In this article, we study the Johnson homomorphisms of basis-conjugating automorphism groups of free groups. We construct obstructions for the surjectivity of the Johnson homomorphisms. By using it, we determine their cokernels of degree up to four and give further observations for degree greater than four. As applications, we give the affirmative answer to the Andreadakis problem for the basis-conjugating automorphism groups of free groups at degree four. Moreover, we calculate twisted first cohomology groups of the braid-permutation automorphism groups of free groups.

A spline is an assignment of polynomials to the vertices of a graph whose edges are labeled by ideals, where the difference of two polynomials labeling adjacent vertices must belong to the corresponding ideal. The set of splines forms a ring. We consider spline rings where the underlying graph is the Cayley graph of a symmetric group generated by a collection of transpositions. These rings generalize the GKM construction for equivariant cohomology rings of flag, regular semisimple Hessenberg and permutohedral varieties. These cohomology rings carry two actions of the symmetric group $S_n$ whose graded characters are both of general interest in algebraic combinatorics. In this paper, we generalize the graded $S_n$-representations from the cohomologies of the above varieties to splines on Cayley graphs of $S_n$ and then (1) give explicit module and ring generators for whenever the $S_n$-generating set is minimal, (2) give a combinatorial characterization of when graded pieces of one $S_n$-representation is trivial, and (3) compute the first degree piece of both graded characters for all generating sets.

If all of the entries of a large $S_n$ character table are covered up and you are allowed to uncover one entry at a time, then how can you quickly identify all of the indexing characters and conjugacy classes? We present a fast algorithmic solution that works even when n is so large that almost none of the entries of the character table can be computed. The fraction of the character table that needs to be uncovered is $O( n^2 \exp({-}2\pi\sqrt{n/6}))$, and for many of these entries we are only interested in whether the entry is zero.

In their celebrated paper [CLR10], Caputo, Liggett and Richthammer proved Aldous’ conjecture and showed that for an arbitrary finite graph, the spectral gap of the interchange process is equal to the spectral gap of the underlying random walk. A crucial ingredient in the proof was the Octopus Inequality — a certain inequality of operators in the group ring $\mathbb{R}\left[{\mathrm{Sym}}_{n}\right]$ of the symmetric group. Here we generalise the Octopus Inequality and apply it to generalising the Caputo–Liggett–Richthammer Theorem to certain hypergraphs, proving some cases of a conjecture of Caputo.

Let ${\mathcal {A}}$ be a unital ${\mathbb {F}}$-algebra and let ${\mathcal {S}}$ be a generating set of ${\mathcal {A}}$. The length of ${\mathcal {S}}$ is the smallest number k such that ${\mathcal {A}}$ equals the ${\mathbb {F}}$-linear span of all products of length at most k of elements from ${\mathcal {S}}$. The length of ${\mathcal {A}}$, denoted by $l({\mathcal {A}})$, is defined to be the maximal length of its generating sets. We show that $l({\mathcal {A}})$ does not exceed the maximum of $\dim {\mathcal {A}} / 2$ and $m({\mathcal {A}})-1$, where $m({\mathcal {A}})$ is the largest degree of the minimal polynomial among all elements of the algebra ${\mathcal {A}}$. As an application, we show that for arbitrary odd n, the length of the group algebra of the dihedral group of order $2n$ equals n.

We settle the question of where exactly do the reduced Kronecker coefficients lie on the spectrum between the Littlewood-Richardson and Kronecker coefficients by showing that every Kronecker coefficient of the symmetric group is equal to a reduced Kronecker coefficient by an explicit construction. This implies the equivalence of an open problem by Stanley from 2000 and an open problem by Kirillov from 2004 about combinatorial interpretations of these two families of coefficients. Moreover, as a corollary, we deduce that deciding the positivity of reduced Kronecker coefficients is ${\textsf {NP}}$-hard, and computing them is ${{{\textsf {#P}}}}$-hard under parsimonious many-one reductions. Our proof also provides an explicit isomorphism of the corresponding highest weight vector spaces.

We revisit Haiman’s conjecture on the relations between characters of Kazdhan–Lusztig basis elements of the Hecke algebra over $S_n$. The conjecture asserts that, for purposes of character evaluation, any Kazhdan–Lusztig basis element is reducible to a sum of the simplest possible ones (those associated to so-called codominant permutations). When the basis element is associated to a smooth permutation, we are able to give a geometric proof of this conjecture. On the other hand, if the permutation is singular, we provide a counterexample.

We study covering numbers of subsets of the symmetric group $S_n$ that exhibit closure under conjugation, known as normal sets. We show that for any $\epsilon>0$, there exists $n_0$ such that if $n>n_0$ and A is a normal subset of the symmetric group $S_n$ of density $\ge e^{-n^{2/5 - \epsilon }}$, then $A^2 \supseteq A_n$. This improves upon a seminal result of Larsen and Shalev (Inventiones Math., 2008), with our $2/5$ in the double exponent replacing their $1/4$.

Our proof strategy combines two types of techniques. The first is ‘traditional’ techniques rooted in character bounds and asymptotics for the Witten zeta function, drawing from the foundational works of Liebeck–Shalev, Larsen–Shalev, and more recently, Larsen–Tiep. The second is a sharp hypercontractivity theorem in the symmetric group, which was recently obtained by Keevash and Lifshitz. This synthesis of algebraic and analytic methodologies not only allows us to attain our improved bounds but also provides new insights into the behavior of general independent sets in normal Cayley graphs over symmetric groups.

We study the zero-sharing behavior among irreducible characters of a finite group. For symmetric groups $\mathsf {S}_n$, it is proved that, with one exception, any two irreducible characters have at least one common zero. To further explore this phenomenon, we introduce the common-zero graph of a finite group G, with nonlinear irreducible characters of G as vertices, and edges connecting characters that vanish on some common group element. We show that for solvable and simple groups, the number of connected components of this graph is bounded above by three. Lastly, the result for $\mathsf {S}_n$ is applied to prove the nonequivalence of the metrics on permutations induced from faithful irreducible characters of the group.

We introduce a new class of permutations, called web permutations. Using these permutations, we provide a combinatorial interpretation for entries of the transition matrix between the Specht and $\operatorname {SL}_2$-web bases of the irreducible $ \mathfrak {S}_{2n} $-representation indexed by $ (n,n) $, which answers Rhoades’s question. Furthermore, we study enumerative properties of these permutations.

Let K be an infinite field of characteristic $p>0$ and let $\lambda, \mu$ be partitions, where $\mu$ has two parts. We find sufficient arithmetic conditions on $p, \lambda, \mu$ for the existence of a nonzero homomorphism $\Delta(\lambda) \to \Delta (\mu)$ of Weyl modules for the general linear group $GL_n(K)$. Also, for each p we find sufficient conditions so that the corresponding homomorphism spaces have dimension at least 2.

Let $\mathcal {C}_n =\left [\chi _{\lambda }(\mu )\right ]_{\lambda , \mu }$ be the character table for $S_n,$ where the indices $\lambda $ and $\mu $ run over the $p(n)$ many integer partitions of $n.$ In this note, we study $Z_{\ell }(n),$ the number of zero entries $\chi _{\lambda }(\mu )$ in $\mathcal {C}_n,$ where $\lambda $ is an $\ell $-core partition of $n.$ For every prime $\ell \geq 5,$ we prove an asymptotic formula of the form

$$ \begin{align*}Z_{\ell}(n)\sim \alpha_{\ell}\cdot \sigma_{\ell}(n+\delta_{\ell})p(n)\gg_{\ell} n^{\frac{\ell-5}{2}}e^{\pi\sqrt{2n/3}}, \end{align*} $$

$\sigma _{\ell }(n)$

$\delta _{\ell }:=(\ell ^2-1)/24,$

$1/\alpha _{\ell }>0$

$L\left (\left ( \frac {\cdot }{\ell } \right ),\frac {\ell -1}{2}\right ).$

$\ell $

$n>\ell ^6/24,$

$\chi _{\lambda }(\mu )=0$

$\lambda $

$\mu $

$\ell $

$Z^*_{\ell }(n)$

$\ell $

$\ell \geq 5$

$$ \begin{align*}Z^*_{\ell}(n)\sim \alpha_{\ell}^2 \cdot \sigma_{\ell}( n+\delta_{\ell})^2 \gg_{\ell} n^{\ell-3}. \end{align*} $$

We show that the Specht ideal of a two-rowed partition is perfect over an arbitrary field, provided that the characteristic is either zero or bounded below by the size of the second row of the partition, and we show this lower bound is tight. We also establish perfection and other properties of certain variants of Specht ideals, and find a surprising connection to the weak Lefschetz property. Our results, in particular, give a self-contained proof of Cohen–Macaulayness of certain h-equals sets, a result previously obtained by Etingof–Gorsky–Losev over the complex numbers using rational Cherednik algebras.

Immanants are functions on square matrices generalizing the determinant and permanent. Kazhdan–Lusztig immanants, which are indexed by permutations, involve $q=1$ specializations of Type A Kazhdan–Lusztig polynomials, and were defined by Rhoades and Skandera (2006, Journal of Algebra 304, 793–811). Using results of Haiman (1993, Journal of the American Mathematical Society 6, 569–595) and Stembridge (1991, Bulletin of the London Mathematical Society 23, 422–428), Rhoades and Skandera showed that Kazhdan–Lusztig immanants are nonnegative on matrices whose minors are nonnegative. We investigate which Kazhdan–Lusztig immanants are positive on k-positive matrices (matrices whose minors of size $k \times k$ and smaller are positive). The Kazhdan–Lusztig immanant indexed by v is positive on k-positive matrices if v avoids 1324 and 2143 and for all noninversions $i< j$ of v, either $j-i \leq k$ or $v_j-v_i \leq k$. Our main tool is Lewis Carroll’s identity.

It is well known that the pair $(\mathcal {S}_n,\mathcal {S}_{n-1})$ is a Gelfand pair where $\mathcal {S}_n$ is the symmetric group on n elements. In this paper, we prove that if G is a finite group then $(G\wr \mathcal {S}_n, G\wr \mathcal {S}_{n-1}),$ where $G\wr \mathcal {S}_n$ is the wreath product of G by $\mathcal {S}_n,$ is a Gelfand pair if and only if G is abelian.

The symmetric group on a set acts transitively on the set of its subsets of a fixed size. We define homomorphisms between the corresponding permutation modules, defined over a field of characteristic two, which generalize the boundary maps from simplicial homology. The main results determine when these chain complexes are exact and when they are split exact. As a corollary we obtain a new explicit construction of the basic spin modules for the symmetric group.

We consider ideals in a polynomial ring that are generated by regular sequences of homogeneous polynomials and are stable under the action of the symmetric group permuting the variables. In previous work, we determined the possible isomorphism types for these ideals. Following up on that work, we now analyze the possible degrees of the elements in such regular sequences. For each case of our classification, we provide some criteria guaranteeing the existence of regular sequences in certain degrees.

Jean-Louis Loday introduced a class of symmetric operads generated by one bilinear operation subject to one relation making each left-normed product of three elements equal to a linear combination of right-normed products: $({{a}_{1}}{{a}_{2}}){{a}_{3}}\,=\,\sum{{{_{\sigma }}_{\in {{s}_{3}}}}\,}{{x}_{\sigma }}\,{{a}_{\sigma \,(1)}}({{a}_{\sigma (2)\,}}{{a}_{\sigma (3)}})$ . Such an operad is called a parametrized one-relation operad. For a particular choice of parameters $\{{{x}_{\sigma }}\}$ , this operad is said to be regular if each of its components is the regular representation of the symmetric group; equivalently, the corresponding free algebra on a vector space $V$ is, as a graded vector space, isomorphic to the tensor algebra of $V$ . We classify, over an algebraically closed field of characteristic zero, all regular parametrized one-relation operads. In fact, we prove that each such operad is isomorphic to one of the following five operads: the left-nilpotent operad defined by the relation $(({{a}_{1}}{{a}_{2}}){{a}_{3}})\,=\,0$ , the associative operad, the Leibniz operad, the dual Leibniz (Zinbiel) operad, and the Poisson operad. Our computational methods combine linear algebra over polynomial rings, representation theory of the symmetric group, and Gröbner bases for determinantal ideals and their radicals.

The chromatic functor of a simple graph is a functorization of the chromatic polynomial. M. Yoshinaga showed that two finitely graphs have isomorphic chromatic functors if and only if they have the same chromatic polynomial. The key ingredient in the proof is the use of stable partitions of graphs. The latter is shown to be closely related to chromatic functors. In this note, we further investigate some interesting properties of chromatic functors associated with simple graphs using stable partitions. Our first result is the determination of the group of natural automorphisms of the chromatic functor, which is, in general, a larger group than the automorphism group of the graph. The second result is that the composition of the chromatic functor associated with a finitely graph restricted to the category FI of finitely sets and injections with the free functor into the category of complex vector spaces yields a consistent sequence of representations of symmetric groups that is representation stable in the sense of Church–Farb.