarily partitions), we have

$$\begin{align*}v(\lambda + \epsilon_k; \sigma) = v(\lambda + \epsilon_k; \sigma + \epsilon_k) - \alpha_{\sigma_k} v(\lambda; \sigma). \end{align*}$$

Proof. This follows applying to the definition of $v(\lambda + \epsilon _k; \sigma )$ the relation

$$\begin{align*}\psi_{\lambda_k +1 - k}^* e^{H^*(\gamma)} = e^{H^*(\gamma)} \psi_{\lambda_k + 1 - k}^* + \gamma \psi^*_{\lambda_k - k} e^{H^*(\gamma)}, \end{align*}$$

for $\gamma = -\alpha _{\sigma _k}$ , which follows from the $\ast $ version of Equation (2.3a).

Lemma 3.4. Suppose $\lambda _k = \lambda _{k-1}$ and $\sigma _k = \sigma _{k-1}$ , then

$$\begin{align*}v(\lambda + \epsilon_k; \sigma) = \beta_{k-1} v(\lambda; \sigma). \end{align*}$$

Proof. This follows from the definition of $v(\lambda; \sigma )$ and the rectification lemma [Reference Iwao, Motegi and ScrimshawIMS24, Lemma 3.6]. Indeed, similar to the previous lemma, we have

$$\begin{align*}\psi^*_{\lambda_k - (k-1)} e^{-H^*(\beta_{k-1})} \psi^*_{\lambda_k + 1 - k} = \beta_{k-1} \psi^*_{\lambda_k - (k-1)} e^{-H^*(\beta_{k-1})} \psi^*_{\lambda_k - k} \end{align*}$$

by the $\ast $ version of Equation (2.3a) and recalling $(\psi ^*_{\lambda _k - k})^2 = 0$ .

Lemma 3.5. Let $\lambda $ be a partition. If $K = \{k_1 < k_2 < \dotsc < k_m\}$ , then

$$\begin{align*}{}^{[\boldsymbol{\alpha},\boldsymbol{\beta}]} \langle \lambda + \epsilon_K \rvert_{\lambda} = \sum_{\mu} C_{\lambda}^{\mu} \cdot {}^{[\boldsymbol{\alpha},\boldsymbol{\beta}]} \langle \mu \rvert, \end{align*}$$

where the coefficients $C_{\lambda }^{\mu }$ are given by

$$\begin{align*}U_{k_n} \dotsm U_{k_2} U_{k_1} \cdot \lambda = \sum_{\mu} C_{\lambda}^{\mu} \cdot \mu. \end{align*}$$

Proof of Theorem 3.2.

Because the operators $\{U_i\}_{i=1}^{\infty }$ satisfy the weak Knuth relations (Lemma 3.1), Fomin-Greene’s theorem [Reference Fomin and GreeneFG98] implies that the noncommutative Schur functions $s_{\lambda }(\mathbf {U} / \boldsymbol {\beta })$ are a commutative algebra (under the natural multiplication) as it is generated (as an algebra) by $e_{i}(\mathbf {U} / \boldsymbol {\beta })$ for $i=0,1,2,\dots $ . Therefore, to prove the theorem, it suffices to show (3.3) for the case when $s_\lambda =e_i$ :

$$\begin{align*}{}^{[\boldsymbol{\alpha},\boldsymbol{\beta}]}\langle \lambda \rvert E_i(a_1, a_2, \ldots) = {}^{[\boldsymbol{\alpha},\boldsymbol{\beta}]}\langle e_{i}(\mathbf{U} / \boldsymbol{\beta}) \cdot \lambda \rvert. \end{align*}$$

By applying the commutator relation in (2.4), we have

$$\begin{align*}{}^{[\boldsymbol{\alpha}, \boldsymbol{\beta}]}\langle \lambda \rvert a_i = \sum_{k=1}^{\infty} {}^{[\boldsymbol{\alpha},\boldsymbol{\beta}]}\langle \lambda + i \epsilon_k \rvert_{\lambda} - \sum_{k=1}^{\infty} \beta_k^i \cdot {}^{[\boldsymbol{\alpha},\boldsymbol{\beta}]}\langle \lambda \rvert = \sum_{k=1}^{\infty} {}^{[\boldsymbol{\alpha},\boldsymbol{\beta}]}\langle \lambda + i \epsilon_k \rvert_{\lambda} - p_i(\boldsymbol{\beta}) \cdot {}^{[\boldsymbol{\alpha},\boldsymbol{\beta}]}\langle \lambda \rvert. \end{align*}$$

Next, write $E_i(p_1, p_2, \ldots; \boldsymbol {\beta }) = e_i({\mathbf {x}} \sqcup \boldsymbol {\beta })$ , and by noting the above equation comes from $p_i({\mathbf {x}} \sqcup \boldsymbol {\beta })$ under the identification $p_i({\mathbf {x}}) \equiv a_i$ , we have

(3.4) $$ \begin{align} {}^{[\boldsymbol{\alpha},\boldsymbol{\beta}]}\langle \lambda \rvert E_i(a_1, a_2, \ldots; \boldsymbol{\beta}) = \sum_{\substack{K \subseteq \mathbb{Z} \\ \left\lvert K \right\rvert = i}} {}^{[\boldsymbol{\alpha},\boldsymbol{\beta}]}\langle \lambda + \epsilon_K \rvert_{\lambda} = {}^{[\boldsymbol{\alpha},\boldsymbol{\beta}]}\langle e_i(\mathbf{U}) \cdot \lambda \rvert, \end{align} $$

where the last equality is by Lemma 3.5. By expanding the plethsym for the left-hand side and solving for ${}^{[\boldsymbol {\alpha },\boldsymbol {\beta }]}\langle \lambda \rvert E_i(a_1, a_2, \ldots )$ , we obtain the desired result.

3.2 Noncommutative pushing operators

We define an operator $u_j^{(\boldsymbol {\alpha },\boldsymbol {\beta })}$ recursively as follows. Consider a partition $\lambda $ , and let k be minimal such that $\lambda _k = \lambda _j$ . Let $\nu := \overline {\mu + \epsilon _j}$ be the smallest partition that contains $\mu + \epsilon _j$ (a box added to row j); that is, we have added a box to all rows $k \leq i \leq j$ . Then

(3.5) $$ \begin{align} u_j^{(\boldsymbol{\alpha},\boldsymbol{\beta})} \cdot \mu = \beta_k \cdots \beta_{j-1} \nu + \alpha_{\mu_j+1} \sum_{i=k}^j \prod_{a=k}^{i-1} (\alpha_{\mu_j+1} + \beta_a) \prod_{a=i}^{j-1} \beta_a u_i^{(\boldsymbol{\alpha},\boldsymbol{\beta})} \cdot \nu. \end{align} $$

Note that the result is well-defined in the completion (by the degree) $\mathbf {k} [\![ \mathcal {P} ]\!]$ since each partition only has finitely many contributions. As we will always be looking for a specific term in this sum, working in the completion will not be consequential. We will show the operators $\mathbf {u}^{(\boldsymbol {\alpha },\boldsymbol {\beta })} = (u_1^{(\boldsymbol {\alpha },\boldsymbol {\beta })}, u_2^{(\boldsymbol {\alpha },\boldsymbol {\beta })}, \ldots )$ correspond to separating out the action of the current operator $a_1$ on ${}_{[\boldsymbol {\alpha },\boldsymbol {\beta }]} \langle \mu \rvert $ .

Lemma 3.6. For any sequences $\mu $ and $\sigma $ (not necessarily a partition), we have

(3.6) $$ \begin{align} {}_{[\boldsymbol{\alpha},\boldsymbol{\beta}]} \langle \mu \rvert_{\sigma} = {}_{[\boldsymbol{\alpha},\boldsymbol{\beta}]}\langle \mu \rvert_{\sigma + \epsilon_j} + \alpha_{\sigma_j+1} {}_{[\boldsymbol{\alpha},\boldsymbol{\beta}]}\langle \mu + \epsilon_j \rvert_{\sigma + \epsilon_j}. \end{align} $$

Proof. We rewrite (2.3b) as

$$\begin{align*}\psi_{\mu_j-j}^* = e^{-H(-\alpha_{\sigma_j+1})} \psi^*_{\mu_j-j} e^{H(-\alpha_{\sigma_j+1})} + \alpha_{\sigma_j+1} e^{-H(-\alpha_{\sigma_j+1})} \psi^*_{\mu_j+1-j} e^{H(-\alpha_{\sigma_j+1})}, \end{align*}$$

and so the claim follows.

Lemma 3.7. For any sequences $\mu $ and $\sigma $ such that $\mu _j = \mu _{j-1}$ and $\sigma _j = \sigma _{j-1}$ , we have

(3.7) $$ \begin{align} {}_{[\boldsymbol{\alpha},\boldsymbol{\beta}]} \langle \mu + \epsilon_j \rvert_{\sigma} = \beta_{j-1} \cdot {}_{[\boldsymbol{\alpha},\boldsymbol{\beta}]}\langle \mu + \epsilon_{j-1} + \epsilon_j \rvert_{\sigma}. \end{align} $$

Lemma 3.8. We have

$$\begin{align*}{}_{[\boldsymbol{\alpha}, \boldsymbol{\beta}]} \langle \mu + \epsilon_j \rvert = {}_{[\boldsymbol{\alpha}, \boldsymbol{\beta}]} \langle u_j^{(\boldsymbol{\alpha},\boldsymbol{\beta})} \cdot \mu \rvert. \end{align*}$$

Proof. Consider $k \leq j$ minimal such that $\mu _j = \mu _k$ and $\sigma _j = \sigma _m$ for all $k \leq m \leq j$ . Recall that for $X = \{i_1, \dotsc , i_m\}$ , we set $\epsilon _X = \epsilon _{i_1} + \cdots + \epsilon _{i_m}$ . Let $\nu = \mu + \epsilon _{[j,k]} = \overline {\mu + \epsilon _j}$ and $v(\mu; \sigma ) := {}_{[\boldsymbol {\alpha },\boldsymbol {\beta }]} \langle \mu \rvert _{\sigma }$ . By repeated applications of (3.7), we have

$$\begin{align*}v(\mu + \epsilon_j; \mu) = \prod_{i=k}^{j-1} \beta_i v(\nu; \mu). \end{align*}$$

Then by applying (3.6) to each $i \in [k, j]$ and then using (3.7), we compute

$$ \begin{align*} v(\mu + \epsilon_j; \mu) & = \prod_{i=k}^{j-1} \beta_i \sum_{X \subseteq [k,j]} \alpha_{\mu_j+1}^{\left\lvert X \right\rvert} v(\nu + \epsilon_X; \nu) \\ & = \prod_{i=k}^{j-1} \beta_i \sum_{X \subseteq [k,j]} \alpha_{\mu_j+1}^{\left\lvert X \right\rvert} \prod_{i \in [k,j) \setminus X} \beta_i v(\overline{\nu + \epsilon_X}; \nu) \\ & = \prod_{i=k}^{j-1} \beta_i \cdot \nu + \prod_{a=k}^{j-1} \beta_a \cdot \alpha_{\mu_j+1} \sum_{i=k}^j \prod_{m=k}^{i-1} (\alpha_{\mu_j+1} + \beta_m) v(\overline{\nu + \epsilon_i}; \nu) \\ & = \prod_{i=k}^{j-1} \beta_i \cdot \nu + \alpha_{\mu_j+1} \sum_{i=k}^j \prod_{m=k}^{i-1} (\alpha_{\mu_j+1} + \beta_m) \prod_{a=i}^{j-1} \beta_a v(\nu + \epsilon_i; \nu). \end{align*} $$

However, this is precisely the recursion formula defining $u_j^{(\boldsymbol {\alpha },\boldsymbol {\beta })}$ , and the claim follows.

For brevity, we will simply write $u_j := u_j^{(\boldsymbol {\alpha },\boldsymbol {\beta })}$ until noted otherwise.

Remark 3.9. Another way to see $\mathbf {u}$ corresponds to the action of the current operator $a_i$ is to first note that $a_1 = \frac {d}{dt} [e^{H(t)}] \big \rvert _{t=0}$ . Therefore, in the expansion of ${}_{[\boldsymbol {\alpha },\boldsymbol {\beta }]} \langle \mu \rvert a_1$ , we can compute the coefficient of ${}_{[\boldsymbol {\alpha },\boldsymbol {\beta }]} \langle \lambda \rvert $ by using (2.6) and computing

$$\begin{align*}{}_{[\boldsymbol{\alpha},\boldsymbol{\beta}]} \langle \mu \rvert a_1 \lvert \lambda \rangle_{[\boldsymbol{\alpha},\boldsymbol{\beta}]} = \frac{d}{dt} \left[ {}_{[\boldsymbol{\alpha},\boldsymbol{\beta}]} \langle \mu \rvert e^{H(t)} \lvert \lambda \rangle_{[\boldsymbol{\alpha},\boldsymbol{\beta}]} \right] \Bigr\rvert_{t=0} = \frac{d}{dt} [ g_{\lambda/\mu}(t; \boldsymbol{\alpha}, \boldsymbol{\beta}) ] \Bigr\rvert_{t=0}. \end{align*}$$

We can give a precise formula by using the combinatorial description given in [Reference Hwang, Jang, Soo Kim, Song and SongHJK⁺25, Def. 4.1] as a single marked reverse plane partition. Indeed, in order for there to be a nonzero contribution, we can only have a single connected component such that the topmost-rightmost box contributes a t. We leave the details to the interested reader.

The $\mathbf {u}^{(\boldsymbol {\alpha },\boldsymbol {\beta })}$ operators do not satisfy the (weak) Knuth relations as demonstrated in the arXiv version of this paper [Reference Iwao, Motegi and ScrimshawIMS23, Ex.-3.11]. However, we can see that $\mathbf {u}^{(0,\boldsymbol {\beta })}$ do, and in fact, this holds in general. This is a straightforward direct computation that refines [Reference IwaoIwa22, Sec. 6.1].

Theorem 3.11. Recall ${}_{[\boldsymbol {\beta }]} \langle \lambda \rvert = {}_{[0,\boldsymbol {\beta }]} \langle \lambda \rvert $ . We have

$$\begin{align*}{}_{[\boldsymbol{\beta}]} \langle \lambda \rvert S_{\mu}(a_1, a_2, \ldots) \equiv {}_{[\boldsymbol{\beta}]} \langle s_{\mu}(\mathbf{u}^{(0,\boldsymbol{\beta})}) \cdot \lambda \rvert\quad \mod{M_{[\boldsymbol{\beta}]}^{\ell\perp}}. \end{align*}$$

Proof. Similarly to the proof of Theorem 3.2, it suffices to prove the theorem for the case when $s_\lambda =e_i$ ( $i=0,1,2,\dots $ ):

$$\begin{align*}{}_{[\boldsymbol{\beta}]}\langle \lambda \rvert E_i(a_1, a_2, \ldots) = {}_{[\boldsymbol{\beta}]} \langle e_i(\mathbf{u}^{(0,\boldsymbol{\beta})}) \cdot \lambda \rvert. \end{align*}$$

To do so, we first compute

$$\begin{align*}{}_{[\boldsymbol{\beta}]}\langle \lambda \rvert a_i = {}_{[\boldsymbol{\beta}]}\langle \lambda \rvert P_i(a_1, a_2, \ldots) \equiv \sum_{j=1}^{\ell} v(\lambda + i \epsilon_j, \lambda)\quad \mod{M_{[\boldsymbol{\beta}]}^{\ell\perp}} \end{align*}$$

from the fact $[e^{H(\gamma )}, a_i] = 0$ for all $i> 0$ . Hence, Lemma 3.8 implies

$$\begin{align*}{}_{[\boldsymbol{\beta}]}\langle \lambda \rvert E_i(a_1, a_2, \ldots) = \sum_{j_1 < \cdots < j_i} {}_{[\boldsymbol{\beta}]} \langle u_{j_i}^{(0,\boldsymbol{\beta})} \cdots u_{j_1}^{(0,\boldsymbol{\beta})} \cdot \lambda \rvert, \end{align*}$$

and the claim follows.

4.1 Pushing operators

Suppose the particles are at positions given by the partition $\mu $ . Then it is easy to see that the action $u_j \cdot \mu $ corresponds to the j-th particle trying to move one step to the right at time i. Indeed, if the j-th particle is also at a site containing smaller particles, then taking the smallest partition containing $\mu + \epsilon _j$ corresponds to pushing the smaller particles. More specifically, if this pushes s particles, then we obtain the scalar $\beta _{j-s} \cdots \beta _{j-1}$ (if $s = 0$ , then it just is the resulting partition).

Example 4.1. Consider $4$ particles. The action

is identified with the particle motion

where the arrows denote the particle being pushed.

We rewrite the action of our noncommutative operators to match the form of Theorem 1.1.

Lemma 4.2. Let $\boldsymbol {\beta } = \boldsymbol {\pi }^{-1}$ and $u_j = u_j^{(0, \pi ^{-1})}$ . Then for any $(j_1, j_2, \dotsc , j_k)$ , we have

$$\begin{align*}u_{j_1} u_{j_2} \cdots u_{j_k} \cdot \mu = \frac{\pi_{j_1} \pi_{j_2} \cdots \pi_{j_k}}{\boldsymbol{\pi}^{\lambda / \mu}} \cdot \lambda. \end{align*}$$

Proof for Case A.

Now let us consider the Case A transition probability for a single time step $\mathsf {P}_A(\lambda |\mu )$ at time i. We can write this as

(4.5) $$ \begin{align} \mathsf{P}_A(\lambda | \mu) = \pi_{j_1} \pi_{j_2} \cdots \pi_{j_k} x_i^k \prod_{j=1}^{\infty} (1 - \pi_j x_i) \end{align} $$

where k is the number of columns in $\lambda / \mu $ and $j_1 \geq j_2 \geq \cdots \geq j_k$ (which is necessarily unique). To match the notation in Theorem 1.1, we specialize $\boldsymbol {\beta } = \boldsymbol {\pi }^{-1}$ . Since we update the particles from largest-to-smallest, the above discussion yields that we can write our time evolution with $\boldsymbol {\pi } = 1$ as

$$\begin{align*}\mathcal{T}_A = \sum_{k=0}^{\infty} h_k(x_i \mathbf{u}) = \sum_{k=0}^{\infty} x_i^k h_k(\mathbf{u}), \end{align*}$$

where we have scaled all the $\mathbf {u}$ operators by $x_i$ to introduce our time-dependent parameters. Indeed, if we apply $h_k(x_i \mathbf {u})$ , then in terms of the particle dynamics, we are moving all particles a total number of k steps with probability (4.5). To see again the particle-dependent parameters for the time evolution, if we bring the $\mathcal {T}_A$ inside the matrix coefficient, we must factor out the total position change factors $\boldsymbol {\pi }^{\lambda /\mu }$ as this cancels with the $\pi _{j_1}^{-1} \cdots \pi _{j_k}^{-1}$ by applying the various $u_j$ to $\mu $ until we get $\lambda $ (see (3.5)). As an example, $u_j \cdot \mu = \pi _{j-s} \cdots \pi _j \cdot {}_{[\boldsymbol {\pi }^{-1}]} \langle u_j \cdot \mu \rvert $ . Therefore, we can write the transition probability (4.5) as

(4.6) $$ \begin{align} \begin{aligned} \mathsf{P}_A(\lambda|\mu) & = \boldsymbol{\pi}^{\lambda/\mu} \prod_{j=1}^{\infty} (1 - \pi_j x_i) \cdot {}_{[\boldsymbol{\pi}^{-1}]} \langle \mathcal{T}_A \cdot \mu | \lambda \rangle_{[\boldsymbol{\pi}^{-1}]} \\ & = \boldsymbol{\pi}^{\lambda/\mu} \prod_{j=1}^{\infty} (1 - \pi_j x_i) \sum_{k=0}^{\infty} x_i^k \cdot {}_{[\boldsymbol{\pi}^{-1}]} \langle h_k(\mathbf{u}) \cdot \mu | \lambda \rangle_{[\boldsymbol{\pi}^{-1}]}. \end{aligned} \end{align} $$

Alternatively we can see (4.6) by noting in the second line, only the term $u_{j_1} u_{j_2} \cdots u_{j_k}$ is nonzero in the pairing by (2.6) and taking this together with Lemma 4.2. Next, we apply Theorem 3.11, (2.5), and (2.7) to rewrite Equation (4.6) as

$$ \begin{align*} \mathsf{P}_A(\lambda|\mu) & = \boldsymbol{\pi}^{\lambda/\mu} \prod_{j=1}^{\infty} (1 - \pi_j x_i) \sum_{k=0}^{\infty} x_i^k \cdot {}_{[\boldsymbol{\pi}^{-1}]} \langle \mu \rvert H_k(a_1, a_2, \dotsc) \lvert \lambda \rangle_{[\boldsymbol{\pi}^{-1}]} \\ & = \boldsymbol{\pi}^{\lambda/\mu} \prod_{j=1}^{\infty} (1 - \pi_j x_i) \cdot {}_{[\boldsymbol{\pi}^{-1}]} \langle \mu \rvert e^{H(x_i)} \lvert \lambda \rangle_{[\boldsymbol{\pi}^{-1}]} = \boldsymbol{\pi}^{\lambda/\mu} \prod_{j=1}^{\infty} (1 - \pi_j x_i) g_{\lambda/\mu}({\mathbf{x}}; \boldsymbol{\pi}^{-1}). \end{align*} $$

This is precisely the claim of Theorem 1.1 for a single time step.

Proof for Case D.

For Case D, we do the analogous proof using $\mathsf {P}_D(\lambda |\mu )$ starting with the time evolution at $\boldsymbol {\rho } = 1$

$$\begin{align*}\mathcal{T}_D = \sum_{k=0}^{\infty} e_k(x_i \mathbf{u}) = \sum_{k=0}^{\infty} x_i e_k(\mathbf{u}), \end{align*}$$

where here we specialize $\boldsymbol {\beta } = \rho ^{-1}$ . Indeed, after adding back in the particle-dependent parameters like before, we compute

$$ \begin{align*} \mathsf{P}_D(\lambda|\mu) & = \frac{\boldsymbol{\rho}^{\lambda/\mu} \cdot {}_{[\boldsymbol{\rho}^{-1}]} \langle \mathcal{T}_D \cdot \mu | \lambda \rangle_{[\boldsymbol{\rho}^{-1}]}}{\prod_{j=1}^{\infty} (1 + \rho_j x_i)} = \frac{\boldsymbol{\rho}^{\lambda/\mu}}{\prod_{j=1}^{\infty} (1 + \rho_j x_i)} \sum_{k=0}^{\infty} x_i^k \cdot {}_{[\boldsymbol{\rho}^{-1}]} \langle \mu \rvert E_k(a_1, a_2, \dotsc) \lvert \lambda \rangle_{[\boldsymbol{\rho}^{-1}]} \\ & = \frac{\boldsymbol{\rho}^{\lambda/\mu}}{\prod_{j=1}^{\infty} (1 + \rho_j x_i)} \cdot {}_{[\boldsymbol{\rho}^{-1}]} \langle \mu \rvert e^{J(x_i)} \lvert \lambda \rangle_{[\boldsymbol{\rho}^{-1}]} = \frac{\boldsymbol{\rho}^{\lambda/\mu}}{\prod_{j=1}^{\infty} (1 + \rho_j x_i)} j_{\lambda'/\mu'}({\mathbf{x}}; \boldsymbol{\rho}^{-1}). \end{align*} $$

Note that the order of the operators from $e_k(\mathbf {u})$ is applied smallest-to-largest and matches the update rule. We can also see the first equality by using Lemma 4.2.

4.2 Blocking operators

Suppose the particles are at positions given by the partition $\mu $ . Then it is easy to see that the action $U_j \cdot \mu $ corresponds to the j-th particle trying to move one step to the right at time i and keeping the other particles fixed. If the move is blocked by the $(j-1)$ -th particle (being at the same position), then we obtain the scalar $\beta _{j-1}$ . Otherwise the particle moves and we simply obtain the resulting partition.

Example 4.3. Consider $4$ particles. The action

(which also equals $U_4 U_2 \cdot (4, 2, 1, 1)$ ) is identified with the particle motion

Note that the fourth particle is blocked but the second particle moves.

We rewrite the noncommutative operator action to match Theorem 1.1.

Lemma 4.4. Let $\beta _j = \rho _{j+1}$ and $U_j = U_j^{(0,\boldsymbol {\beta })}$ . Then for any $(j_1, j_2, \dotsc , j_k)$ , we have

$$\begin{align*}U_{j_1} U_{j_2} \cdots U_{j_k} \cdot \mu = \frac{\rho_{j_1} \rho_{j_2} \cdots \rho_{j_k}}{\boldsymbol{\rho}^{\lambda / \mu}} \cdot \lambda. \end{align*}$$

Proof for Case B.

Let us consider the transition probability for a single time step $\mathsf {P}_B(\lambda ' | \mu ')$ in Case B starting at time i. This proof is largely analogous to that for Case A. Here, we use $U_i := U_i^{(0,\boldsymbol {\alpha })}$ and specialize $\alpha _j = \rho _{j+1}$ . Since we update particles from smallest-to-largest, the above description means one time evolution is given by

$$\begin{align*}\mathcal{T}_B = \sum_{k=0}^{\infty} e_k(x_i \mathbf{U}) = \sum_{k=0}^{\infty} x_i^k e_k(\mathbf{U}). \end{align*}$$

From the analogous arguments as in Case A or the above discussion, we need to multiply by $\boldsymbol {\rho }^{\lambda / \mu }$ to move the time evolution inside the matrix coefficient and account for the movement of all of the particles. Additionally, we scale each $U_j$ by $x_i$ to introduce the time parameters. Therefore, we have

(4.7) $$ \begin{align} \mathsf{P}_B(\lambda' | \mu') = \frac{\boldsymbol{\rho}^{\lambda/\mu} \cdot {}^{[\boldsymbol{\alpha}]} \langle \mathcal{T}_B \cdot \mu | \lambda \rangle^{[\boldsymbol{\alpha}]}}{\prod_{j=1}^{\infty} (1 + \rho_j x_i)} = \sum_{k=0}^{\infty} \frac{\boldsymbol{\rho}^{\lambda/\mu} x_i^k \cdot {}^{[\boldsymbol{\alpha}]} \langle e_k(\mathbf{U}) \cdot \mu | \lambda \rangle^{[\boldsymbol{\alpha}]}}{\prod_{j=1}^{\infty} (1 + \rho_j x_i)}. \end{align} $$

We can also see Equation (4.7) by using Lemma 4.4 with the fact that for any $\ell \geq \ell (\lambda )$ , we have

$$\begin{align*}\mathsf{P}_B(\lambda | \mu) = \prod_{j=1}^{\infty} (1 + \rho_j x_i) \sum_{k=0}^{\infty} \sum_{\substack{j_1 < \cdots < j_k \\ U_{j_k} \cdots U_{j_1} \cdot \mu' = \ast \cdot \lambda'}} \rho_{j_k} \cdots \rho_{j_1} \cdot x_i^k, \end{align*}$$

where $\ast $ represents any nonzero constant; note that $U_{j_1} \cdots U_{j_k} \cdot \mu ' = \ast \cdot \lambda '$ is simply saying the movement of the particles from $\mu $ to $\lambda $ is given by moving (with blocking) the $j_1, j_2, \dotsc , j_k$ particles (in that order). Next, we use (2.5) and Theorem 3.2 to compute

$$ \begin{align*} J_{\lambda'/\!\!/\mu'}({\mathbf{x}}; \boldsymbol{\alpha}) = {}^{[\boldsymbol{\alpha}]} \langle \mu \rvert e^{J(x_i)} \lvert \lambda \rangle^{[\boldsymbol{\alpha}]} & = \sum_{m=0}^{\infty} x_i^m \cdot {}^{[\boldsymbol{\alpha}]} \langle \mu \rvert E_m(a_1, a_2, \ldots) \lvert \lambda \rangle^{[\boldsymbol{\alpha}]} \\ & = \sum_{m=0}^{\infty} x_i^m \cdot {}^{[\boldsymbol{\alpha}]} \langle e_m(\mathbf{U} / \boldsymbol{\alpha}) \cdot \mu | \lambda \rangle^{[\boldsymbol{\alpha}]} \\ & = \sum_{k=0}^{\infty} \sum_{m=k}^{\infty} (-1)^{k-m} x_i^m h_{m-k}(\boldsymbol{\alpha}) \cdot {}^{[\boldsymbol{\alpha}]} \langle e_k(\mathbf{U}) \cdot \mu | \lambda \rangle^{[\boldsymbol{\alpha}]} \\ & = \sum_{k=0}^{\infty} x_i^k \prod_{j=1}^{\infty} (1 + \alpha_j x_i)^{-1} \cdot {}^{[\boldsymbol{\alpha}]} \langle e_k(\mathbf{U}) \cdot \mu | \lambda \rangle^{[\boldsymbol{\alpha}]}. \end{align*} $$

Therefore, comparing this with Equation (4.7) (recall that $\alpha _j = \rho _{j+1}$ ), we have Theorem 1.1,

$$\begin{align*}\mathsf{P}_B(\lambda | \mu) = \prod_{i=1}^n (1 + \rho_1 x_i)^{-1} \boldsymbol{\rho}^{\lambda/\mu} J_{\lambda'/\!\!/\mu'}({\mathbf{x}};\boldsymbol{\alpha}), \end{align*}$$

for one time step.

Alternatively, let us examine Equation (4.7). Necessarily we must have $\lambda / \mu $ being a vertical strip (equivalently $\lambda ' / \mu '$ being a horizontal strip) as otherwise both sides are $0$ , so we now assume $\lambda / \mu $ is a vertical strip. Suppose $m = \left \lvert \lambda \right \rvert - \left \lvert \mu \right \rvert $ particles move, and so each term in the sum is $0$ unless $k \geq m$ . Furthermore, when $k>m$ , we only get a nonzero contribution from the j such that $\lambda _{j-1} = \mu _j$ , where necessarily $j> 1$ . Let $J = \{ j \in \mathbb {Z}_{>1} \mid \lambda _{j-1} = \mu _j \}$ be the (infinite) set of all such indices, and so we have

$$ \begin{align*} \sum_{k=0}^{\infty} x_i^k \cdot {}^{[\boldsymbol{\alpha}]} \langle e_k(\mathbf{U}) \cdot \mu | \lambda \rangle^{[\boldsymbol{\alpha}]} & = \sum_{k=m}^{\infty} x_i^k \sum_{\substack{X \subseteq J \\ \left\lvert X \right\rvert = k-m}} \prod_{j \in X} \rho_j = \sum_{k=m}^{\infty} x_i^k e_{k-m}(\boldsymbol{\rho}_J) \\ & = x_i^m \sum_{k=0}^{\infty} x_i^k e_k(\boldsymbol{\rho}_J) = x_i^m \prod_{j \in J} (1 + \rho_j x_i), \end{align*} $$

where $\boldsymbol {\rho }_J = \{\rho _j \mid j \in J\}$ (since $e_k$ is a symmetric function, we do not need to worry about the order). Therefore, we have

$$\begin{align*}\mathsf{P}_B(\lambda|\mu) = \frac{\boldsymbol{\rho}^{\lambda/\mu} x_i^m}{1 + \rho_1 x_i} \prod_{j \in \widetilde{J}} (1 + \alpha_j x_i)^{-1}, \end{align*}$$

where $\widetilde {J} = \{ j \in \mathbb {Z}_{>0} \mid \lambda _j \neq \mu _{j+1} \}$ (note that we have also shifted the indices). From the combinatorial description of $J_{\lambda ' /\!\!/ \mu '}({\mathbf {x}}_1; \boldsymbol {\alpha })$ , we have obtained Theorem 1.1 for a single time step.

For the alternative proof using the combinatorial description of $G_{\lambda /\!\!/\mu }({\mathbf {x}}; \boldsymbol {\beta })$ , some slightly more detailed analysis about the motion of the particle is needed. This is discussed in Section 5.3. Note that it only depends on $\lambda $ and $\mu $ , not on the motion of any other particles.

5.1 Case A: Geometric pushing

This case was discussed in [Reference Motegi and ScrimshawMS25] by going through the last passage percolation (LPP) model. To make this combinatorially explicit, we simply note that the resulting matrix $[G(k,n)]_{k,n}$ is the analog of a Gelfand–Tsetlin pattern for the reverse plane partition. More precisely, the n-th column gives the shape of the entries at most n. From this description, we essentially have a combinatorial proof of Case A of Theorem 1.1. The only other ingredient needed is to note that in $\boldsymbol {\pi }^{\lambda /\mu } g_{\lambda /\mu }({\mathbf {x}}; \boldsymbol {\pi }^{-1})$ , we get a contribution of $\pi _j x_i$ for a box with an i in row j; recalling that we align the entries at the bottom of merged cells. In terms of the more classical description of reverse plane partitions, we are only counting boxes with an i that is not above another box with label i. The remaining factor of $\prod _{j=1}^{\ell } \prod _{i=1}^n (1 - \pi _j x_i)$ comes from the normalization factor in the geometric distribution.

We can make this very precise with a direct correlation between the movement of particles and entries in the reverse plane partition. An entry i in row j (that is not a merged box) corresponds to the j-th particle moving a step at time i. Thus (ignoring the common normalization factor), by conditioning on the j-th particle at time i moving k steps, then the reverse plane partition has k entries with value i at row j (with no i directly below it). The general case of multiple particles moving follows fundamental facts of conditional probability. Hence, we have shown the one step transition probability at time i is given by

$$\begin{align*}\mathsf{P}_{A,1}(\lambda | \mu) = \prod_{j=1}^{\ell} (1 - \pi_j x_i) \boldsymbol{\pi}^{\lambda/\mu} \prod_{j=1}^{\ell-1} \pi_j^{-r_j} \operatorname{\mathrm{wt}}(T) =\prod_{j=1}^{\ell} (1 - \pi_j x_i) \boldsymbol{\pi}^{\lambda/\mu} g_{\lambda/\mu}(x_i;\boldsymbol{\pi}^{-1}) , \end{align*}$$

where T is the unique reverse plane partition of skew shape $\lambda / \mu $ with all boxes filled with i. In some more detail about the expression, recall $r_j$ is the number of boxes of T in row j that have been merged with the box below, and those boxes correspond to the moves of the j-th particle, automatically pushed from the particle behind it. This means that the number $r_j$ corresponds to the distance coming from the push and has to be subtracted from the actual total distance $\lambda _j-\mu _j$ of the j-th particle’s move in the power of $\pi _j$ . This gives the factor $ \prod _{j=1}^{\ell -1} \pi _j^{\lambda _j-\mu _j-r_j} \times \pi _\ell ^{\lambda _\ell -\mu _\ell }=\boldsymbol {\pi }^{\lambda /\mu } \prod _{j=1}^{\ell -1} \pi _j^{-r_j} $ . We also have a factor which is a power of $x_i$ , and the power is equal to the total degree of $\boldsymbol {\pi }^{\lambda /\mu } \prod _{j=1}^{\ell -1} \pi _j^{-r_j} $ . We conclude the precise factor is $x_i^{|\lambda /\mu |-\sum _{j=1}^{\ell -1} r_j}$ . Since all boxes of the reverse plane partition are filled with the same number, we note $|\lambda /\mu |-\sum _{j=1}^{\ell -1} r_j$ is equal to the total number of columns of the skew shape $\lambda /\mu $ , and $x_i^{|\lambda /\mu |-\sum _{j=1}^{\ell -1} r_j}$ is exactly $\operatorname {\mathrm {wt}}(T)$ for one variable case (recall we have fused boxes for the weight).

Hence, we have that the reverse plane partition exactly encodes the movement of all of the particles at time $i = 0, 1, \dotsc , n$ by the branching rules (Proposition 2.3) and the Markov property as described at the beginning of Section 4 (or basic facts of conditional probability).

Example 5.1. We consider Case A with $\lambda = 31$ and $n = 2$ . Hence, we are considering $\ell $ particles the move over two time steps. Since all but the first two particles are fixed, we can ignore them. We have the following reverse plane partitions and states, where we have drawn the merged boxes in gray. An arrow denotes that a particle was pushed.

Below each configuration, we have written the conditional probability except for the normalization constant in the geometric distribution $(1 - \pi _j x_i)$ . Note that these are precisely the terms appearing in $\boldsymbol {\pi }^{\lambda } g_{\lambda }({\mathbf {x}}; \boldsymbol {\pi }^{-1})$ .

5.2 Case D: Bernoulli pushing

Algebraically, this case is simply applying the involution $\omega $ defined by $\omega s_{\lambda } = s_{\lambda '}$ . As such, we should expect the movement of the j-th particle to correspond to the entries in the j-th column, as opposed to the j-th row in Case A. Indeed, this is the case, but we need to reformulate the parameters for the particle process. Recalling from [Reference Patrias and PylyavskyyPP16, Thm. 5.11] (see also [Reference YeliussizovYel17, Prop. 3.4]), we can write $J_{\lambda /\mu }({\mathbf {x}}; \alpha ) = G_{\lambda /\mu }\big ({\mathbf {x}}; \alpha / (1 + \alpha {\mathbf {x}}) \big )$ , it becomes natural to instead consider $\pi _j x_i$ as a rate at which the particle moves rather than the success probability, which becomes $\frac {\pi _j x_i}{1 + \pi _j x_i}$ . We also see this in the normalization factor as

$$\begin{align*}\omega \left( \prod_{j=1}^{\ell} \prod_{i=1}^n (1 - \pi_j x_i) \right) = \prod_{j=1}^{\ell} \prod_{i=1}^n (1 + \pi_j x_i)^{-1}. \end{align*}$$

Since we are considering it as a rate, we use a different set of parameters $\boldsymbol {\rho }$ since we simply require $\rho _j x_i> 0$ for all i and j, as opposed to $\pi _j x_i \in (0, 1)$ . The remainder of the proof is that after we factor out the denominators, we make the same observation as in Case A that an entry i in column j corresponds to the j-th particle moving one step at time i and contributes $\rho _j x_i$ to the (unnormalized) probability.

Example 5.2. Let us consider Case D for $\lambda = 31$ with $n = 2$ . Thus, three particles (all others are fixed) move over two time steps. The possible terms, states, and valued-set tableaux are

where $\pi _{ji} = \frac {\rho _j x_i}{1 + \rho _j x_i}$ and $\sigma _{ji} = 1 - \pi _{ji} = \frac {1}{1 + \rho _j x_i}$ . Once we factor out the denominators, we have $\boldsymbol {\rho }^{\lambda '/\mu '} j_{\lambda /\mu }({\mathbf {x}}; \boldsymbol {\rho }^{-1})$ . In the factors considered below each state, taking it as a $\{0,1\}$ -matrix by setting $\pi _{ij} = 1$ and $\sigma _{ij} = 0$ , we obtain the transposed LPP $\{0,1\}$ -matrix.

Furthermore, the above description of the weight contributions makes a clear connection with $01$ -matrices in [Reference Dieker and WarrenDW08]. Indeed, we can equate each state with a $01$ -matrix $[M_{ij}]_{i,j}$ by setting $M_{ij} = 1$ if and only if we use $\pi _{ij}$ (and necessarily $M_{ij} = 0$ corresponds to $\sigma _{ij}$ ). Since all entries of the matrix and particle motions are done independently, the aforementioned correspondence means the transition probabilities are equal.

5.3 Case C: Geometric blocking

Now we consider the TASEP with the more classical blocking behavior, and as before, we will proceed by conditioning on the motion of particles. As we have the blocking behavior, we expect the first particle to behave differently than all of the subsequent particles. This justifies why the normalization factor only involves $\pi _1$ rather than all $\pi _j$ . Like in Case A, we will identify entries i in row j corresponding to the j-th particle moving at time i. As the first particle will never be blocked, it simply moves at times $i_1 \leq i_2 \leq \cdots \leq i_{\lambda _1}$ .

Next, let us consider the movement of the j-th particle for $j> 1$ . Suppose the first (resp. second) particle at time $i-1$ is $p_{j-1}$ (resp. $p_j$ ). Let p be the position of the second particle at time i. Recall that we update the position of the j-th particle before the $(j-1)$ -th particle. Therefore if $p_j = p_{j-1}$ , then the second particle does not move, which can be phrased as the probability the second particle is at position $p = p_j$ at time i is $ 1 = \sum _{k=0}^{\infty } P_G(w_{j,i} = k). $ Now suppose $p_j < p_{j-1}$ , and if $p < p_{j-1}$ , then this occurs with probability $(1 - \pi _j x_i)(\pi _j x_i)^{p-p_j}$ as there is no blocking behavior. Lastly, if $p = p_{j-1}$ , then the probability is

$$\begin{align*}(1 - \pi_j x_i) \sum_{k=p_{j-1}-p_j}^{\infty} (\pi_j x_i)^k = (\pi_j x_i)^{p_{j-1}-p_j}. \end{align*}$$

Therefore, we want to identify the motion of the particles with semistandard tableaux like for the classical TASEP (with geometric jumping). Yet, whenever the j-th particle does not move the maximal possible distance it can, there are two terms contributing to the probability. We split this into two separate terms that we encode into tableaux, and we do so by having the $-(\pi _j x_i)^{p_{j-1} - p_j + 1}$ term correspond to adding an extra i to a box in row $j-1$ (since we take $\beta _{j-1} = \pi _j$ , recalling $j> 1$ ). As such, the motion of the particles is constructed from a set-valued tableau T by using the semistandard tableau $\min (T)$ built from the smallest entries in each box of T. Indeed, we can only add an i to a box in the j-th row whenever $\Lambda ^{(i)}_{j-1}> \Lambda ^{(i)}_j$ , where $\Lambda ^{(i)}$ is the positions of the particles at time i, which is equivalent to $(\Lambda ^{(i)})_{i=1}^n$ being the Gelfand–Tselin pattern corresponding to $\min (T)$ . Hence, we still have the normalization factor $\prod _{i=1}^n (1 - \pi _1 x_i)$ , which completes the proof of Theorem 1.1 in this case.

As another way to see this, consider evolution of the particles from time $t=i-1$ with positions $(\mu _1,\dots ,\mu _\ell )$ to time $t=i$ with positions $(\lambda _1,\dots ,\lambda _\ell )$ . The preceding discussion means that when the j-th particle ( $j=2,\dots ,\ell $ ) is blocked by the $(j-1)$ -th particle, which corresponds to the case $\lambda _j=\mu _{j-1}$ , summing up conditional probabilities gives $(\pi _j x_i)^{\lambda _j-\mu _j}$ . If it is not blocked ( $\lambda _j \neq \mu _{j-1}$ ), the conditional probability is $(\pi _j x_i)^{\lambda _j-\mu _j}(1-\pi _j x_i)$ . The unified expression for the factor coming from the move of the j-th particle is $(\pi _j x_i)^{\lambda _j-\mu _j}(1-(1-\delta _{\lambda _j, \mu _{j-1}}) \pi _j x_i)$ , where $\delta _{ij}$ is the Kronecker delta: $\delta _{ij}=1$ if $i \neq j$ and 0 otherwise. As for the first particle, there is nothing to block its motion and we have the factor $(1-\pi _1 x_i)(\pi _1 x_i)^{\lambda _1-\mu _1}$ . Hence we have

$$ \begin{align*} \mathsf{P}_{C,1}(\lambda | \mu)&= (1-\pi_1 x_i)(\pi_1 x_i)^{\lambda_1-\mu_1} \prod_{j=2}^\ell (\pi_j x_i)^{\lambda_j-\mu_j}(1-(1-\delta_{\lambda_j, \mu_{j-1}}) \pi_j x_i) \\ &=(1-\pi_1 x_i) \boldsymbol{\pi}^{\lambda/\mu} x_i^{|\lambda/\mu|} \prod_{j=2}^\ell (1-(1-\delta_{\lambda_j, \mu_{j-1}}) \pi_j x_i). \end{align*} $$

We also have the refinement of [Reference YeliussizovYel17, Prop. 8.8]:

(5.1) $$ \begin{align} G_{\lambda /\!\!/ \mu}(x_i;\boldsymbol{\pi})=x_i^{|\lambda/\mu|} \prod_{j=2}^\ell (1-(1-\delta_{\lambda_j, \mu_{j-1}}) \pi_j x_i). \end{align} $$

We can see (5.1) holds by the analogous refinement of the combinatorial description in [Reference YeliussizovYel19, Thm. 4.6] by using $\beta _j$ if a contribution to $\beta $ occurs on the j-th row. Alternatively, we can deduce (5.1) by applying (2.1) and the combinatorial description of $G_{\lambda /\mu }(x_i; \boldsymbol {\pi })$ (where there is a unique semistandard tableau).

Example 5.3. Consider $\lambda = 631$ with $n = 5$ . A minimal tableau for particle motions is

The set-valued tableau T with the largest degree with this minimal tableau $\min (T)$ is

Every other set-valued tableau $T'$ with $\min (T') = \min (T)$ is formed by removing some of the extra (bold) entries in T; in other words, each of these bold entries can be chosen to be added independently to yield a valid set-valued tableau $T'$ with $\min (T') = \min (T)$ . Note that each of the bold entries correspond to a case where a particle does not move its maximal possible distance. Note that at time $i = 4$ , the third particle has moved its maximum possible distance since the update order is left-to-right and it becomes blocked by the second particle. Thus, we do not have an bold $\mathbf {\color {darkred}4}$ appearing in the second row. We can also consider infinitely other particles, which are all blocked, and so they do not contribute to the probability.

5.4 Case B: Bernoulli blocking

Essentially this is modifying Case C in the same way as we modified Case A to obtain Case D. While in this case, things are seemingly very different since $J_{\lambda }({\mathbf {x}}_n; \boldsymbol {\alpha })$ is always a formal power series, we see this on the probability side by considering the power series expansion of the rate

$$\begin{align*}\frac{1}{1 + \rho_j x_i} = \sum_{k=0}^{\infty} (-\rho_j x_i)^k. \end{align*}$$

Thus, if there are k extra entries of i in column $j-1$ , then this corresponds to choosing $(-\rho _j x_i)^k$ in the above expansion. Like Case C, the movement of the particles for a multiset-valued tableau T is described by the semistandard tableaux $\min (T)$ formed by taking the smallest entries in each box. In contrast to Case C, we can always repeat any entry i in $\min (T)$ (within its box) an arbitrary number of times and the result remains a multiset-valued tableau; this reflects that we multiply every entry by $(1 + \rho _j x_i)^{-1}$ .

Example 5.5. Let us take $\lambda = 4311$ with $n =5$ . Then an example of the correspondence between a semistandard tableau $\min (T)$ and the particle motions is

Any multiset valued tableau with $\min (T)$ will have a corresponding set-valued tableau (which does not change $\min (T)$ ) whose entries are a subset of each box of

We have written every entry in bold to indicate (and emphasize) that we can repeat every entry as many times as we desire.

6.1 Pushing

We begin by stating a straightforward extension of [Reference Motegi and ScrimshawMS25, Cor. 3.14], which can be proven using the lattice model given therein. However, we will sketch a proof using our free fermion presentation. To state the claim, we need the flagged Schur function, which we denote by $s_{\lambda ,\phi }({\mathbf {x}})$ for a flagging $\phi $ . Let $\lambda $ be a partition and $\phi $ a flagging (a sequence of integers). The flagged Schur function $s_{\lambda ,\phi }({\mathbf {x}})$ is defined by

$$\begin{align*}s_{\lambda,\phi}({\mathbf{x}}) = \sum_T \operatorname{\mathrm{wt}}(T), \end{align*}$$

where the sum is over all semistandard Young tableaux T of shape $\lambda $ such that the entries in row i are at most $\phi _i$ .

Proposition 6.1. Let $\ell = \ell (\lambda )$ . We have

(6.1) $$ \begin{align} s_{\lambda,\phi}({\mathbf{x}}, \boldsymbol{\beta}_{\ell}) = \sum_{\mu \subseteq \lambda} \boldsymbol{\beta}^{\lambda-\mu} g_{\mu}({\mathbf{x}}; \boldsymbol{\beta}), \end{align} $$

where $\phi $ is the flagging such that the entries in the i-th row are at most $\beta _i$ .

Proof. From [Reference IwaoIwa23, Ex. 2.6], the flagged Schur function can be written as

$$\begin{align*}s_{\lambda,\phi}({\mathbf{x}}_n, \boldsymbol{\beta}_{\ell}) = \langle \emptyset \rvert e^{H({\mathbf{x}}_n)} \lvert \lambda \rangle_{(\boldsymbol{\beta})}, \qquad\qquad \text{ where } \lvert \lambda \rangle_{(\boldsymbol{\beta})} := \prod^{\rightarrow}_{1 \leq i \leq \ell} \left( e^{H(\beta_i)} \psi_{\lambda_i-i} \right) \lvert -\ell \rangle. \end{align*}$$

We want to show that

(6.2) $$ \begin{align} \lvert \lambda \rangle_{(\boldsymbol{\beta})} = e^{H(\beta_1)} \lvert \lambda \rangle_{[\vec{\boldsymbol{\beta}}]} = \sum_{\mu \subseteq \lambda} \boldsymbol{\beta}^{\lambda / \mu} \lvert \mu \rangle_{[\boldsymbol{\beta}]}, \end{align} $$

where $\vec {\boldsymbol {\beta }} = (\beta _2, \beta _3, \ldots )$ denotes the shifted parameters and the last equality is the branching rule [Reference Motegi and ScrimshawMS25, Cor. 3.19] (which we will provide a free fermionic proof). The first equality is immediate from the definitions. We will prove the last equality in (6.2) by using (2.6), which reduces the claim to showing ${}_{[\boldsymbol {\beta }]} \langle \mu | \lambda \rangle _{(\boldsymbol {\beta })} = \boldsymbol {\beta }^{\lambda - \mu }$ for $\mu \subseteq \lambda $ and $0$ otherwise. Indeed, the claim follows from a similar proof to that in [Reference Iwao, Motegi and ScrimshawIMS24, Thm. 3.10] except we now have the i-th diagonal entry equal to $\beta _i^{\lambda _i - \mu _i}$ due to the shift. Hence, Equation (6.2) yields

(6.3) $$ \begin{align} s_{\lambda,\phi}({\mathbf{x}}_n, \boldsymbol{\beta}_{\ell}) = \langle \emptyset \rvert e^{H({\mathbf{x}}_n)} \lvert \lambda \rangle_{(\boldsymbol{\beta})} = \sum_{\mu \subseteq \lambda} \boldsymbol{\beta}^{\lambda - \mu} \langle \emptyset \rvert e^{H({\mathbf{x}}_n)} \lvert \mu \rangle_{[\boldsymbol{\beta}]} = \sum_{\mu \subseteq \lambda} \boldsymbol{\beta}^{\lambda-\mu} g_{\mu}({\mathbf{x}}_n; \boldsymbol{\beta}), \end{align} $$

where we used (2.7) with the fact ${}_{[\boldsymbol {\beta }]} \langle \emptyset \rvert = \langle \emptyset \rvert $ for the last equality.

Next we compute formulas for the multipoint distribution. Using the fact that we must have $G(i, n) \geq G(i+1, n)$ , for $\ell $ particles, the k-point distribution is equivalent to the $\ell $ -point distribution. In particular, we consider $1 = i_1 \geq i_2 \geq \cdots \geq i_k$ , and we have

$$\begin{align*}\mathsf{P}(G(i_k,n) \leq \lambda_{i_k}, \dotsc, G(i_1,n) \leq \lambda_{i_1}) = \mathsf{P}(G(\ell,n) \leq \lambda_{\ell}, \dotsc, G(1,n) \leq \lambda_1), \end{align*}$$

where $\lambda _j = \lambda _{i_k}$ with k maximal such that $i_k \leq j$ . The ordering on $\{i_k\}$ does not lose any generality, but we do require knowledge about the maximum distance the first particle can move. As such, we will use the notation

$$\begin{align*}\mathsf{P}_{\leq,n}(\lambda | \nu) := \mathsf{P}(G(\ell,n) \leq \lambda_{\ell}, \dotsc, G(1,n) \leq \lambda_1 | \mathbf{G}(0) = \nu). \end{align*}$$

Theorem 6.3. For Case A, the multipoint distribution with $\ell $ particles is given by

$$\begin{align*}\mathsf{P}_{\leq,n}(\lambda | \nu) = \boldsymbol{\pi}^{\lambda/\nu} \prod_{i=1}^{\ell} \prod_{j=1}^n (1 - \pi_i x_j) \cdot \det \big[ h_{\lambda_i - \nu_j + j - i}({\mathbf{x}} \sqcup \boldsymbol{\pi}_i^{-1} / \boldsymbol{\pi}_{j-1}^{-1}) \big]_{i,j=1}^{\ell}. \end{align*}$$

Proof. Using (6.2), we compute

$$ \begin{align*} \mathsf{P}_{\leq,n}(\lambda|\nu) & = \prod_{i=1}^{\ell} \prod_{j=1}^n (1 - \pi_i x_j) \sum_{\nu \subseteq \mu \subseteq \lambda} \boldsymbol{\pi}^{\mu / \nu} g_{\mu/\nu}({\mathbf{x}}_n; \boldsymbol{\pi}^{-1}) \\ & = \boldsymbol{\pi}^{\lambda/\nu} \prod_{i=1}^{\ell} \prod_{j=1}^n (1 - \pi_i x_j) \cdot {}_{[\boldsymbol{\pi}^{-1}]} \langle \nu \rvert e^{H({\mathbf{x}}_n)} \lvert \lambda \rangle_{(\boldsymbol{\pi}^{-1})}. \end{align*} $$

Next we apply Wick’s theorem to compute ${}_{[\boldsymbol {\pi }^{-1}]} \langle \nu \rvert e^{H({\mathbf {x}}_n)} \lvert \lambda \rangle _{(\boldsymbol {\pi }^{-1})} = \det [ \langle -\ell \rvert P_j Q_i \lvert -\ell \rangle ]_{i,j=1}^{\ell }$ , where

$$ \begin{align*} P_j & := e^{H(\boldsymbol{\pi}_{j-1}^{-1})} \psi^*_{\nu_j-j}e^{-H(\boldsymbol{\pi}_{j-1}^{-1})} = \sum_{m=0}^{\infty} h_m(\emptyset / \boldsymbol{\pi}_{j-1}^{-1}) \psi^*_{\nu_j-j+m}, \\ Q_i & := e^{H({\mathbf{x}} \sqcup \boldsymbol{\pi}_i^{-1})} \psi_{\lambda_i-i} e^{-H({\mathbf{x}} \sqcup \boldsymbol{\pi}_i^{-1})} = \sum_{k=0}^{\infty} h_k({\mathbf{x}} \sqcup \boldsymbol{\pi}_i^{-1}) \psi_{\lambda_i-i-k}. \end{align*} $$

Therefore, the entries in the determinant are (cf. [Reference IwaoIwa23, Prop. 2.4])

$$ \begin{align*} \langle -\ell \rvert P_j Q_i \lvert -\ell \rangle & = \sum_{m=0}^{\infty} \sum_{k=0}^{\infty} h_m(\emptyset / \boldsymbol{\pi}_{j-1}^{-1}) h_k({\mathbf{x}} \sqcup \boldsymbol{\pi}_i^{-1}) \cdot \langle -\ell \rvert \psi^*_{\nu_j-j+m} \psi_{\lambda_i-i-k} \lvert -\ell \rangle \\ & = \sum_{k=0}^{\infty} h_{\lambda_i - \nu_j + j - i - k}(\emptyset / \boldsymbol{\pi}_{j-1}^{-1}) h_k({\mathbf{x}} \sqcup \boldsymbol{\pi}_i^{-1}) \\ & = h_{\lambda_i - \nu_j + j - i}({\mathbf{x}} \sqcup \boldsymbol{\pi}_i^{-1} / \boldsymbol{\pi}_{j-1}^{-1}), \end{align*} $$

yielding the claim.

Alternative proof.

We give another proof of Theorem 6.3 using the skew Cauchy formula (Theorem 2.4). If we take $\mu = \emptyset $ and the specializations $\boldsymbol {\alpha } = 0$ and ${\mathbf {x}} = \boldsymbol {\pi }^{-1}$ , we obtain

$$\begin{align*}\sum_{\lambda \supseteq \nu} \boldsymbol{\pi}^{\lambda} g_{\lambda/\nu}({\mathbf{y}}; \boldsymbol{\pi}) = \prod_i \frac{1}{1 - \pi_1 y_i} g_{\nu}({\mathbf{x}}; \boldsymbol{\beta}) \boldsymbol{\pi}^{\nu} \end{align*}$$

since $G_{\lambda }(\boldsymbol {\pi }^{-1}; \boldsymbol {\pi }) = \boldsymbol {\pi }^{\lambda }$ by directly examining the bi-alternate formula [Reference Hwang, Jang, Soo Kim, Song and SongHJK⁺24, Reference Hwang, Jang, Soo Kim, Song and SongHJK⁺25] (see also [Reference Fujii, Nobukawa and ShimazakiFNS23] for a description of other ways to verify this identity). The claim follows from the Jacobi–Trudi formula (see, e.g., [Reference Hwang, Jang, Soo Kim, Song and SongHJK⁺24, Thm. 6.1] or [Reference Iwao, Motegi and ScrimshawIMS24, Thm. 4.1]).

Unless otherwise stated, we will henceforth use the flagging $\phi $ from Proposition 6.1 for our flagged Schur functions. As a special case of Theorem 6.3 using Proposition 6.1, we obtain

$$ \begin{align*} \mathsf{P}_{\leq,n}(\lambda | \emptyset) & = \prod_{i=1}^{\ell} \prod_{j=1}^n (1 - \pi_i x_j) \sum_{\mu \subseteq \lambda} \boldsymbol{\pi}^{\mu} g_{\mu}({\mathbf{x}}_n; \boldsymbol{\pi}^{-1}) = \boldsymbol{\pi}^{\lambda} \prod_{i=1}^{\ell} \prod_{j=1}^n (1 - \pi_i x_j) s_{\lambda,\phi}({\mathbf{x}}_n, \boldsymbol{\pi}_{\ell}^{-1}). \end{align*} $$

Furthermore, this recovers [Reference Motegi and ScrimshawMS25, Thm. 4.26] by taking $\lambda $ to be an $\ell \times m$ rectangle, which allows us to forget about the flagging $\phi $ .

Remark 6.4. We want to compare Theorem 6.3 to [Reference Johansson and RahmanJR22, Thm. 2]. We begin by following [Reference Iwao, Motegi and ScrimshawIMS24, Thm. 4.18] with the substitution $w \mapsto z^{-1}$ to write the entries of the determinant as the integral

$$\begin{align*}h_{\lambda_i-\nu_j-i+j}({\mathbf{x}}_n \sqcup \boldsymbol{\pi}_i^{-1} / \boldsymbol{\pi}_{j-1}^{-1}) = \frac{1}{2\pi\mathbf{i}} \oint_{\gamma_r} \frac{\prod_{k=1}^{j-1} (1 - \pi_k^{-1} z^{-1})}{\prod_{k=1}^i (1 - \pi_k^{-1} z^{-1})} \frac{z^{\lambda_i - \mu_j - i + j}}{\prod_{k=1}^n (1 - x_k z^{-1})} \, \frac{dz}{z}, \end{align*}$$

where $\gamma $ is a counterclockwise oriented circle of radius $r> \left \lvert x_1 \right \rvert , \left \lvert \pi _1^{-1} \right \rvert , \left \lvert x_2 \right \rvert , \left \lvert \pi _2^{-1} \right \rvert , \ldots $ and $\mathbf {i} = \sqrt {-1}$ . Hence, we can express $\mathsf {P}(G(\ell ,n) \leq \lambda _{\ell }, \dotsc , G(1,n) \leq \lambda _1 | \mathbf {G}(0) = \nu ) = \det [F_{ij}]_{i,j=1}^{\ell }$ , where

(6.4) $$ \begin{align} F_{ij} = \frac{1}{2\pi\mathbf{i}} \oint_{\gamma_r} \frac{(\pi_i z)^{\lambda_i}}{(\pi_j z)^{\mu_j}} \frac{\prod_{k=1}^{j-1} (z - \pi_k^{-1})}{\prod_{k=1}^i (z - \pi_k^{-1})} \prod_{k=1}^n \frac{1 - x_k \pi_i}{1 - x_k z^{-1}} \, dz. \end{align} $$

This is the transpose of the expression in [Reference Johansson and RahmanJR22, Thm. 2] after noting that their requirement is $G(i,n) < \lambda _i + 1$ (this is an equivalent condition since $\lambda _i \in \mathbb {Z}_{\geq 0}$ ) and they use weakly increasing sequences (i.e., the order of the particles is reversed). In particular, we must multiply the integrand by $\prod _{k=1}^{\ell } \frac {z - \pi _k^{-1}}{z - \pi _k^{-1}}$ and reindex the particles and matrix by $j \mapsto \ell + 1 - j$ (so $\pi _j \mapsto \pi _{\ell +1-j}$ as well). A similar transformation shows that [Reference Johansson and RahmanJR22, Thm. 1] is equivalent to the integral formula from [Reference Iwao, Motegi and ScrimshawIMS24, Thm. 4.18] with Theorem 1.1.

Example 6.5. Let us consider $\lambda = (2,1)$ and $\nu = \emptyset $ for Case A. Thus we take $\ell = 2$ , and we assume that $n \geq 2$ . In this computation, we will essentially ignore the normalization factor $C = \prod _{i=1}^{\ell } \prod _{j=1}^n (1 - \pi _i x_j)$ , as that factor clearly cancels. By the definition and Theorem 1.1,

$$ \begin{align*} C^{-1} \cdot \mathsf{P}_{\leq, n}(\lambda | \emptyset) & = C^{-1} \cdot \mathsf{P}(G(2,n) \leq \lambda_2, G(1,n) \leq \lambda_1) \\ & = C^{-1} \cdot \big( \mathsf{P}(\mathbf{G}(n) = (2,1)) + \mathsf{P}(\mathbf{G}(n) = (2)) + \mathsf{P}(\mathbf{G} = (1,1)) \\ & \hspace{40pt} + \mathsf{P}(\mathbf{G}(n) = (1)) + \mathsf{P}(\mathbf{G}(n) = \emptyset) \big) \\ & = \pi_1^2 \pi_2 g_{21}({\mathbf{x}}_n, \boldsymbol{\pi}^{-1}) + \pi_1^2 g_{2}({\mathbf{x}}_n, \boldsymbol{\pi}^{-1}) + \pi_1 \pi_2 g_{11}({\mathbf{x}}_n, \boldsymbol{\pi}^{-1}) \\ & \hspace{20pt} + \pi_1 g_{1}({\mathbf{x}}_n, \boldsymbol{\pi}^{-1}) + g_{\emptyset}({\mathbf{x}}_n, \boldsymbol{\pi}^{-1}) \\ & = \pi_1^2 \pi_2 (s_{21} + \pi_1^{-1} s_2) + \pi_1^2 s_2 + \pi_1 \pi_2 (s_{11} + \pi_1^{-1} s_1) + \pi_1 s_1 + 1 \\ & = \pi_1^2 \pi_2 s_{21} + (\pi_1 \pi_2 + \pi_1^2) s_2 + \pi_1 \pi_2 s_{11} + (\pi_1 + \pi_2) s_1 + 1, \end{align*} $$

where we have written $s_{\lambda } = s_{\lambda }({\mathbf {x}}_n)$ for brevity. Next, by the branching rule for flagged Schur functions, we see that

$$ \begin{align*} \pi_1^2 \pi_2 s_{21,\phi}({\mathbf{x}}_n, \boldsymbol{\pi}_2^{-1}) & = \pi_1^2 \pi_2 \big( s_{21}({\mathbf{x}}_n) s_{21/21,\phi}(\boldsymbol{\pi}_2^{-1}) + s_{2}({\mathbf{x}}_n) s_{21/2,\phi}(\boldsymbol{\pi}_2^{-1}) + s_{11}({\mathbf{x}}_n) s_{21/11,\phi}(\boldsymbol{\pi}_2^{-1}) \\ & \hspace{40pt} + s_{1}({\mathbf{x}}_n) s_{21/1,\phi}(\boldsymbol{\pi}_2^{-1}) + s_{\emptyset}({\mathbf{x}}_n) s_{21/\emptyset,\phi}(\boldsymbol{\pi}_2^{-1}) \big) \\ & = \pi_1^2 \pi_2 \big( s_{21} + (\pi_1^{-1} + \pi_2^{-1}) s_{2} + \pi_1^{-1}s_{11} + (\pi_1^{-2} + \pi_1^{-1} \pi_2^{-1}) s_{1} + \pi_1^{-2} \pi_2^{-1} \big) \\ & = \pi_1^2 \pi_2 s_{21} + (\pi_1 \pi_2 + \pi_1^2) s_2 + \pi_1 \pi_2 s_{11} + (\pi_1 + \pi_2) s_1 + 1. \end{align*} $$

Finally, let us compute the determinant from Theorem 6.3:

$$ \begin{align*} \pi_1^2 \pi_2 \det \begin{bmatrix} h_2({\mathbf{x}}, \pi_1^{-1}) & h_3({\mathbf{x}}) \\ h_0({\mathbf{x}}, \pi_1^{-1}, \pi_2^{-1}) & h_1({\mathbf{x}}, \pi_2^{-1}) \end{bmatrix} & = \pi_1^2 \pi_2 ( h_2 + \pi_1^{-1} h_1 + \pi_1^{-2}) \cdot (h_1 + \pi_2^{-1}) - \pi_1^2 \pi_2 h_3 \cdot 1 \\ & = \pi_1^2 \pi_2 (h_{21} - h_3) + \pi_1 \pi_2 h_{11} + (\pi_2 + \pi_1) h_1 + \pi_1^2 h_2 + 1 \\ & = \pi_1^2 \pi_2 s_{21} + (\pi_1 \pi_2 + \pi_1^2) s_2 + \pi_1 \pi_2 s_{11} + (\pi_2 + \pi_1) s_1 + 1. \end{align*} $$

Theorem 6.6. For Case D, the multipoint distribution with $\ell $ particles is given by

$$\begin{align*}\mathsf{P}_{\leq,n}(\lambda | \nu) = \boldsymbol{\rho}^{\lambda/\nu} \prod_{i=1}^{\ell} \prod_{j=1}^n (1 - \rho_i x_j)^{-1} \cdot \det \big[ e_{\lambda_i - \nu_j + j - i}({\mathbf{x}} \sqcup -\boldsymbol{\rho}_{j-1}^{-1} / {-\boldsymbol{\rho}_i^{-1}}) \big]_{i,j=1}^{\ell}. \end{align*}$$

Example 6.7. For this example, we consider the Case D TASEP. Consider $\lambda = (2, 1, 1)$ and $\nu = (1)$ , and so we take $\ell = 3$ and $n \geq 3$ . Note that $\lambda ' = (3,1)$ and $\nu ' = (1)$ , We will (again) ignore the normalization factor $C = \prod _{i=1}^{\ell } \prod _{j=1}^n (1 - \rho _i x_j)^{-1}$ as it clearly is present in all formulas. By the definition and Theorem 1.1, we have

$$ \begin{align*} C^{-1} \cdot \mathsf{P}_{\leq, n}(\lambda | \nu) & = \rho_1 \rho_2 \rho_3 j_{31/1}({\mathbf{x}}; \boldsymbol{\rho}^{-1}) + \rho_1 \rho_2 j_{21/1}({\mathbf{x}}; \boldsymbol{\rho}^{-1}) + \rho_2 \rho_3 j_{3/1}({\mathbf{x}}; \boldsymbol{\rho}^{-1}) \\ & \hspace{20pt} + \rho_2 j_{2/1}({\mathbf{x}}; \boldsymbol{\rho}^{-1}) + \rho_1 j_{11/1}({\mathbf{x}}; \boldsymbol{\rho}^{-1}) + j_{1/1}({\mathbf{x}}; \boldsymbol{\rho}^{-1}) \\ & = \rho_1 \rho_2 \rho_3 (s_3 + s_{21} + \rho_2^{-1} s_2 + \rho_2^{-1} s_{11}) + \rho_1 \rho_2 (s_2 + s_{11}) \\ & \hspace{20pt} + \rho_2 \rho_3 (s_2 + \rho_2^{-1} s_1) + \rho_2 s_1 + \rho_1 s_1 + 1 \\ & = \rho_1 \rho_2 \rho_3 s_3 + \rho_1 \rho_2 \rho_3 s_{21} + (\rho_1 \rho_2 + \rho_1 \rho_3 + \rho_2 \rho_3) s_2 \\ & \hspace{20pt} + (\rho_1 \rho_2 + \rho_1 \rho_3) s_{11} + (\rho_1 + \rho_2 + \rho_3) s_1 + 1. \end{align*} $$

The determinant formula from Theorem 6.6 yields

$$ \begin{align*} & \rho_1 \rho_2 \rho_3 \det \begin{bmatrix} e_1({\mathbf{x}} / (-\rho_1^{-1})) & e_3({\mathbf{x}}) & e_4({\mathbf{x}}, -\rho_2^{-1}) \\ e_{-1}({\mathbf{x}} / {-\boldsymbol{\rho}_2^{-1}}) & e_1({\mathbf{x}} / (-\rho_2^{-1})) & e_2({\mathbf{x}}) \\ e_{-2}({\mathbf{x}} / {-\boldsymbol{\rho}_3^{-1}}) & e_0({\mathbf{x}} / (-\rho_2^{-1}, -\rho_3^{-1})) & e_1({\mathbf{x}} / (-\rho_3^{-1}) \\ \end{bmatrix} \\ & \qquad = \rho_1 \rho_2 \rho_3 \det \begin{bmatrix} e_1 + \rho_1^{-1} & e_3 & e_4 - \rho_2^{-1} e_3 \\ 0 & e_1 + \rho_2^{-1} & e_2 \\ 0 & 1 & e_1 + \rho_3^{-1} \\ \end{bmatrix} \\ & \qquad = \rho_1 \rho_2 \rho_3 (e_1 + \rho_1^{-1}) (e_1 + \rho_2^{-1}) (e_1 + \rho_3^{-1}) - \rho_1 \rho_2 \rho_3 e_2 (e_1 + \rho_1^{-1}) \cdot 1 \\ & \qquad = \rho_1 \rho_2 \rho_3 e_{111} + (\rho_1 \rho_2 + \rho_1 \rho_3 + \rho_2 \rho_3) e_{11} + (\rho_1 + \rho_2 + \rho_3) e_1 + 1 - \rho_1 \rho_2 \rho_3 e_{21} - \rho_2 \rho_3 e_2 \\ & \qquad = \rho_1 \rho_2 \rho_3 (s_3 + 2s_{21} + s_{111}) + (\rho_1 \rho_2 + \rho_1 \rho_3 + \rho_2 \rho_3) (s_2 + s_{11}) + (\rho_1 + \rho_2 + \rho_3) s_1 + 1 \\ & \qquad \hspace{20pt} - \rho_1 \rho_2 \rho_3 (s_{21} + s_{111}) - \rho_2 \rho_3 s_{11} \\ & \qquad = \rho_1 \rho_2 \rho_3 s_3 + \rho_1 \rho_2 \rho_3 s_{21} + (\rho_1 \rho_2 + \rho_1 \rho_3 + \rho_2 \rho_3) s_2 \\ & \qquad \hspace{20pt} + (\rho_1 \rho_2 + \rho_1 \rho_3) s_{11} + (\rho_1 + \rho_2 + \rho_3) s_1 + 1. \end{align*} $$

Analogously to Equation (6.4), in Case D we have $\mathsf {P}_{\leq ,n}(\lambda | \nu ) = \det [F_{ij}]_{i,j=1}^{\ell }$ with

(6.5) $$ \begin{align} F_{ij} = \frac{1}{2\pi\mathbf{i}} \oint_{\gamma_r} \frac{(\rho_i z)^{\lambda_i}}{(\rho_j z)^{\mu_j}} \frac{\prod_{k=1}^{j-1} (z - \rho_k^{-1})}{\prod_{k=1}^i (z - \rho_k^{-1})} \prod_{k=1}^n \frac{1 - x_k z^{-1}}{1 - x_k \rho_i} \, dz. \end{align} $$

6.2 Blocking

Next, let us consider Case C, and recall that $\beta _i = \pi _{i+1}$ . We begin with some preparatory formulas.

Note that $g_{\lambda }(\pi _1; \boldsymbol {\beta }) = \boldsymbol {\pi }^{\lambda }$ from the combinatorial description. Taking the skew Cauchy formula (Theorem 2.4) with the specializations $\boldsymbol {\alpha } = 0$ and ${\mathbf {y}} = \boldsymbol {\pi }_1$ , we obtain

(6.6) $$ \begin{align} \sum_{\lambda \supseteq \nu, \mu} G_{\lambda/\!\!/\mu}({\mathbf{x}}; \boldsymbol{\beta}) \boldsymbol{\pi}^{\lambda/\nu} = \prod_i \frac{1}{1 - \pi_1 x_i} \sum_{\eta \subseteq \nu \cap \mu} G_{\nu/\!\!/\eta}({\mathbf{x}}; \boldsymbol{\beta}) \boldsymbol{\pi}^{\mu/\eta}. \end{align} $$

We will use the notation

$$\begin{align*}\mathsf{P}_{\geq,n}(\nu | \mu) := \mathsf{P}(G(\ell,n) \geq \nu_{\ell}, \dotsc, G(1,n) \geq \nu_1 | \mathbf{G}(0) = \mu). \end{align*}$$

Using Equation (6.6), we obtain an expression for the multipoint distribution for Case C as

(6.7a) $$ \begin{align} \mathsf{P}_{\geq,n}(\nu | \mu) & = \prod_{j=1}^n (1 - \pi_1 x_j) \sum _{\lambda \supseteq \nu,\mu} \boldsymbol{\pi}^{\lambda/\mu} G_{\lambda /\!\!/ \mu}({\mathbf{x}}_n; \boldsymbol{\beta}) \end{align} $$

(6.7b) $$ \begin{align} & = \sum _{\eta \subseteq \nu \cap \mu} \boldsymbol{\pi}^{\nu/\eta} G_{\nu /\!\!/ \eta}({\mathbf{x}}_n; \boldsymbol{\beta}). \end{align} $$

Note that if $\nu = \mu $ , then we obtain a Cauchy–Littlewood type identity with Grothendieck polynomials from the first equality (6.7a) since the total probability is $1$ . That is, we have

(6.8) $$ \begin{align} \sum_{\lambda} G_{\lambda /\!\!/ \mu}({\mathbf{x}}_n; \boldsymbol{\beta}) \boldsymbol{\pi}^{\lambda} = \prod_{i=1}^n \frac{1}{1 - \pi_1 x_i} \boldsymbol{\pi}^{\mu}, \end{align} $$

which is also the skew Pieri rule [Reference Iwao, Motegi and ScrimshawIMS24, Eq. (4.7a)] (which refines [Reference YeliussizovYel19, Thm. 7.10]) with the skew Pieri $\nu = \emptyset $ and the same specializations $\boldsymbol {\alpha } = 0$ and ${\mathbf {y}} = \boldsymbol {\pi }_1$ we used for the skew Cauchy formula. Another special case is when $\mu = \emptyset $ , where we obtain a single $\boldsymbol {\pi }^{\nu } G_{\nu }({\mathbf {x}}_n; \boldsymbol {\beta })$ in (6.7b). This can be expressed as a determinant by the Jacobi–Trudi formula [Reference Iwao, Motegi and ScrimshawIMS24, Thm. 4.1]. Moreover, this is just the specialization $\boldsymbol {\alpha } = 0$ and ${\mathbf {y}} = \boldsymbol {\pi }_1$ of the skew Pieri formula [Reference Iwao, Motegi and ScrimshawIMS24, Eq. (4.7b)].

Noting that we have used the skew Cauchy identity in computing (6.7b), we want to evaluate

(6.9) $$ \begin{align} {}^{[\boldsymbol{\beta}]} \langle \mu \rvert e^{H^*(\pi_1)} e^{H({\mathbf{x}}_n)} \lvert \nu \rangle^{[\boldsymbol{\beta}]} = \prod_{i=1}^{\ell} \prod_{j=1}^n \frac{1}{1-\beta_i x_j} \cdot {}^{[\boldsymbol{\beta}]} \langle \mu \rvert e^{H^*(\pi_1 / \boldsymbol{\beta}_{\ell})} e^{H({\mathbf{x}}_n)} e^{H^*(\boldsymbol{\beta}_{\ell})} \lvert \nu \rangle^{[\boldsymbol{\beta}]}. \end{align} $$

By applying Wick’s theorem similar to the proof of Theorem 6.3 (cf. the proof of [Reference Iwao, Motegi and ScrimshawIMS24, Thm. 4.1]), we obtain the following formula for the general case.

Theorem 6.8. For Case C, the multipoint distribution with $\ell $ particles is given by

$$\begin{align*}\mathsf{P}_{\geq,n}(\nu|\mu) = \prod_{j=2}^{\ell} \prod_{i=1}^n (1-\pi_j x_i)^{-1} \det \big[ h_{\nu_i - \mu_j - i + j}\big({\mathbf{x}} /\!\!/ (\boldsymbol{\pi}_i / \boldsymbol{\beta}_j \big) \big]_{i,j=1}^{\ell}. \end{align*}$$

We remark that the left-hand side of (6.9) is using the vector obtained by applying the $*$ anti-involution to (6.2):

$$\begin{align*}{}^{(\boldsymbol{\pi})} \langle \nu \rvert := \sum_{\lambda \subseteq \nu} \boldsymbol{\pi}^{\nu/\lambda} \cdot {}^{[\boldsymbol{\pi}]} \langle \lambda \rvert. \end{align*}$$

However, our computation for Theorem 6.8 is not simply the $*$ version of Proposition 6.1 as we need to take the pairing with $\lvert \mu \rangle ^{[\boldsymbol {\beta }]}$ , not $\lvert \mu \rangle ^{[\boldsymbol {\pi }]}$ .

We also provide another determinant formula for the multipoint distribution by using the standard probability-theoretic computation to sum over determinants with matrix elements given by integrals.

Theorem 6.9. For Case C, the multipoint distribution with $\ell $ particles is given by

$$ \begin{gather*} \mathrm{P}_{\ge,n}(\nu|\mu) = \prod_{i=1}^\ell \prod_{j=1}^n (1-\pi_i x_j) \boldsymbol{\pi}^{\nu/\mu} \det [ C_{ij} ]_{i,j=1}^\ell, \\ \text{where } C_{ij} = \begin{cases} \displaystyle \oint_{\widetilde{\gamma}_r} \frac{1 }{ (1-\beta_1 w^{-1}) \prod_{k=1}^{j-1} (1-\beta_k w^{-1}) \prod_{m=1}^n (1-x_m w) w^{\nu_1-\mu_j-1+j} } \frac{dw}{2 \pi \mathbf{i} w} & \text{if } i = 1, \\[15pt] \displaystyle \oint_{\widetilde{\gamma}_r} \frac{\prod_{k=1}^{i-2} (1-\beta_k w^{-1}) }{ \prod_{k=1}^{j-1} (1-\beta_k w^{-1}) \prod_{m=1}^n (1-x_m w) w^{\nu_i-\mu_j-i+j} } \frac{dw}{2 \pi \mathbf{i} w} & \text{if } i \ge 2, \end{cases} \end{gather*} $$

with the contour $\widetilde {\gamma }_r$ being a circle centered at the origin with radius r satisfying $0 < r < \left \lvert x_m^{-1} \right \rvert $ for $m = 1, \dotsc , n$ and $r> \left \lvert \beta _1 \right \rvert , \left \lvert \beta _2 \right \rvert , \ldots $ .

Proof. Using Theorem 1.1 and the integral formula [Reference Iwao, Motegi and ScrimshawIMS24, Thm. 4.19], we write the multipoint distribution as

$$ \begin{align*} \mathrm{P}_{\ge,n}(\nu|\mu) & = \prod_{i=1}^\ell \prod_{j=1}^n (1-\pi_i x_j) \sum_{\lambda_1=\nu_1}^\infty \sum_{\lambda_2=\nu_2}^{\lambda_1} \cdots \sum_{\lambda_\ell=\nu_\ell}^{\lambda_{\ell-1}} \boldsymbol{\pi}^{-\mu} \\ & \hspace{20pt} \times \det \Bigg[ \oint_{\widetilde{\gamma}_r} \frac{\pi_i^{\lambda_i} \prod_{k=1}^{i-1} (1-\beta_k w^{-1}) }{ \prod_{k=1}^{j-1} (1-\beta_k w^{-1}) \prod_{m=1}^n (1-x_m w) w^{\lambda_i-\mu_j-i+j} } \frac{dw}{2 \pi \mathbf{i} w} \Bigg]_{i,j=1}^\ell. \end{align*} $$

Inserting the sum $\displaystyle \sum _{\lambda _\ell =\nu _\ell }^{\lambda _{\ell -1}}$ into the $\ell $ -th row, the matrix element in the j-th column becomes

(6.10) $$ \begin{align} \sum_{\lambda_\ell=\nu_\ell}^{\lambda_{\ell-1}} \oint_{\widetilde{\gamma}_r} & \frac{\pi_\ell^{\lambda_\ell} \prod_{k=1}^{\ell-1} (1-\beta_k w^{-1}) }{ \prod_{k=1}^{j-1} (1-\beta_k w^{-1}) \prod_{m=1}^n (1-x_m w) w^{\lambda_\ell-\mu_j-\ell+j} } \frac{dw}{2 \pi \mathbf{i} w} \nonumber \\ = & \oint_{\widetilde{\gamma}_r} \frac{\pi_\ell^{\nu_\ell} \prod_{k=1}^{\ell-2} (1-\beta_k w^{-1}) }{ \prod_{k=1}^{j-1} (1-\beta_k w^{-1}) \prod_{m=1}^n (1-x_m w) w^{\nu_\ell-\mu_j-\ell+j} } \frac{dw}{2 \pi \mathbf{i} w} \nonumber \\ & \hspace{20pt} - \oint_{\widetilde{\gamma}_r} \frac{\pi_\ell^{\lambda_{\ell-1}+1} \prod_{k=1}^{\ell-2} (1-\beta_k w^{-1}) }{ \prod_{k=1}^{j-1} (1-\beta_k w^{-1}) \prod_{m=1}^n (1-x_m w) w^{\lambda_{\ell-1}-\mu_j-(\ell-1)+j} } \frac{dw}{2 \pi \mathbf{i} w}. \end{align} $$

The second term in (6.10) can be eliminated using the $(\ell -1)$ -th row; hence, the matrix elements in the $\ell $ -th row after performing the first sum can be written as

$$\begin{align*}\oint_{\widetilde{\gamma}_r} \frac{\pi_\ell^{\nu_\ell} \prod_{k=1}^{\ell-2} (1-\beta_k w^{-1}) }{ \prod_{k=1}^{j-1} (1-\beta_k w^{-1}) \prod_{m=1}^n (1-x_m w) w^{\nu_\ell-\mu_j-\ell+j} } \frac{dw}{2 \pi \mathbf{i} w}. \end{align*}$$

Iterating this process, we obtain our claim.

For Case B, we again apply the $\omega $ involution, which replaces $e^{H({\mathbf {x}}_n)}$ with $e^{J({\mathbf {x}}_n)}$ , but otherwise the proof is similar (compare the proofs of Theorem 6.3 and Theorem 6.6). Therefore, we obtain the following.

Theorem 6.10. For Case B, the multipoint distribution with $\ell $ particles is given by

$$\begin{align*}\mathsf{P}_{\geq,n}(\nu|\mu) = \prod_{i=2}^{\ell} \prod_{j=1}^n (1-\rho_i x_j) \det \big[ e_{\nu_i - \mu_j - i + j}({\mathbf{x}}_n /\!\!/ ({-\boldsymbol{\rho}_j}/{-\boldsymbol{\rho}_i})) \big]_{i,j=1}^{\ell}. \end{align*}$$

In both of these cases, we can obtain contour integral formulas for the entries in matrices of the multipoint distribution determinants by following [Reference Iwao, Motegi and ScrimshawIMS24, Thm. 4.19] (see also Remark 6.4).

7.1 Blocking

We will consider continuous Markov process where particles have independent exponential clocks, where the j-th particle’s clock has rate $\pi _j$ , and when the clock rings, if the $(j-1)$ -th particle is not at the same site, then the j-th particle jumps one step to the right. We take the limit $p \to 0$ in the integral formula [Reference Iwao, Motegi and ScrimshawIMS24, Thm. 4.19], and using the classical formula $e^x = \lim _{n\to \infty } (1 + x/n)^n$ , we obtain the following.

Corollary 7.1. The continuous time limit of Case C, for $t \in (0, \infty )$ , is

$$\begin{align*}\mathsf{P}_C(\mathbf{G}(t) = \lambda| \mathbf{G}(0) = \mu) = \prod_{j=1}^{\ell} e^{-\pi_j t} \boldsymbol{\pi}^{\lambda/\mu} \det\left[ \oint_{\widetilde{\gamma}_r} \frac{\prod_{k=1}^{i-1} (1 - \beta_k w^{-1}) e^{tw}}{\prod_{k=1}^{j-1} (1 - \beta_k w^{-1}) w^{(\lambda_i - i) - (\mu_j - j)}} \frac{dw}{2\pi\mathbf{i} w} \right]_{i,j=1}^{\ell}, \end{align*}$$

where the contour $\widetilde {\gamma }_r$ is a circle centered at the origin with radius r satisfying $r> \left \lvert \beta _1 \right \rvert , \left \lvert \beta _2 \right \rvert , \ldots $ .

In Corollary 7.1, if we substitute $w^{-1} = z$ , note the j-th particle in the fermionic positioning is at $\nu _j - j$ from the bosonic positions $\nu $ , reindexing the particles $j \mapsto \ell + 1 - j$ (as in Remark 6.4), and taking the transpose of the determinant, we recover [Reference Rákos and SchützRS06, Thm. 1]. Furthermore, by taking the same limit of the multipoint distribution from Theorem 6.9, we recover [Reference Rákos and SchützRS06, Cor.]. On the other hand, these satisfy the corresponding TASEP Kolmogorov’s equation [Reference Rákos and SchützRS06, Eq. (3)] from Theorem 1.1 and boundary conditions [Reference Rákos and SchützRS06, Eq. (2)] as expected:

(7.1) $$ \begin{align} \frac{d}{dt} \mathsf{P}_C(\mathbf{G}(t)=\lambda | \mathbf{G}(0)=\mu) &= \sum_{s=1}^{\ell} \pi_s \mathsf{P}_C(\mathbf{G}(t)=\lambda | \mathbf{G}(0)=\mu) \nonumber\\ &\quad - \sum_{s=1}^{\ell} \pi_s \mathsf{P}_C(\mathbf{G}(t)=\lambda-\epsilon_s | \mathbf{G}(0)=\mu), \end{align} $$

(7.2) $$ \begin{align} \pi_s \mathsf{P}_C(\mathbf{G}(t)=\lambda-\epsilon_s | \mathbf{G}(0)=\mu) &= \pi_{s+1} \mathsf{P}_C(\mathbf{G}(t)=\lambda | \mathbf{G}(0)=\mu) \qquad \text{if } \lambda_s=\lambda_{s+1}. \end{align} $$

See the arXiv version [Reference Iwao, Motegi and ScrimshawIMS23, Thm.-7.2] for more details.

7.2 Pushing

We will consider continuous Markov process where particles have independent exponential clocks, where the j-th particle’s clock has rate $\pi _j$ , and when the clock rings, all particles $j' \leq j$ at the site of the j-th particle move one step to the right. We can similarly describe the transition probability as a determinant of contour integrals for the pushing behavior.

Corollary 7.2. The continuous time limit of Case A, for $t \in (0, \infty )$ , is

$$\begin{align*}\mathsf{P}_A(\mathbf{G}(t) = \lambda| \mathbf{G}(0) = \mu) = \prod_{j=1}^{\ell} e^{-\pi_j t} \boldsymbol{\pi}^{\lambda/\mu} \det\left[ \oint_{\gamma_r} \frac{\prod_{k=1}^{j-1} (1 - \pi_k^{-1} w) e^{tw}}{\prod_{k=1}^{i-1} (1 - \pi_k^{-1} w) w^{(\lambda_i - i) - (\mu_j - j)}} \frac{dw}{2\pi\mathbf{i} w} \right]_{i,j=1}^{\ell}. \end{align*}$$

By the same computations as in the blocking behavior case, we can also show the continuous time limit of Case A also satisfies the Kolmogorov’s equation [Reference Rákos and SchützRS06, Eq. (3)], but the boundary conditions are different due:

$$ \begin{align*} \pi_s \mathsf{P}_A(\mathbf{G}(t) = \lambda + \epsilon_{s+1} | \mathbf{G}(0) = \mu) = \pi_{s+1} \mathsf{P}_A(\mathbf{G}(t) = \lambda | \mathbf{G}(0) = \mu) \qquad \text{ if } \lambda_s = \lambda_{s+1}. \end{align*} $$