The centralizer of two numbers under the natural action of $S_k$ on [k], the maximal parabolic subgroup of $S_k$, and generalized patterns

关于对称群中心化子子群的计数

On counting centralizer subgroups of symmetric groups

本文研究对称群 $S_{2m}$ 的中心化子子群结构，证明了使 $gHg^{-1}\cap H$ 大小为 $m$ 的多项式的排列 $g$ 密度为零。

A Note on Element Centralizers in Finite Coxeter Groups

The normalizer $N_W(W_J)$ of a standard parabolic subgroup $W_J$ of a finite Coxeter group $W$ splits over the parabolic subgroup with complement $N_J$ consisting of certain minimal length coset representatives of $W_J$ in $W$. In this note we show that (with the exception of a small number of cases arising from a situation in Coxeter groups of type $D_n$) the centralizer $C_W(w)$ of an element $w \in W$ is in a similar way a semidirect product of the centralizer of $w$ in a suitable small parabolic subgroup $W_J$ with complement isomorphic to the normalizer complement $N_J$.

McKay Centralizer Algebras

For a finite subgroup $G$ of the special unitary group $SU_2$, we study the centralizer algebra $Z_k(G) = End_G(V^{\otimes k})$ of $G$ acting on the $k$-fold tensor product of its defining representation $V= \mathbb{C}^2$. These subgroups are in bijection with the simply-laced affine Dynkin diagrams. The McKay correspondence relates the representation theory of these groups to the associated Dynkin diagram, and we use this connection to show that the structure and representation theory of $Z_k(G)$ as a semisimple algebra is controlled by the combinatorics of the corresponding Dynkin diagram.

Avoiding maximal parabolic subgroups of S_k

We find an explicit expression for the generating function of the number of permutations in S_n avoiding a subgroup of S_k generated by all but one simple transpositions. The generating function turns out to be rational, and its denominator is a rook polynomial for a rectangular board.

Growth in Chevalley groups relatively to parabolic subgroups and some applications

Given a Chevalley group ${\mathbf G}(q)$ and a parabolic subgroup $P\subset {\mathbf G}(q)$, we prove that for any set $A$ there is a certain growth of $A$ relatively to $P$, namely, either $AP$ or $PA$ is much larger than $A$. Also, we study a question about intersection of $A^n$ with parabolic subgroups $P$ for large $n$. We apply our method to obtain some results on a modular form of Zaremba's conjecture from the theory of continued fractions and make the first step towards Hensley's conjecture about some Cantor sets with Hausdorff dimension greater than $1/2$.

典型群中抛物子群的幂幺根的多项式性质

Polynomial properties of unipotent radicals of parabolic subgroups in classical groups

本文研究辛群、正交群和酉群中极大标准抛物子群的幂幺根，给出其次数为$q^e$的不可约特征标数量的显式公式。