Labelling the character tables of symmetric and alternating groups

关于 $F_4$ 型 Weyl 群不可约字符标号的研究

On the labelling of characters of Weyl groups of type $F_4$

本文指出 $F_4$ 型 Weyl 群字符标号两种约定的问题，并通过 CHEVIE 实验发现 Spaltenstein 在 $E_7$ 型广义 Springer 对应表中的错误。

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.

Promotion on oscillating and alternating tableaux and rotation of matchings and permutations

Using Henriques' and Kamnitzer's cactus groups, Schützenberger's promotion and evacuation operators on standard Young tableaux can be generalised in a very natural way to operators acting on highest weight words in tensor products of crystals. For the crystals corresponding to the vector representations of the symplectic groups, we show that Sundaram's map to perfect matchings intertwines promotion and rotation of the associated chord diagrams, and evacuation and reversal. We also exhibit a map with similar features for the crystals corresponding to the adjoint representations of the general l…

cactus群在标准Young表形上的最大传递性

Maximal transitivity of the cactus group on standard Young tableaux

cactus群在标准Young表形上的作用2-传递性取决于形状是否自转置或单钩，且其像为置换群或交替群，扩展了Schubert calculus问题的Galois群研究。

Character values and decomposition matrices of symmetric groups

By exploiting relationships between the values taken by ordinary characters of symmetric groups we prove two theorems in the modular representation theory of the symmetric group. 1. The decomposition matrices of symmetric groups in odd characteristic have distinct rows. In characteristic 2 the rows of a decomposition matrix labelled by the different partitions $λ$ and $μ$ are equal if and only if $λ$ and $μ$ are conjugate. An analogous result is proved for Hecke algebras. 2. A Specht module for the symmetric group $S_n$, defined over an algebraically closed field of odd characteristic, is deco…

关于对称群与交错群中元素极小多项式的研究

On Minimal Polynomials of Elements in Symmetric and Alternating Groups

本文完整刻画了对称群与交错群不可约表示下满足循环长度整除最大循环长度的置换元素的极小多项式，并由此给出 Swanson 定理、Klyachko 结果及 Sundaram 猜想的新证明。