Dimensions of the irreducible representations of the symmetric and alternating group

Noncommutative irreducible characters of the symmetric group and noncommutative Schur functions

In the Hopf algebra of symmetric functions, Sym, the basis of Schur functions is distinguished since every Schur function is isomorphic to an irreducible character of a symmetric group under the Frobenius characteristic map. In this note we show that in the Hopf algebra of noncommutative symmetric functions, Nsym, of which Sym is a quotient, the recently discovered basis of noncommutative Schur functions exhibits that every noncommutative Schur function is isomorphic to a noncommutative irreducible character of a symmetric group when working in noncommutative character theory. We simultaneousl…

非对角对称交替符号矩阵

Off-diagonally symmetric alternating sign matrices

本研究给出了奇数阶非对角对称交替符号矩阵（OSASM）计数的乘积公式，还代数证明了偶数阶OSASM的对称性质，解决了相关的全部三个计数猜想。

The intrinsic hyperplane arrangement in an arbitrary irreducible representation of the symmetric group

For every irreducible complex representation~$π_λ$ of the symmetric group~$§_n$, we construct, in a canonical way, a so-called intrinsic hyperplane arrangement~$\A_λ$ in the space of~$π_λ$. This arrangement is a direct generalization of the classical braid arrangement (which is the special case of our construction corresponding to the natural representation of~$§_n$), has a natural description in terms of invariant subspaces of Young subgroups, and enjoys a number of remarkable properties.

Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups

The partition algebra $\mathsf{P}_k(n)$ and the symmetric group $\mathsf{S}_n$ are in Schur-Weyl duality on the $k$-fold tensor power $\mathsf{M}_n^{\otimes k}$ of the permutation module $\mathsf{M}_n$ of $\mathsf{S}_n$, so there is a surjection $\mathsf{P}_k(n) \to \mathsf{Z}_k(n) := \mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k}),$ which is an isomorphism when $n \ge 2k$. We prove a dimension formula for the irreducible modules of the centralizer algebra $\mathsf{Z}_k(n)$ in terms of Stirling numbers of the second kind. Via Schur-Weyl duality, these dimensions equal the multiplicities …

Sign conjugacy classes of the alternating groups

A conjugacy class $C$ of a finite group $G$ is a sign conjugacy class if every irreducible character of $G$ takes value 0, 1 or -1 on $C$. In this paper we classify the sign conjugacy classes of alternating groups.

Half-turn symmetric alternating sign matrices and Tokuyama type factorisation for orthogonal group characters

Half turn symmetric alternating sign matrices (HTSASMs) are special variations of the well-known alternating sign matrices which have a long and fascinating history. HTSASMs are interesting combinatorial objects in their own right and have been the focus of recent study. Here we explore counting weighted HTSASMs with respect to a number of statistics to derive an orthogonal group version of Tokuyama's factorisation formula, which involves a deformation and expansion of Weyl's denominator formula multiplied by a general linear group character. Deformations of Weyl's original denominator formula…

Vertically symmetric alternating sign matrices and a multivariate Laurent polynomial identity

In 2007, the first author gave an alternative proof of the refined alternating sign matrix theorem by introducing a linear equation system that determines the refined ASM numbers uniquely. Computer experiments suggest that the numbers appearing in a conjecture concerning the number of vertically symmetric alternating sign matrices with respect to the position of the first 1 in the second row of the matrix establish the solution of a linear equation system similar to the one for the ordinary refined ASM numbers. In this paper we show how our attempt to prove this fact naturally leads to a more …

交错群与对称群的顶点本原 s-弧传递有向图中 s 的界

Bounding $s$ for vertex-primitive $s$-arc-transitive digraphs of alternating and symmetric groups

本文针对 Praeger 提出的有限顶点本原 s-弧传递有向图的上界问题，在群为交错群或对称群的情形下证明了 s ≤ 2。

Symmetry classes of alternating-sign matrices under one roof

In a previous article [math.CO/9712207], we derived the alternating-sign matrix (ASM) theorem from the Izergin-Korepin determinant for a partition function for square ice with domain wall boundary. Here we show that the same argument enumerates three other symmetry classes of alternating-sign matrices: VSASMs (vertically symmetric ASMs), even HTSASMs (half-turn-symmetric ASMs), and even QTSASMs (quarter-turn-symmetric ASMs). The VSASM enumeration was conjectured by Mills; the others by Robbins [math.CO/0008045]. We introduce several new types of ASMs: UASMs (ASMs with a U-turn side), UUASMs (t…

关于对称群不可约表示的最大维数

On the maximal dimension of an irreducible representation of the symmetric group

本文证明了对称群 S_N 不可约表示的最大维数 d_N 满足渐近公式 d_N = sqrt(N!) e^{-(𝔡+o(1))√N}，其中 𝔡>0 为常数，解决了 Vershik–Kerov 在 1985 年提出的问题。

Subalgebras of Solomon's descent algebra based on alternating runs

The number of alternating runs is a natural permutation statistic. We show it can be used to define some commutative subalgebras of the symmetric group algebra, and more precisely of the descent algebra. The Eulerian peak algebras naturally appear as subalgebras of our run algebras. We also calculate the orthogonal idempotents for run algebras in terms of noncommutative symmetric functions.

Enumeration of Symmetry Classes of Alternating Sign Matrices and Characters of Classical Groups

An alternating sign matrix is a square matrix with entries 1, 0 and -1 such that the sum of the entries in each row and each column is equal to 1 and the nonzero entries alternate in sign along each row and each column. To some of the symmetry classes of alternating sign matrices and their variations, G. Kuperberg associate square ice models with appropriate boundary conditions, and give determinanat and Pfaffian formulae for the partition functions. In this paper, we utilize several determinant and Pfaffian identities to evaluate Kuperberg's determinants and Pfaffians, and express the round p…

On the base size of the symmetric and the alternating group acting on partitions

Given three positive integers $n,a,b$ with $n=ab$, we determine the base size of the symmetric group and of the alternating group of degree $n$ in their action on the set of partitions into $b$ parts having cardinality $a$.

Specht modules decompose as alternating sums of restrictions of Schur modules

Schur modules give the irreducible polynomial representations of the general linear group $\mathrm{GL}_t$. Viewing the symmetric group $\mathfrak{S}_t$ as a subgroup of $\mathrm{GL}_t$, we may restrict Schur modules to $\mathfrak{S}_t$ and decompose the result into a direct sum of Specht modules, the irreducible representations of $\mathfrak{S}_t$. We give an equivariant Möbius inversion formula that we use to invert this expansion in the representation ring for $\mathfrak{S}_t$ for $t$ large. In addition to explicit formulas in terms of plethysms, we show the coefficients that appear alternat…

On the Doubly Refined Enumeration of Alternating Sign Matrices and Totally Symmetric Self-Complementary Plane Partitions

We prove the equality of doubly refined enumerations of Alternating Sign Matrices and of Totally Symmetric Self-Complementary Plane Partitions using integral formulae originating from certain solutions of the quantum Knizhnik--Zamolodchikov equation.

Basic sets for the double covering groups of the symmetric and alternating groups in odd characteristic

In this paper, following the methods of http://arxiv.org/abs/1305.7449v2, we show that the double covering groups of the symmetric and alternating groups have p-basic sets for any odd prime p.

Unitriangular basic sets for blocks of the symmetric and alternating groups of small weights

We study the existence of unitriangular basic sets for the symmetric group which behave nicely with respect to the Mullineux involution. Such sets give a natural labelling for the modular irreducible representations. We show that, for any odd prime p, the p-blocks of the symmetric group with weight 2 have stable unitriangular basic sets which we describe by studying the combinatorics of partitions in these blocks.

Note on parity and the irreducible characters of the symmetric group

The object of this short note is to prove a theorem and present a conjecture for the number of even entries in the character table of the symmetric group.

Flag-transitive point-primitive non-symmetric 2-(v,k,2) designs with alternating socle

This paper studies flag-transitive point-primitive non-symmetric $2$-($v,k,2$) designs. We prove that if $\mathcal{D}$ is a non-trivial non-symmetric $2$-$(v,k,2)$ design admitting a flag-transitive point-primitive automorphism group $G$ with $Soc(G)=A_{n}$ for $n\geq5$, then $\mathcal{D}$ is a $2$-$(6,3,2)$ or $2$-$(10,4,2)$ design.

Irreducible Symmetric Group Characters of Rectangular Shape

We give a new formula for the values of an irreducible character of the symmetric group S_n indexed by a partition of rectangular shape. Some observations and a conjecture are given concerning a generalization to arbitrary shapes.

Refined Enumeration of Symmetry Classes of Alternating Sign Matrices

We prove refined enumeration results on several symmetry classes as well as related classes of alternating sign matrices with respect to classical boundary statistics, using the six-vertex model of statistical physics. More precisely, we study vertically symmetric, vertically and horizontally symmetric, vertically and horizontally perverse, off-diagonally and off-antidiagonally symmetric, vertically and off-diagonally symmetric, quarter turn symmetric as well as quasi quarter turn symmetric alternating sign matrices. Our results prove conjectures of Fischer, Duchon and Robbins.

Vertex-primitive s-arc-transitive digraphs of alternating and symmetric groups

A fascinating problem on digraphs is the existence problem of the finiteupper bound on s for all vertex-primitive s-arc-transitive digraphs except directed cycles (which is known to be reduced to the almost simple groups case). In this paper,we prove that s 2 for all G-vertex-primitive s-arc-transitive digraphs with G an (insoluble) alternating or symmetric group, which makes an important progress towards a solution of the problem. The proofs involves some methods that may be used to investigate other almost simple groups cases.

Enumerations of half-turn symmetric alternating-sign matrices of odd order

It was shown by Kuperberg that the partition function of the square-ice model related to half-turn symmetric alternating-sign matrices of even order is the product of two similar factors. We propose a square-ice model whose states are in bijection with half-turn symmetric alternating-sign matrices of odd order. The partition function of the model is expressed via the above mentioned factors. The contributions to the partition function of the states corresponding to the alternating-sign matrices having 1 or -1 as the central entry are found and the related enumerations are obtained.

Constructing all irreducible Specht modules in a block of the symmetric group

For any prime p, we construct, and simultaneously count, all of the complex Specht modules in a given p-block of the symmetric group which remain irreducible when reduced modulo p. We call the Specht modules with this property p-irreducible modules. Recently Fayers has proven a conjecture of James and Mathas that provides a characterization of the partitions that correspond to the p-irreducible modules. In this paper we present a method for decomposing the partitions corresponding to p-irreducible modules, and we use this decomposition to construct and count all of the partitions corresponding…

The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions

We consider the groundstate wavefunction of the quantum symmetric antiferromagnetic XXZ chain with open and twisted boundary conditions at $Δ=-{1/2}$, along with the groundstate wavefunction of the corresponding O($n$) loop model at $n=1$. Based on exact results for finite-size systems, sums involving the wavefunction components, and in some cases the largest component itself, are conjectured to be directly related to the total number of alternating sign matrices and plane partitions in certain symmetry classes.

Divisibility graph for symmetric and alternating groups

Let $X$ be a non-empty set of positive integers and $X^*=X\setminus \{1\}$. The divisibility graph $D(X)$ has $X^*$ as the vertex set and there is an edge connecting $a$ and $b$ with $a, b\in X^*$ whenever $a$ divides $b$ or $b$ divides $a$. Let $X=cs~{G}$ be the set of conjugacy class sizes of a group $G$. In this case, we denote $D(cs~{G})$ by $D(G)$. In this paper we will find the number of connected components of $D(G)$ where $G$ is the symmetric group $S_n$ or is the alternating group $A_n$.

对称群和交错群双覆盖中每个元素的最小多项式

Minimal Polynomials in Spin Representations of Symmetric and Alternating Groups

本文研究了对称群和交错群双覆盖中元素的最小多项式，探讨其在不可约旋转变换表示中的性质。

Irreducible characters of the symmetric group and exponential growth

We consider sequences of degrees of ordinary irreducible $S_n$-characters. We assume that the corresponding Young diagrams have rows and columns bounded by some linear function of $n$ with leading coefficient less than one. We show that any such sequence has at least exponential growth and we compute an explicit bound.

Duality on hypermaps with symmetric or alternating monodromy group

Duality is the operation that interchanges hypervertices and hyperfaces on oriented hypermaps. The duality index measures how far a hypermap is from being self-dual. We say that an oriented regular hypermap has \emph{duality-type} $\{l,n\}$ if $l$ is the valency of its vertices and $n$ is the valency of its faces. Here, we study some properties of this duality index in oriented regular hypermaps and we prove that for each pair $n$, $l \in \mathbb{N}$, with $n,l \geq 2$, it is possible to find an oriented regular hypermap with extreme duality index and of duality-type $\{l,n \}$, even if we are…

p-Saturations of Welter's Game and the Irreducible Representations of Symmetric Groups

We establish a relation between the Sprague-Grundy function $\text{sg}$ of a $p$-saturation of Welter's game and the degrees of the ordinary irreducible representations of symmetric groups. In this game, a position can be viewed as a partition $λ$. Let $ρ^λ$ be the irreducible representation of $\text{Sym}(|λ|)$ indexed by $λ$. For every prime $p$, we show the following results: (1) the degree of $ρ^λ$ is prime to $p$ if and only if $\text{sg}(λ) = |λ|$; (2) the restriction of $ρ^λ$ to $\text{Sym}(\text{sg}(λ))$ has an irreducible component with degree prime to $p$. Further, for every integer …

关于对称群和交错群的二面体分支系数的正性

On the Positivity of Dihedral Branching Coefficients of the Symmetric and Alternating Groups

研究确定了对称群和交错群的二面体分支系数的正性条件，并推广了关于循环子群的已知结果。

Algorithmically distinguishing irreducible characters of the symmetric group

Suppose that $χ_λ$ and $χ_μ$ are distinct irreducible characters of the symmetric group $S_n$. We give an algorithm that, in time polynomial in $n$, constructs $π\in S_n$ such that $χ_λ(π)$ is provably different from $χ_μ(π)$. In fact, we show a little more. Suppose $f=χ_λ$ for some irreducible character $χ_λ$ of $S_n$, but we do not know $λ$, and we are given only oracle access to $f$. We give an algorithm that determines $λ$, using a number of queries to $f$ that is polynomial in $n$. Each query can be computed in time polynomial in $n$ by someone who knows $λ$.

生成对称群与交错群的公正回避与达成博弈

Impartial avoidance and achievement games for generating symmetric and alternating groups

本研究分析对称群与交错群上的两种公正博弈，通过计算尼姆数确定其胜负结果，为有限群生成集博弈提供了理论判定。

一个新的对称函数恒等式及其在对称群特征值中的应用

A New Symmetric Function Identity With an Application to symmetric group character values

本文研究对称群特征和猜想，通过Schur函数与幂和对称函数内积分析Ev(λ)上的特征交错和，发现幂和对称多项式交错和与单项式对称多项式乘积的恒等式，证明了该特征交错和的一个特例为0。

关于对称群和塞格里夫超代数不可约表示的注记

A note on irreducible representations of symmetric groups and Sergeev superalgebras

本文通过钩长公式明确构造对称群和塞格里夫超代数的不可约表示，并给出特征p>0下的维数公式。

Quantum Knizhnik-Zamolodchikov Equation, Totally Symmetric Self-Complementary Plane Partitions and Alternating Sign Matrices

We present multiresidue formulae for partial sums in the basis of link patterns of the polynomial solution to the level 1 U_q(\hat sl_2) quantum Knizhnik--Zamolodchikov equation at generic values of the quantum parameter q. These allow for rewriting and generalizing a recent conjecture [Di Francesco '06] connecting the above to generating polynomials for weighted Totally Symmetric Self-Complementary Plane Partitions. We reduce the corresponding conjectures to a single integral identity, yet to be proved.

A closed character formula for symmetric powers of irreducible representations

We prove a closed character formula for the symmetric powers $S^N V(λ)$ of a fixed irreducible representation $V(λ)$ of a complex semi-simple Lie algebra $\mathfrak{g}$ by means of partial fraction decomposition. The formula involves rational functions in rank of $\mathfrak{g}$ many variables which are easier to determine than the weight multiplicities of $S^N V(λ)$ themselves. We compute those rational functions in some interesting cases. Furthermore, we introduce a residue-type generating function for the weight multiplicities of $S^N V(λ)$ and explain the connections between our character f…

Alternating sign matrices and totally symmetric plane partitions

We introduce a new family $\mathcal{A}_{n,k}$ of Schur positive symmetric functions, which are defined as sums over totally symmetric plane partitions. In the first part, we show that, for $k=1$, this family is equal to a multivariate generating function involving $n+3$ variables of objects that extend alternating sign matrices (ASMs), which have recently been introduced by the authors. This establishes a new connection between ASMs and a class of plane partitions, thereby complementing the fact that ASMs are equinumerous with totally symmetric self-complementary plane partitions as well as wi…

Irreducible representations of the symmetric groups from slash homologies of p-complexes

In the 40s, Mayer introduced a construction of (simplicial) $p$-complex by using the unsigned boundary map and taking coefficients of chains modulo $p$. We look at such a $p$-complex associated to an $(n-1)$-simplex; in which case, this is also a $p$-complex of representations of the symmetric group of rank $n$ - specifically, of permutation modules associated to two-row compositions. In this article, we calculate the so-called slash homology - a homology theory introduced by Khovanov and Qi - of such a $p$-complex. We show that every non-trivial slash homology group appears as an irreducible …

对称群和交错群的双重覆盖的旋特征的同余

Congruences for spin characters of the double covers of the symmetric and alternating groups

本文从算术角度研究对称群和交错群双重覆盖的旋特征生成函数，利用经典解析工具和初等生成函数操作，推导出许多 Ramanujan-like 同余。

广义对称群的不可约特征标

Irreducible characters of the generalized symmetric group

本文通过顶点代数方法重新证明了Murnaghan-Nakayama规则，并提出新的迭代公式计算广义对称群的特征标，发现其与对称群的模特征标存在数值关系。

对角对称交替符号矩阵

Diagonally symmetric alternating sign matrices

本文研究对角对称交替符号矩阵（DSASMs）的计数，得到其数量及相关统计量生成函数的Pfaffian公式，引入与DSASM一一对应的六顶点模型变体并研究相关主题。

Diagonally and antidiagonally symmetric alternating sign matrices of odd order

We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang-Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbin…

Alternating Catalan numbers and curves with triple ramification

It is known that the monodromy group of each cover of a general curve of genus g>3 equals either the symmetric or the alternating group. The classical Catalan numbers count the minimal degree covers (with symmetric monodromy) of a general curve of even genus. We solve the analogous problem for the alternating group and we determine the number of alternating covers of minimal degree 2g+1 of a general curve of genus g.

对称群不可约表示所关联的内蕴超平面配置中的表示稳定性

Representation stability in the intrinsic hyperplane arrangements associated to irreducible representations of the symmetric-groups

本文从表示稳定性角度研究对称群不可约表示的内蕴超平面配置，理论证明了超平面补的表示稳定性定理，并通过模拟统计了平坦的枚举结果。

Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order

For each $α\in \{0,1,-1 \}$, we count diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order with a maximal number of $α$'s along the diagonal and the antidiagonal, as well as DASASMs of fixed odd order with a minimal number of $0$'s along the diagonal and the antidiagonal. In these enumerations, we encounter product formulas that have previously appeared in plane partition or alternating sign matrix counting, namely for the number of all alternating sign matrices, the number of cyclically symmetric plane partitions in a given box, and the number of vert…