2.1 Finitely generated semigroups

A subset $\Gamma \subseteq {{\mathbb {Z}}}^n$ is a semigroup if $\vec 0\in \Gamma$ and $\Gamma$ is closed under addition. A finitely generated semigroup $\Gamma$ generates a closed convex polyhedral cone $\Gamma _{\mathbb {Q}}\subseteq {\mathbb {Q}}^n$ :

\begin{equation*} \Gamma _{\mathbb {Q}}=\{x\in {\mathbb {Q}}^n:\exists t\in {{\mathbb {Z}}}_{\gt 0}\ tx\in \Gamma \}. \end{equation*}

The subgroup of ${{\mathbb {Z}}}^n$ generated by $\Gamma$ is

\begin{equation*} \Gamma _{{\mathbb {Z}}}=\{x-y\,:\,x,y\in \Gamma \}. \end{equation*}

The semigroup $\Gamma$ is saturated if $\Gamma =\Gamma _{{\mathbb {Z}}}\cap \Gamma _{\mathbb {Q}}$ .

2.2 GIT-semigroups

We recall the GIT-perspective of [Reference RessayreRes10]. Let $G$ be a complex reductive group acting on an irreducible projective variety $X$ . Let ${\operatorname {Pic}}^G(X)$ be the group of $G$ -linearized line bundles. Given ${\mathcal L}\in {\operatorname {Pic}}^G(X)$ , let ${\operatorname {H}}^0(X,{\mathcal L})$ be the space of sections of $\mathcal L$ ; it is a $G$ -module. Let ${\operatorname {H}}^0(X,{\mathcal L})^G$ be the subspace of invariant sections. Define

\begin{equation*} {\operatorname {GIT-semigroup}}(G,X)=\{{\mathcal L}\in {\operatorname {Pic}}^G(X)\,:\, {\operatorname {H}}^0(X,{\mathcal L})^G\neq \{0\}\}. \end{equation*}

This is a semigroup since $X$ being irreducible says the product of two nonzero $G$ -invariant sections is a nonzero $G$ -invariant section. The saturated version of it is

\begin{equation*} {\operatorname {GIT-sat}}(G,X)=\{{\mathcal L}\in {\operatorname {Pic}}^G(X)\otimes {\mathbb {Q}}: \exists t\gt 0, \ {\operatorname {H}}^0(X,{\mathcal L}^{\otimes t})^G\neq \{0\}\}. \end{equation*}

2.3 The tensor semigroup

Let $\mathfrak {g}$ be a semisimple complex Lie algebra, with fixed Borel subalgebra $\mathfrak b$ and Cartan subalgebra ${\mathfrak t}\subset {\mathfrak b}$ . Denote by $\Lambda ^+({\mathfrak {g}})\subset {\mathfrak t}^*$ the semigroup of the dominant weights. It is contained in the weight lattice $\Lambda ({\mathfrak {g}})\simeq {{\mathbb {Z}}}^r$ , where $r$ is the rank of $\mathfrak {g}$ . Given $\lambda \in \Lambda ^+({\mathfrak {g}})$ , denote by $V_{\mathfrak {g}}(\lambda )$ (or simply $V(\lambda )$ ) the irreducible representation of $\mathfrak {g}$ with highest weight $\lambda$ . Let $V(\lambda )^*$ be the dual representation. Consider the semigroup

\begin{equation*} {{\mathfrak {g}}\operatorname {-semigroup}}=\{(\lambda ,\mu ,\nu )\in (\Lambda ^+({\mathfrak {g}}))^3\;:\;V(\nu )^*\subset V(\lambda )\otimes V(\mu )\}, \end{equation*}

and the generated cone ${\mathfrak {g}}\operatorname {-sat}$ in $(\Lambda ({\mathfrak {g}})\otimes {\mathbb {Q}})^3$ . When ${\mathfrak {g}}={\mathfrak {sp}}(2m,{\mathbb {C}})$ we have $V(\nu )^*\simeq V(\nu )$ and ${\mathfrak {g}}\operatorname {-semigroup}$ is what we denoted by $\mathfrak {sp}$ -semigroup $(m)$ in the introduction. The set ${\mathfrak {g}}\operatorname {-semigroup}$ spans the rational vector space $(\Lambda ({\mathfrak {g}})\otimes {\mathbb {Q}})^3$ , or equivalently, the cone ${\mathfrak {g}}\operatorname {-sat}$ has nonempty interior. The group $({{\mathfrak {g}}\operatorname {-semigroup}})_{{\mathbb {Z}}}$ is well known (see, for example, [Reference Pasquier and RessayrePR13, Theorem 1.4]):

\begin{equation*} ({{\mathfrak {g}}\operatorname {-semigroup}})_{{\mathbb {Z}}}=\{(\lambda ,\mu ,\nu )\in (\Lambda ({\mathfrak {g}}))^3\;:\; \lambda +\mu +\nu \in \Lambda _R({\mathfrak {g}})\}, \end{equation*}

where $\Lambda _R({\mathfrak {g}})$ is the root lattice of $\mathfrak {g}$ .

We now interpret ${\mathfrak {g}}\operatorname {-semigroup}$ in terms of § 2.2. Consider the semisimple, simply connected algebraic group $G$ with Lie algebra $\mathfrak {g}$ . Denote by $B$ and $T$ the connected subgroups of $G$ with Lie algebras $\mathfrak b$ and $\mathfrak t$ , respectively. The character groups $X(B)=X(T)=\Lambda ({\mathfrak {g}})$ of $B$ and $T$ coincide. For $\lambda \in X(T)$ , ${\mathcal L}_\lambda$ is the unique $G$ -linearized line bundle on the flag variety $G/B$ such that $B$ acts on the fiber over $B/B$ with weight $-\lambda$ .

Assume $X=(G/B)^3$ . Then ${\operatorname {Pic}}^G(X)$ identifies with $X(T)^3$ . For $(\lambda ,\mu ,\nu )\in X(T)^3$ , define ${\mathcal L}_{(\lambda ,\mu ,\nu )}$ . By the Borel–Weil theorem,

(10) \begin{align} {\operatorname {H}}^0(X,{\mathcal L}_{(\lambda ,\mu ,\nu )})=V(\lambda )^*\otimes V(\mu )^*\otimes V(\nu )^*. \end{align}

In particular, ${\operatorname {GIT-semigroup}}(G,X)\simeq {{\mathfrak {g}}\operatorname {-semigroup}}$ .

Given three parabolic subgroups $P,Q$ and $R$ containing $B$ , we consider more generally $X=G/P\times G/Q\times G/R$ . Then ${\operatorname {Pic}}^G(X)$ identifies with $X(P)\times X(Q)\times X(R)$ which is a subgroup of $X(T)^3$ . Moreover,

\begin{equation*} {\operatorname {GIT-semigroup}}(G,X)={\operatorname {GIT-semigroup}}(G,(G/B)^3)\cap (X(P)\times X(Q)\times X(R)). \end{equation*}

2.4 Schubert calculus

We need notation for the cohomology ring ${\operatorname {H}}^*(G/P,{{\mathbb {Z}}})$ ; $P\supset B$ being a parabolic subgroup. Let $W$ (respectively, $W_P$ ) be the Weyl group of $G$ (respectively, $P$ ). Let $\ell \,:\,W\longrightarrow {\mathbb {N}}$ be the Coxeter length, defined with respect to the simple reflections determined by the choice of $B$ . Let $W^P$ be the minimal length representatives of the cosets in $W/W_P$ .

For a closed irreducible subvariety $Z\subset G/P$ , let $[Z]$ be its class in ${\operatorname {H}}^*(G/P,{{\mathbb {Z}}})$ , of degree $2(\dim (G/P)-\dim (Z))$ . For $v \in W^P$ , set

\begin{equation*}\sigma _v=[\overline {BvP/P}]\end{equation*}

( $\dim (\overline {BvP/P})=\ell (v)$ ). Then

\begin{equation*} {\operatorname {H}}^*(G/P,{{\mathbb {Z}}})=\bigoplus _{v\in W^P}{{\mathbb {Z}}}\sigma _v. \end{equation*}

Let $w_0$ be the longest element of $W$ and $w_{0,P}$ be the longest element of $W_P$ . Set $v^\vee =w_0vw_{0,P}$ and $\sigma _v=\sigma ^{v^\vee }$ ; $\sigma ^v$ and $\sigma _v$ are Poincaré dual.

Let $\rho$ be the half sum of the positive roots of $G$ . To any one-parameter subgroup $\tau :{{\mathbb {C}}}^*\to T$ , associate the parabolic subgroup (see [Reference Mumford, Fogarty and KirwanMFK94])

\begin{equation*} P(\tau )=\Big\{g\in G\,:\,\lim _{t\to 0}\tau (t)g\tau (t{^{-1}}) \mbox { exists}\Big\}. \end{equation*}

Fix such a $\tau$ such that $P=P(\tau )$ .

For $v\in W^P$ , define the BK-degree of $\sigma ^v\in {\operatorname {H}}^*(G/P,{{\mathbb {Z}}})$ to be

\begin{equation*} {\operatorname {BK-deg}}(\sigma ^v):=\langle v{^{-1}}(\rho )-\rho ,\tau \rangle .\end{equation*}

Let $v_1,v_2$ and $v_3$ in $W^P$ . By [Reference Belkale and KumarBK06, Proposition 17], if $\sigma ^{v_3}$ appears in the product $\sigma ^{v_1} \cdot \sigma ^{v_2}$ then

(11) \begin{align} {\operatorname {BK-deg}}(\sigma ^{v_3})\leqslant {\operatorname {BK-deg}}(\sigma ^{v_1})+{\operatorname {BK-deg}}(\sigma ^{v_2}). \end{align}

In other words, the BK-degree filters the cohomology ring. Let $\odot _0$ denote the associated graded product on ${\operatorname {H}}^*(G/P,{{\mathbb {Z}}})$ .

2.5 Well-covering pairs

In [Reference RessayreRes10], ${\operatorname {GIT-sat}}(G,X)$ is described in terms of well-covering pairs. When $X=(G/B)^3$ , it recovers the description made by Belkale and Kumar [Reference Belkale and KumarBK06]. We now discuss the case when $X=G/P\times G/Q\times G/R$ is the product of three partial flag varieties of $G$ .

Let $\tau$ be a dominant one-parameter subgroup of $T$ . The centralizer $G^\tau$ of the image of $\tau$ in $G$ is a Levi subgroup. Moreover, $ P(\tau )$ is the parabolic subgroup generated by $B$ and $G^\tau$ . Let $C$ be an irreducible component of the fixed set $X^\tau$ of $\tau$ in $X$ . It is well known that $C$ is the $(G^\tau )^3$ -orbit of some $T$ -fixed point:

(12) \begin{align} C=G^\tau u{^{-1}} P/P\times G^\tau v{^{-1}} Q/Q \times G^\tau w{^{-1}} R/R, \end{align}

with $u\in {W_P}\backslash W/W_{P(\tau )}$ , and similarly for $v$ and $w$ . Set

\begin{equation*} C^+=P(\tau ) u{^{-1}} P/P\times P(\tau ) v{^{-1}} Q/Q \times P(\tau ) w{^{-1}} R/R. \end{equation*}

Then the closure of $C^+$ is a Schubert variety (for $G^3$ ) in $X$ . By [Reference RessayreRes10, Proposition 11], the pair $(C,\tau )$ is well covering if and only if

(13) \begin{align} [\overline {PuP(\tau )/P(\tau )}]\odot _0 [\overline {QvP(\tau )/P(\tau )}]\odot _0 [\overline {RwP(\tau )/P(\tau )}]=[pt] \in {\operatorname {H}}^*(G/P(\tau ),{{\mathbb {Z}}}). \end{align}

It is said to be dominant if

(14) \begin{align} [\overline {PuP(\tau )/P(\tau )}]\cdot [\overline {QvP(\tau )/P(\tau )}]\cdot [\overline {RwP(\tau )/P(\tau )}]\neq 0 \in {\operatorname {H}}^*(G/P(\tau ),{{\mathbb {Z}}}). \end{align}

In this paper, the reader can take these characterizations as definitions of well-covering and dominant pairs. They are used in [Reference RessayreRes10] to produce inequalities for the GIT-cones.

In the case $P=Q=R=B$ , there is a more precise statement. The fact that the inequalites define the cone is due to Belkale and Kumar [Reference Belkale and KumarBK06, Theorem 28]. The irredundancy is [Reference RessayreRes10, Theorem B]. Let $\alpha$ be a simple root of $G$ . Denote by $P^\alpha$ the associated maximal parabolic subgroup of $G$ containing $B$ . Denote by $\varpi _{\alpha ^\vee }$ the associated fundamental one-parameter subgroup of $T$ characterized by $\langle \varpi _{\alpha ^\vee },\beta \rangle =\delta _{\alpha \beta }$ (Kronecker delta) for any simple root $\beta$ .

Theorem 2.3 ([Reference Belkale and KumarBK06, Theorem 28], [Reference RessayreRes10, Theorem B]). Here $X=(G/B)^3$ . Let $(\lambda ,\mu ,\nu )\in X(T)^3$ be dominant. Then, ${\mathcal L}_{(\lambda ,\mu ,\nu )}\in {\operatorname {GIT-sat}}(G,X)$ if and only if for any simple root $\alpha$ , for any $u,v,w$ in $W^{P^\alpha }$ such that

(15) \begin{align} [\overline {BuP^\alpha /P^\alpha }]\odot _0 [\overline {BvP^\alpha /P^\alpha }]\odot _0 [\overline {BwP^\alpha /P^\alpha }]=[pt] \in {\operatorname {H}}^*(G/P^\alpha ,{{\mathbb {Z}}}), \\[-30pt] \nonumber \end{align}

(16) \begin{align} \langle u\varpi _{\alpha ^\vee },\lambda \rangle +\langle v\varpi _{\alpha ^\vee },\mu \rangle +\langle w \varpi _{\alpha ^\vee },\nu \rangle \leqslant 0. \\[-8pt] \nonumber \end{align}

Moreover, this list of inequalities is irredundant.

Theorem 2.3 can be obtained from Proposition 2.1(3) by showing that it is sufficient to consider the one-parameter subgroups $\tau$ equal to $\varpi _{\alpha ^\vee }$ for some simple root $\alpha$ . See the proof of Theorem 5.1 below for a similar argument.

2.6 The eigencone

A relationship between ${\mathfrak {g}}\operatorname {-sat}$ and projections of coadjoint orbits was discovered by Heckman [Reference HeckmanHec82]. Theorem 2.4 below interprets ${\mathfrak {g}}\operatorname {-sat}$ in terms of eigenvalues.

Fix a maximal compact subgroup $U$ of $G$ such that $T\cap U$ is a Cartan subgroup of $U$ . Let $\mathfrak u$ and $\mathfrak t$ denote the Lie algebras of $U$ and $T$ , respectively. Let ${\mathfrak t}^+$ be the Weyl chamber of $\mathfrak t$ corresponding to $B$ . Let $\sqrt {-1}$ denote the usual complex number. It is well known that $\sqrt {-1}{\mathfrak t}^+$ is contained in $\mathfrak u$ and that the map

(17) \begin{align} \begin{array}{c@{\quad}c@{\quad}c} {\mathfrak t}^+&\longrightarrow &{\mathfrak u}/U\\ \xi &\longmapsto &U.(\sqrt {-1}\xi ) \end{array} \end{align}

is a homeomorphism. Here $U$ acts on $\mathfrak u$ by the adjoint action. Consider the set

\begin{equation*} \Gamma (U):=\{(\xi ,\zeta ,\eta )\in ({\mathfrak t}^+)^3\,:\, U.(\sqrt {-1}\xi )+U.(\sqrt {-1}\zeta )+U.(\sqrt {-1}\eta )\ni 0\}. \end{equation*}

Let ${\mathfrak u}^*$ (respectively, ${\mathfrak t}^*$ ) denote the dual (respectively, complex dual) of $\mathfrak u$ (respectively, $\mathfrak t$ ). Let ${\mathfrak t}^{*+}$ denote the dominant chamber of ${\mathfrak t}^*$ corresponding to $B$ . By taking the tangent map at the identity, one can embed $X(T)^+$ in ${\mathfrak t}^{*+}$ . Note that this embedding induces a rational structure on the complex vector space ${\mathfrak t}^*$ . Moreover, it allows the tensor cone ${\mathfrak {g}}\operatorname {-sat}$ to be embedded in $({\mathfrak t}^{*+})^3$ .

The Cartan-Killing form allows ${\mathfrak t}^+$ and ${\mathfrak t}^{*+}$ to be identified. In particular, $\Gamma (U)$ also embeds in $({\mathfrak t}^{*+})^3$ ; the subset of $({\mathfrak t}^{*+})^3$ thus obtained is denoted by $\tilde \Gamma (U)$ to avoid any confusion. The following result is well known; see, for example, [Reference KumarKum14, Theorem 5] and the references therein.

3.1 The root system of type $C$

Let $V={{\mathbb {C}}}^{2n}$ with the standard basis $(\vec e_1,\vec e_2,\ldots ,\vec e_{2n})$ . Let $J_n$ be the $n\times n$ `anti-diagonal’ identity matrix and define a skew-symmetric bilinear form $\omega (\bullet ,\bullet ): V\times V\to {{\mathbb {C}}}$ using the block matrix $\Omega :=\left [\begin{matrix} 0 & J_n \\ -J_n & 0 \end{matrix}\right ]$ . By definition, the symplectic group $G={\operatorname {Sp}}(2n,{\mathbb {C}})$ is the group of automorphisms of $V$ that preserve this bilinear form.

Given an $n\times n$ matrix $A=(A_{ij})_{1\leqslant i,j\leqslant n}$ , define ${}^TA$ by $({}^TA)=A_{n+1-j\,n+1-i}$ , obtained from $A$ by reflection across the antidiagonal. The Lie algebra ${\mathfrak {sp}}(2n,{\mathbb {C}})$ is the set of matrices $M\in \textsf {Mat}_{2n\times 2n}({\mathbb {C}})$ such that ${}^tM\Omega +\Omega M=0$ ; namely,

(18) \begin{align} {\mathfrak {sp}}(2n,{\mathbb {C}})= \left \{ \begin{pmatrix} A&B\\ C&-{}^TA \end{pmatrix} : \begin{array}{l} A,B,C \mbox { of size }n\times n,\\ {}^TB=B \mbox { and } {}^TC=C \end{array} \right \} \end{align}

which has the complex dimension $2n^2+n$ . The Lie algebra ${\mathfrak u}(2n,{\mathbb {C}})$ of the unitary group $U(2n,{\mathbb {C}})$ is the set of anti-Hermitian matrices. Thus, (18) gives

(19) \begin{align} {\mathfrak {sp}}(2n,{\mathbb {C}})\cap {\mathfrak u}(2n,{\mathbb {C}})= \left \{ \begin{pmatrix} A&B\\ -{}^t\bar B&-{}^TA \end{pmatrix} : {}^t\bar A=-A \mbox { and } {}^TB=B \right \}, \end{align}

which has real dimension $2n^2+n$ . As a consequence, $U(2n)\cap {\operatorname {Sp}}(2n,{\mathbb {C}})$ is a maximal compact subgroup of ${\operatorname {Sp}}(2n,{\mathbb {C}})$ .

Let $B$ be the Borel subgroup of upper triangular matrices in $G$ . Let

\begin{equation*}T=\{{\operatorname {diag}}(t_1,\ldots ,t_{n},t_{n}^{-1},\ldots ,t_1^{-1})\,:\,t_i\in {\mathbb {C}}^*\}\end{equation*}

be the maximal torus contained in $B$ . For $i\in [n]$ , let $\varepsilon _i$ denote the character of $T$ that maps ${\operatorname {diag}}(t_1,\ldots ,t_{n},t_{n}^{-1},\ldots ,t_1^{-1})$ to $t_i$ ; then $X(T)=\oplus _{i=1}^n{{\mathbb {Z}}}\varepsilon _i$ . Here

\begin{equation*} \begin{array}{c} \Phi ^+=\{\varepsilon _i\pm \varepsilon _j\,:\,1\leqslant i\lt j\leqslant n\}\cup \{2\varepsilon _i\,:\,1\leqslant i\leqslant n\}, \\[6pt] \Delta =\{\alpha _1=\varepsilon _1-\varepsilon _2,\,\alpha _2=\varepsilon _2-\varepsilon _3,\ldots ,\, \alpha _{n-1}=\varepsilon _{n-1}-\varepsilon _{n},\,\alpha _{n}=2\varepsilon _{n}\},\\[6pt] X(T)^+=\displaystyle\left\{\sum \nolimits_{i=1}^{n}\lambda _i \varepsilon _i\,:\, \lambda _1\geqslant \cdots \geqslant \lambda _{n}\geqslant 0\right\}={\operatorname {Par}}_n. \end{array} \end{equation*}

For $i\in [2n]$ , set $\overline {i}=2n+1-i$ . The Weyl group $W$ of $G$ may be identified with a subgroup of the Weyl group $S_{2n}$ of ${\operatorname {SL}}(V)$ . More precisely,

\begin{equation*} W=\{w\in S_{2n}\,:\,w(\overline {i})=\overline {w(i)} \ \ \forall i\in [2n]\}. \end{equation*}

Observe that $T\cap U(2n,{\mathbb {C}})$ has real dimension $n$ and is a maximal torus of $U(2n)\cap {\operatorname {Sp}}(2n,{\mathbb {C}})$ . The bijection (17) implies that any matrix $M_1=\begin{pmatrix} A&B\\ -{}^t\bar B&-{}^TA \end{pmatrix}$ in ${\mathfrak {sp}}(2n,{\mathbb {C}})\cap {\mathfrak u}(2n,{\mathbb {C}})$ (see (19)) is diagonalizable with eigenvalues in $\sqrt {-1}{{\mathbb {R}}}$ . Moreover, with the eigenvalues in nonincreasing order, we get

\begin{equation*} \lambda (\sqrt {-1}M_1)\in \{(\lambda _1\geqslant \cdots \geqslant \lambda _n\geqslant -\lambda _n\geqslant \cdots \geqslant -\lambda _1)\,:\,\lambda _i\in {{\mathbb {R}}}\}. \end{equation*}

For $\lambda =(\lambda _1,\dots ,\lambda _n)\in {\operatorname {Par}}_n$ , set $\hat \lambda =(\lambda _1,\dots ,\lambda _n,-\lambda _n,\dots ,-\lambda _1)$ . Now, Theorems 1.1 with $m=n$ and Theorem 2.4 give an interpretation of ${\operatorname {NL-sat}}(n)$ in terms of eigenvalues:

Proposition 3.1. Let $\lambda ,\mu ,\nu \in {\operatorname {Par}}_n$ . Then $(\lambda ,\mu ,\nu )\in {\operatorname {NL-sat}}(n)$ if and only if there exist three matrices $M_1,M_2,M_3\in {\mathfrak {sp}}(2n,{\mathbb {C}})\cap {\mathfrak u}(2n,{\mathbb {C}})$ such that $M_1+M_2+M_3=0$ and

\begin{equation*} (\hat \lambda ,\hat \mu ,\hat \nu )=(\lambda (\sqrt {-1}M_1), \lambda (\sqrt {-1}M_2), \lambda (\sqrt {-1}M_3)). \end{equation*}

3.2 Isotropic Grassmannians and Schubert classes

Our reference for this subsection is [Reference RessayreRes12, § 5]. For $r=1,\dots ,n$ , the one-parameter subgroup $\varpi _{\alpha _r^\vee }$ is given by

\begin{equation*}\varpi _{\alpha _r^\vee } (t)={\operatorname {diag}}(t,\dots ,t,1,\dots ,1,t{^{-1}},\dots ,t{^{-1}}),\end{equation*}

where $t$ and $t{^{-1}}$ occur $r$ times.

A subspace $W\subseteq V$ is isotropic if for all $\vec v, \vec v^{\prime}\in W$ , $\omega (\vec v, \vec v^{\prime})=0$ . Given an $r$ -subset $I\subset [2n]$ , we set $F_I={\operatorname {Span}}(\vec e_i:i\in I)$ . Clearly, $F_I$ is isotropic if and only if $I\cap \bar I=\emptyset$ , where $\bar I=\{\bar i\,:\,i\in I\}$ . Now, $P^{\alpha _r}$ is the stabilizer of the isotropic subspace $F_{\{1,\dots ,r\}}$ . Thus, $G/P^{\alpha _r}={\operatorname {Gr}}_\omega (r,2n)$ is the Grassmannian of isotropic $r$ -dimensional vector subspaces of $V$ .

Let ${\mathcal S}(r,N)$ denote the set of subsets of $\{1,\dots ,N\}$ with $r$ elements. Set

\begin{equation*} \operatorname {Schub}({\operatorname {Gr}}_\omega (r,2n)):=\{I\in {\mathcal S}(r,2n)\,:\, I\cap \overline {I}=\emptyset \}. \end{equation*}

If $I=\{i_1\lt \cdots \lt i_r\}\in \operatorname {Schub}({\operatorname {Gr}}_\omega (r,2n))$ , let $i_{\overline {k}}:=\overline {i_k}$ for $k\in [r]$ , and $\{i_{r+1}\lt \cdots \lt i_{\overline {r+1}}\}=[2n]-(I\cup \overline {I})$ . Therefore, $w_I=(i_1,\dots ,i_{2n})\in S_{2n}$ is the element of $W^{P^{\alpha _r}}$ corresponding to $F_I$ ; that is, $F_I=w_I P^{\alpha _r}/P^{\alpha _r}$ .

Set

\begin{equation*} \operatorname {Schub}^{\prime}({\operatorname {Gr}}_\omega (r,2n)):= \left \{ \begin{array}{l@{\quad}l} (A,A^{\prime})\,:\, &A\in {\mathcal S}(a,n),\, A^{\prime}\in {\mathcal S}(a^{\prime},n)\mbox { for some $a$ and $a^{\prime}$}\\ &\mbox{s.t.}\;\;a+a^{\prime}=r\mbox { and } A\cap A^{\prime}=\emptyset \end{array} \right \}. \end{equation*}

This map is a bijection:

(20) \begin{align} \begin{array}{c@{\quad}c@{\quad}l} \operatorname {Schub}({\operatorname {Gr}}_\omega (r,2n))&\longrightarrow &\operatorname {Schub}^{\prime}({\operatorname {Gr}}_\omega (r,2n))\\ I&\longmapsto &(\bar I\cap [n],I\cap [n]). \end{array} \end{align}

Recall from the introduction the definition of $\tau (I)$ and hence $\tau (A)$ and $\tau (A^{\prime})$ . The relationship between these three partitions is depicted in Figure 1. In particular, note that $\tau (A^{\prime})\subseteq \tau (I)$ .

Given $I\in \operatorname {Schub}({\operatorname {Gr}}_\omega (r,2n))$ for some $1\leqslant r\leqslant n$ , set $I^2\in {\mathcal S}(r,2r)$ and $I^0\in {\mathcal S}(r,2n-r)$ to be the unique $r$ -element sets, such that

\begin{equation*}\tau (I^2) = \tau (I,I\cup \bar {I})\text { and }\tau (I^0) = \tau (I,[2n]- \bar {I}).\end{equation*}

Figure 1. $\tau (I)$ , $\tau (A)$ and $\tau (A^{\prime})\; (\tau (A^{\prime})\subseteq \tau (I))$ .

Example 3.2. Let $I = \{1,3,5\} \in \operatorname {Schub}({\operatorname {Gr}}_\omega (3,8))$ . Then $w_I = 13527468\in S_8$ ,

\begin{equation*} \tau (I^2) = \tau (\{1,3,5\},\{1,3,4,5,6,8\}) = 100 \text { and } \tau (I^0) = \tau (\{1,3,5\},\{1,2,3,5,7\}) = 110. \end{equation*}

Thus $I^2 = \{1,2,4\},I^0 = \{1,3,4\}$ . By Equation (20), $A = \bar {I}\cap [n] = \{4\}$ and $A^{\prime} = I\cap [n] = \{1,3\}$ .

Now, by [Reference Belkale and KumarBK10, Proposition 32],

(21) \begin{align} {\operatorname {codim}}(\overline {BF_I})= |\tau (I^0)^{\vee [(2n-2r)^r]}| + 1/2(|\tau (I^2)^{\vee [r^r]}| + |I\cap [n]|). \end{align}

Moreover,

(22) \begin{align} \dim ({\operatorname {Gr}}_\omega (r,2n))= r(2n-2r)+\frac {r(r+1)}{2}. \end{align}

Let $I\in \operatorname {Schub}({\operatorname {Gr}}_\omega (r,2n))$ and $A=\bar I\cap [n]$ , $A^{\prime}=I\cap [n]$ be the corresponding pair in $\operatorname {Schub}^{\prime}({\operatorname {Gr}}_\omega (r,2n))$ . Set

(23) \begin{align} \tau ^0(A,A^{\prime})=\tau (I^0),\quad \tau ^2(A,A^{\prime})=\tau (I^2). \end{align}

For later use, observe that

(24) \begin{align} \tau (I)=\tau (I^0)+\tau (I^2). \end{align}

While the above discussion defines $\tau ^0(A,A^{\prime}), \tau ^2(A,A^{\prime})$ through the bijection (20), we emphasize that these partitions from Theorem 1.2 can be defined explicitly.

Definition-Lemma 3.4. Set $a=|A|$ and $a^{\prime}=|A^{\prime}|$ . Write $A=\{\alpha _1\lt \cdots \lt \alpha _a\}$ and $A^{\prime}=\{\alpha ^{\prime}_1\lt \cdots \lt \alpha ^{\prime}_{a^{\prime}}\}$ . Then

\begin{equation*} \begin{array}{rl@{\quad}rl} \tau ^2(A,A^{\prime})_k&=a+|A^{\prime}\cap [\alpha _k,n]|&\forall k&=1,\dots ,a;\\[4pt] \tau ^2(A,A^{\prime})_{l+a}&=|A\cap [\alpha ^{\prime}_{a^{\prime}+1-l}]|&\forall l&=1,\dots ,a^{\prime};\\[4pt] \tau ^0(A,A^{\prime})_k&=n-a-a^{\prime}+|[\alpha _k,n] - (A\cup A^{\prime})| &\forall k&=1,\dots ,a;\\[4pt] \tau ^0(A,A^{\prime})_{l+a}&=|[\alpha ^{\prime}_{a^{\prime}+1-l}]- (A\cup A^{\prime})|&\forall l&=1,\dots ,a^{\prime}. \end{array} \end{equation*}

Proof. Write $I=A^{\prime}\cup \bar A=\{i_1\lt \cdots \lt i_r\}$ with $r=a+a^{\prime}$ . By definition,

\begin{equation*}\tau ^2(A,A^{\prime})_k:=\tau (I^2)_k=|\bar I\cap [i_{a+a^{\prime}+1-k}]|\quad \text {for $1\leqslant k\leqslant a+a^{\prime}$}.\end{equation*}

If $k\leqslant a$ , then $i_{a+a^{\prime}+1-k}\in \bar A\subset [n+1,2n]$ , $i_{a+a^{\prime}+1-k}=\overline {\alpha _k}$ and $\bar I\cap [i_{a+a^{\prime}+1-k}]=A\cup \overline {A^{\prime}\cap [\alpha _k,n]}$ . The first assertion follows.

If $k=a+l$ for some positive $l$ , then $i_{a+a^{\prime}+1-k}\in A^{\prime}\subseteq [n]$ , $i_{a+a^{\prime}+1-k}=\alpha ^{\prime}_{a^{\prime}+1-l}$ and $\bar I\cap [i_{a+a^{\prime}+1-k}]=A\cap [\alpha ^{\prime}_{a^{\prime}+1-l}]$ .

Similarly,

\begin{equation*} \tau ^0(A,A^{\prime})_k:=\tau (I^0)_k=|[i_{a+a^{\prime}+1-k}]\cap ([2n]-(I\cup \bar I)| \quad\text{for $1\leqslant k\leqslant a+a^{\prime}$.} \end{equation*}

If $k\leqslant a$ , then $[i_{a+a^{\prime}+1-k}]\cap ([2n]-(I\cup \bar I))= ([n]-(A\cup A^{\prime}))\cup \overline {[\alpha _k,n]-(A\cup A^{\prime})}$ (a disjoint union). This proves the third claim.

If $k=a+l$ with some positive $l$ , then $\alpha ^{\prime}_{a^{\prime}+1-l}=i_{a+a^{\prime}+1-k}\in [n]$ ; the last assertion follows.

3.3 The parabolic subgroup $P_0$

Fix $m\geqslant n$ . Let $P_0$ be the subgroup of ${\operatorname {Sp}}(2m,{\mathbb {C}})$ of matrices

(25) \begin{align} \left(\begin{array}{c@{\quad}c@{\quad}c} T_1&*&*\\ 0&A&*\\ 0&0&T_2 \end{array} \right), \end{align}

where $T_1$ and $T_2$ are $n\times n$ upper-triangular matrices and $A$ is a matrix in ${\operatorname {Sp}}(2m-2n,{\mathbb {C}})$ . $P_0$ is the standard parabolic subgroup of ${\operatorname {Sp}}(2m,{\mathbb {C}})$ corresponding to the simple roots $\{\alpha _{n+1},\dots ,\alpha _{m}\}$ . A character $\lambda =\sum _{i=1}^m\lambda _i\varepsilon _i\in X(T)$ extends to $P_0$ if and only if $\lambda _{n+1}=\cdots =\lambda _m=0$ . Thus the set of dominant characters of $X(P_0)$ identifies with ${\operatorname {Par}}_n$ . Hence,

(26) \begin{align} {{\mathfrak {sp}}\operatorname {-sat}}(m)\cap ({\operatorname {Par}}_n^{\mathbb {Q}})^3={\operatorname {GIT-sat}}({\operatorname {Sp}}(2m,{\mathbb {C}}),({\operatorname {Sp}}(2m,{\mathbb {C}})/P_0)^3). \end{align}

Let $\operatorname {Schub}^{P_0}({\operatorname {Gr}}_\omega (r,2m))$ be the set of $I\in \operatorname {Schub}({\operatorname {Gr}}_\omega (r,2m))$ such that the Schubert variety $\overline {BF_I}$ is $P_0$ -stable. Then $I\in \operatorname {Schub}^{P_0}({\operatorname {Gr}}_\omega (r,2m))$ if and only if $w_I\in W^{P^{\alpha _r}}$ and $s_{\alpha _i}w_I\leqslant w_I$ (cover in Bruhat order for $W$ ) for all $i\in [n+1,m]$ . Since the simple transposition $s_{\alpha _i}$ swaps $i$ and $\bar {i}$ with $i+i$ and $\bar {i+1}$ respectively if $i\lt m$ , and swaps $m$ with $\bar m$ if $i = m$ , we have

(27) \begin{align} I\in \operatorname {Schub}^{P_0}({\operatorname {Gr}}_\omega (r,2m)) \iff I\cap [n+1,2m-n]=[k,2m-n] \,\text{for some $k\geqslant m+1$.} \end{align}

7.1 Representations of ${\operatorname {GL}}(n,{\mathbb {C}})$

The irreducible rational representations $V(\lambda )$ of ${\operatorname {GL}}(n,{\mathbb {C}})$ are indexed by their highest weight

\begin{equation*}\lambda \in \Lambda _n^+=\{(\lambda _1\geqslant \cdots \geqslant \lambda _n)\,:\,\lambda _i\in {{\mathbb {Z}}}\}\supset {\operatorname {Par}}_n.\end{equation*}

One has tensor product multiplicities $c_{\lambda ,\mu }^{\nu }$ defined for any $\lambda ,\mu ,\nu \in \Lambda _n^+$ by

(37) \begin{align} V(\lambda )\otimes V(\mu )=\bigoplus _{\nu \in \Lambda _n^+} V(\nu )^{\oplus c_{\lambda ,\mu }^\nu }. \end{align}

When $\lambda ,\mu ,\nu \in {\operatorname {Par}}_n$ , $c_{\lambda ,\mu }^\nu$ is the Littlewood–Richardson coefficient (which is why we use the same notation).

The dual representation $V(\lambda )^*$ has highest weight

\begin{equation*}\lambda ^*=(-\lambda _n\geqslant \cdots \geqslant -\lambda _1)\in \Lambda _n^+.\end{equation*}

Moreover, for any $a\in {{\mathbb {Z}}}$ ,

(38) \begin{align} V(\lambda +a^n)=({\mathrm {det}})^a\otimes V(\lambda ). \end{align}

Consequently, for any $\lambda ,\mu ,\nu \in \Lambda ^+_n$ ,

(39) \begin{align} c_{\lambda ,\mu }^\nu =c_{\lambda +a^n,\mu +b^n}^{\nu +(a+b)^n}=c_{\lambda ^*+a^n,\mu ^*+b^n}^{\nu ^*+(a+b)^n}; \end{align}

this is [Reference Briand, Orellana and RosasBOR15, Theorem 4]. For $a$ and $b$ big enough, formula (39) implies that $c_{\lambda ,\mu }^\nu$ is a Littlewood–Richardson coefficient.

Let $\nu ^t$ denote the conjugate of $\nu$ . Since $c^{\nu }_{\lambda ,\mu } = c^{\nu ^t}_{\lambda ^t,\mu ^t}$ , by (39),

(40) \begin{align} c^{\nu }_{\lambda ,\mu } = c^{\nu ^t}_{\lambda ^t,\mu ^t} = c^{(\nu ^t)^{\vee [(n+m)^{a+b}]}}_{(\lambda ^t)^{\vee [n^{a+b}]},(\mu ^t)^{\vee [m^{a+b}]}} = c^{(\nu ^{\vee [(a+b)^{n+m}]})^t}_{(\lambda ^{\vee [(a+b)^n]})^t,(\mu ^{\vee [(a+b)^m]})^t} = c^{\nu ^{\vee [(a+b)^{n+m}]}}_{\lambda ^{\vee [(a+b)^n]},\mu ^{\vee [(a+b)^m]}}, \end{align}

for any $m\geqslant \ell (\mu )$ .

7.2 Six-fold Newell–Littlewood coefficients

Let $p$ , $q$ and $m$ be positive integers such that $p+q\leqslant m$ . Following R. Howe, Tan and Willenbring [Reference Howe, Tan and WillenbringHT05], to any $\lambda ^+\in {\operatorname {Par}}_p$ and $\lambda ^-\in {\operatorname {Par}}_q$ we associate the following element in $\Lambda _m^+$ :

\begin{equation*} [\lambda ^+,\lambda ^-]_m = (\lambda ^+_1,\lambda ^+_2,\ldots , \lambda ^+_p,\underbrace {0,\ldots ,0}_{m-p-q},-\lambda ^-_{q},\ldots ,-\lambda ^-_1).\end{equation*}

Let $V(\lambda )\boxtimes V(\mu )$ be the irreducible representation of ${\operatorname {GL}}(n,{\mathbb {C}})\times {\operatorname {GL}}(n,{\mathbb {C}})$ , where $\boxtimes$ refers to the external tensor product. View ${\operatorname {GL}}(n,{\mathbb {C}})\subset {\operatorname {GL}}(n,{\mathbb {C}})\times {\operatorname {GL}}(n,{\mathbb {C}})$ under the diagonal embedding. The associated branching coefficient is

\begin{equation*}[V(\nu ): V(\lambda )\boxtimes V(\mu )]:= \dim {\operatorname {Hom}}_{{\operatorname {GL}}(n,{\mathbb {C}})}(V(\nu ), V(\lambda )\boxtimes V(\mu )|_{{\operatorname {GL}}(n,{\mathbb {C}})}).\end{equation*}

Proposition 7.1 ([Reference Howe, Tan and WillenbringHT05, § 2.1.1], [Reference KingKin71]). Let $\lambda ^\pm$ , $\mu ^\pm$ and $\nu ^\pm$ be six partitions. Let $p,q,r$ and $s$ be four nonnegative integers such that

\begin{equation*} \begin{array}{l@{\quad}l@{\quad}l} \ell (\lambda ^+)\leqslant p,& \ell (\mu ^+)\leqslant r,&\ell (\nu ^+)\leqslant p+r,\\ \ell (\lambda ^-)\leqslant q,&\ell (\mu ^-)\leqslant s,&\ell (\nu ^-)\leqslant q+s.\\ \end{array} \end{equation*}

Let $m$ be a positive integer such that $m\geqslant p+q+r+s$ . Then

\begin{align*} N_{\mu ^+,\nu ^+,\lambda ^+,\mu ^-,\nu ^-,\lambda ^-}= & \ \dim {\operatorname {Hom}}_{{\operatorname {GL}}(n,{\mathbb {C}})}(V(\nu ), V(\lambda )\boxtimes V(\mu ))\\ = & \ c_{[\lambda ^+,\lambda ^-]_m, [\mu ^+,\mu ^-]_m}^{[\nu ^+,\nu ^-]_m}. \end{align*}

Conversely, any Littlewood–Richardson coefficient is a six-fold Newell–Littlewood coefficient. More precisely, $c_{\lambda ,\mu }^\nu =N_{\mu ,\nu ,\lambda ,\emptyset ,\emptyset ,\emptyset }$ , which corresponds to the case when $\lambda ^-=\mu ^-=\nu ^-=\emptyset$ in Proposition 7.1.

We now use Proposition 7.1 to rephrase Theorem 1.5. Fix $A$ , $A^{\prime}$ , $B$ , $B^{\prime}$ , $C$ and $C^{\prime}$ subseets of $[n]$ satisfying the two first conditions of Theorem 1.5. Set $p=|B^{\prime}|$ , $r=|C^{\prime}|$ , $q=n-p$ and $s=n-q$ . Observe that $|A|=p+r$ , $|B|\leqslant q$ , $|C|\leqslant s$ and $|A^{\prime}|\leqslant n-p-r\leqslant p+q$ . Finally, set $m=2n=p+q+r+s$ . Since $\ell (\tau (A))\leqslant |A|$ , Proposition 7.1 implies that

\begin{equation*} N_{\tau (A), \tau (C^{\prime}), \tau (B), \tau (A^{\prime}), \tau (C), \tau (B^{\prime})}=c_{[\tau (B^{\prime}),\tau (B)]_m, [\tau (C^{\prime}),\tau (C)]_m}^{[\tau (A),\tau (A^{\prime})]_m} \end{equation*}

is a Littlewood–Richardson coefficient for ${\operatorname {GL}}_{2n}({\mathbb {C}})$ . In particular, in Theorem 1.5, condition $(3)$ can be replaced by:

1. (3^′) $c_{[\tau (B^{\prime}),\tau (B)]_m, [\tau (C^{\prime}),\tau (C)]_m}^{[\tau (A),\tau (A^{\prime})]_m}\gt 0.$

We now observe that Proposition 7.1 and Knutson–Tao saturation [Reference Knutson and TaoKT99] imply the saturation result for the six-fold Newell–Littlewood-coefficients from the introduction (Proposition 1.6).

8.1 Extended Horn inequalities

We recall the following notion from [Reference Gao, Orelowitz and YongGOY20b].

Definition 8.1. An extended Horn inequality on ${\operatorname {Par}}_n^3$ is

(41) \begin{align} 0\leqslant |\lambda _A|-|\lambda _{A^{\prime}}|+|\mu _B|-|\mu _{B^{\prime}}|+|\nu _C|-|\nu _{C^{\prime}}| \end{align}

where $A,A^{\prime},B,B^{\prime},C,C^{\prime} \subseteq [n]$ satisfy the following assertions:

1. (I) $A \cap A^{\prime}= B \cap B^{\prime} = C \cap C^{\prime} = \emptyset$ ;

2. (II) $|A| = |B^{\prime}|+|C^{\prime}|,|B| =|A^{\prime}|+|C^{\prime}|, |C| = |A^{\prime}|+|B^{\prime}|$ ;

3. (III) there exists $A_1,A_2,B_1,B_2,C_1,C_2\subseteq [n]$ such that

   1. (i) $|A_1|=|A_2|=|A^{\prime}|, |B_1|=|B_2|=|B^{\prime}|, |C_1|=|C_2|=|C^{\prime}|$ ,

   2. (ii) $c^{\tau (A^{\prime})}_{\tau (A_1),\tau (A_2)}, c^{\tau (B^{\prime})}_{\tau (B_1),\tau (B_2)}, c^{\tau (C^{\prime})}_{\tau (C_1),\tau (C_2)}\gt 0$ ,

   3. (iii) $c^{\tau (A)}_{\tau (B_1),\tau (C_2)}, c^{\tau (B)}_{\tau (C_1),\tau (A_2)}, c^{\tau (C)}_{\tau (A_1),\tau (B_2)}\gt 0.$

Definition 8.2. The extended Horn cone is

(42) \begin{align} {\operatorname {EH}}(n):=\{(\lambda ,\mu ,\nu )\in ({\operatorname {Par}}_n^{{\mathbb {Q}}})^3\,:\, \mbox {inequalities (41) are satisfied}\}. \end{align}

Let

\begin{equation*}\overline {{\operatorname {EH}}}(n)={\operatorname {EH}}(n)\cap \{(\lambda ,\mu ,\nu )\in ({\operatorname {Par}}_n)^3: |\lambda |+|\mu |+|\nu |\mbox { is even}\}.\end{equation*}

We will prove a weakened version of Conjecture 8.3.

Consequently, we are able to answer an issue raised in [Reference Gao, Orelowitz and YongGOY20b, § 1].

Corollary 8.5 is analogous to the situation in Zelevinsky’s [Reference ZelevinskyZel99], before [Reference Knutson and TaoKT99].

The following shows that Theorem 1.5 is equivalent to Theorem 8.4.

Proof. Definition 8.1 implies

\begin{equation*}c_{\tau (B_1),\tau (C_2)}^{\tau (A)} c_{\tau (C_2), \tau (C_1)}^{\tau (C^{\prime})} c_{\tau (C_1), \tau (A_2)}^{\tau (B)} c_{\tau (A_2), \tau (A_1)}^{\tau (A^{\prime})} c_{\tau (A_1), \tau (B_2)}^{\tau (C)} c_{\tau (B_2), \tau (B_1)}^{\tau (B^{\prime})}\gt 0.\end{equation*}

Since $\tau (A_1),\,\tau (A_2)\dots$ have length at most $n$ , this implies $N_{\tau (A),\tau (C^{\prime}),\tau (B),\tau (A^{\prime}),\tau (C),\tau (B^{\prime})}\neq 0$ , and thus $(\tau (A), \tau (C^{\prime}), \tau (B), \tau (A^{\prime}), \tau (C), \tau (B^{\prime}))\in \operatorname {NL}^6\!\operatorname {-sat}(r)$ .

Conversely, if $(\tau (A), \tau (C^{\prime}), \tau (B), \tau (A^{\prime}), \tau (C), \tau (B^{\prime}))\in \operatorname {NL}^6\!\operatorname {-sat}(r)$ , by Proposition 1.6, $N_{\tau (A),\tau (C^{\prime}),\tau (B),\tau (A^{\prime}),\tau (C),\tau (B^{\prime})}\neq 0$ . Therefore, there exists $\alpha _1,\alpha _2,\ldots ,\alpha _6\in {\operatorname {Par}}_n$ such that

\begin{equation*}c_{\alpha _1,\alpha _2}^{\tau (A)}c_{\alpha _2,\alpha _3}^{\tau (C^{\prime})}c_{\alpha _3,\alpha _4}^{\tau (B)} c_{\alpha _4,\alpha _5}^{\tau (A^{\prime})}c_{\alpha _5,\alpha _6}^{\tau (C)}c_{\alpha _6,\alpha _1}^{\tau (B^{\prime})}\gt 0.\end{equation*}

Set $a=|A|$ . Then, the Young diagram of $\tau (A)$ is contained in the rectangle $a\times (n-a)$ . But the nonvanishing of $c_{\alpha _1,\alpha _2}^{\tau (A)}$ implies that $\alpha _1\subset \tau (A)$ . Hence, there exists $B_1\subseteq [n]$ such that $\tau (B_1)=\alpha _1$ . Similarly, we can pick $C_2,C_1,A_2,A_1,B_2\subseteq [n]$ such that $\tau (C_2)=\alpha _2, \tau (C_1)=\alpha _3$ , etc., that satisfy Definition 8.1.

8.2 Proof of Theorem 1.5

( $\Rightarrow$ ) By Lemma 8.6, ${{\operatorname {EH}}}(n)$ is the cone defined by the inequalities in Theorem 1.5. Now, ${\operatorname {NL-sat}}(n)\subseteq {{\operatorname {EH}}}(n)$ is immediate from [Reference Gao, Orelowitz and YongGOY20b, Theorem 1].

( $\Leftarrow$ ) Fix an inequality (4) associated to $(A,A^{\prime},B,B^{\prime},C,C^{\prime})$ appearing in Theorem 1.2. We now show the even stronger statement that

(43) \begin{align} N_{\tau (A),\tau (C^{\prime}),\tau (B),\tau (A^{\prime}),\tau (C),\tau (B^{\prime})}\neq 0. \end{align}

This would imply that the inequality appears in Theorem 1.5, completing the proof.

Set $a=|A|, a^{\prime}=|A^{\prime}|, b=|B|, b^{\prime}=|B^{\prime}|, c=|C|, c^{\prime}=|C^{\prime}|$ and $r=|A|+|A^{\prime}|$ . Let $I\in \operatorname {Schub}({\operatorname {Gr}}_\omega (r,2n))$ be associated to $(A,A^{\prime})\in \operatorname {Schub}^{\prime}({\operatorname {Gr}}_\omega (r,2n))$ , under (20). Similarly define $J$ and $K$ . By (24) and condition (3) in Theorem 1.2, the semigroup property of nonzero Littlewood–Richardson coefficients implies

(44) \begin{align} c^{\tau (K)}_{\tau (I)^{\vee [(2n-r)^r]},\tau (J)^{\vee [(2n-r)^r]}}\neq 0. \end{align}

Fix a nonnegative integer $k$ . Note that $c=a^{\prime}+b^{\prime}$ by condition (2) in Theorem 1.2, and $c_{(a^{\prime})^k,(b^{\prime})^k}^{(c^k)}=1$ . Using the semigroup property once more, one gets

(45) \begin{align} c^{\tau (K)+(c^k)}_{\tau (I)^{\vee [(2n-r)^r]+((a^{\prime})^k)},\tau (J)^{\vee [(2n-r)^r]}+((b^{\prime})^k)}\gt 0. \end{align}

Observing Figure 1,

(46) \begin{align} (\tau (I)^\vee )^t = [\tau (A)^t, \tau (A^{\prime})^t]_{2n-r}+((a^{\prime})^{2n-r}). \end{align}

Similarly,

(47) \begin{align} (\tau (I)^\vee +(a^{\prime})^k)^t = [\tau (A)^t, \tau (A^{\prime})^t]_{m}+((a^{\prime})^{m}) \end{align}

and

(48) \begin{align} (\tau (K)+(c^k))^t = [\tau (C^{\prime})^t, \tau (C)^t]_{m}+(c^{m}), \end{align}

where $m = 2n-r+k$ . Now, by (39), (40), (47) and (48), condition (45) implies

(49) \begin{align} c^{[\tau (C^{\prime})^t, \tau (C)^t]_{m}}_{[\tau (A)^t, \tau (A^{\prime})^t]_m, [\tau (B)^t, \tau (B^{\prime})^t]_m}\gt 0. \end{align}

On the other hand, for $k$ (and hence $m$ ) that is big enough, we can apply Proposition 7.1 to get

(50) \begin{align} N_{\tau (A)^t,\tau (C^{\prime})^t,\tau (B)^t,\tau (A^{\prime})^t,\tau (C)^t,\tau (B^{\prime})^t} = c^{[\tau (C^{\prime})^t,\tau (C)^t]_{m}}_{[\tau (A)^t,\tau (A^{\prime})^t]_{m}, [\tau (B)^t,\tau (B^{\prime})^t]_{m}}. \end{align}

Since the six-fold Newell–Littlewood coefficient are invariant by conjugating the partitions, (49) and (50) imply (43) as expected.