The Computational Complexity of Rules for the Character Table of S_n

Congruences in character tables of symmetric groups

If $λ$ and $μ$ are two non-empty Young diagrams with the same number of squares, and $\boldsymbolλ$ and $\boldsymbolμ$ are obtained by dividing each square into $d^2$ congruent squares, then the corresponding character value $χ_{\boldsymbolλ}(\boldsymbolμ)$ is divisible by $d!$.

String of columns rewriting and confluence of the jeu de taquin

Schützenberger's jeu de taquin is an algorithm on the structure of tableaux, which transforms a skew tableau into a Young one by local transformation rules on the columns of the tableaux. This algorithm defines an equivalence relation on tableaux compatible with the plactic congruence, and gives a proof of the Littlewood-Richardson rule on Schur polynomials. In this article, we introduce the notion of string of columns rewriting system as mechanism of transformations of glued sequences of columns. We describe the execution of the jeu de taquin algorithm as rewriting paths of a string of column…

Submatrices of character tables and basic sets

In this investigation of character tables of finite groups we study basic sets and associated representation theoretic data for complementary sets of conjugacy classes. For the symmetric groups we find unexpected properties of characters on restricted sets of conjugacy classes, like beautiful combinatorial determinant formulae for submatrices of the character table and Cartan matrices with respect to basic sets; we observe that similar phenomena occur for the transition matrices between power sum symmetric functions to bounded partitions and the $k$-Schur functions introduced by Lapointe and M…

On even entries in the character table of the symmetric group

We show that almost every entry in the character table of $S_n$ is even as $n\to\infty$. This resolves a conjecture of Miller. We similarly prove that almost every entry in the character table of $S_n$ is zero modulo $3,5,7,11,$ and $13$ as $n\to\infty$, partially addressing another conjecture of Miller.

特征标表中的高阶同余

Higher Congruences in Character Tables

受Peluse和Soundararajan相关研究启发，本文研究有限群特征标表两列模素数幂同余的条件问题。

具有对称性的超 Robinson-Schensted-Knuth 对应关系及超 Littlewood-Richardson 规则

A super Robinson-Schensted-Knuth correspondence with symmetry and the super Littlewood-Richardson rule

本文提出了一种超 Robinson-Schensted-Knuth 对应关系，用于超表aux 上的符号字母，通过 Schensted 插入算法的超版本进行构建，并给出了超 RSK 对应关系的几何解释，最终推导出超 Schur 函数在有限符号字母上的组合版本的超 Littlewood-Richardson 规则。

Almost all entries in the character table of the symmetric group are multiples of any given prime

We show that almost every entry in the character table of $S_N$ is divisible by any fixed prime as $N\to\infty$. This proves a conjecture of Miller.

对称群特征表中的零

Zeros in the character table of the symmetric group

研究对称群特征表中零的类型，验证猜想并回答相关问题

Murnaghan-Nakayama规则与Hecke-Clifford代数的自旋双迹

Murnaghan-Nakayama rule and spin bitrace for the Hecke-Clifford Algebra

本文为Hecke-Clifford代数导出Pfaffian型Murnaghan-Nakayama规则，基于Frobenius公式和顶点算子，进而得到不可约特征标的一般公式、迭代公式、特殊划分的紧凑公式、对称性质及自旋双迹的组合公式，并列出n≤7的特征标表。

On Characters of a Class of P-polynomial table algebras and applications

In this paper, we study the characters of homogeneous monotonic P-polynomial table algebras with finite dimension d>=5. We then apply them to association schemes. To this end, we develop some methods using tridiagonal matrices and Z-transform. Moreover, we calculate the eigenvalues of a special tridiagonal matrix which is found through the first intersection matrix of P-polynomial table algebras.

对称群特征标表中零元的大规模蒙特卡洛模拟

Large-scale Monte Carlo simulations for zeros in character tables of symmetric groups

本文通过大规模蒙特卡洛模拟逼近大型对称群特征标表中零元的密度，发现与先前结论相反，几乎所有零元均属于单一经典类型。

Schur Superpolynomials: Combinatorial Definition and Pieri Rule

Schur superpolynomials have been introduced recently as limiting cases of the Macdonald superpolynomials. It turns out that there are two natural super-extensions of the Schur polynomials: in the limit $q=t=0$ and $q=t\rightarrow\infty$, corresponding respectively to the Schur superpolynomials and their dual. However, a direct definition is missing. Here, we present a conjectural combinatorial definition for both of them, each being formulated in terms of a distinct extension of semi-standard tableaux. These two formulations are linked by another conjectural result, the Pieri rule for the Schu…

Set-valued tableaux 与 Gelfand-Zetlin 多胞体的胞腔

Set-valued tableaux and cells of Gelfand-Zetlin polytopes

本文针对Grassmannian Grothendieck多项式的两种组合规则（基于set-valued tableaux和基于Gelfand-Zetlin多胞体有效胞腔）构造显式双射，逐项匹配求和项，证明两者等价，并研究晶体算子在胞腔上的性质。

Pieri and Murnaghan--Nakayama type Rules for Chern classes of Schubert Cells

We develop Pieri type as well as Murnaghan--Nakayama type formulas for equivariant Chern--Schwartz--MacPherson classes of Schubert cells in the classical flag variety. These formulas include as special cases many previously known multiplication formulas for Chern--Schwartz--MacPherson classes or Schubert classes. We apply the equivariant Murnaghan--Nakayama formula to the enumeration of rim hook tableaux.