Permutation Representations of the Orbits of the Automorphism Group of a Finite Module over Discrete Valuation Ring

On extendability of permutations

Let $V$ be a left vector space over a division ring and let ${\mathcal P}(V)$ be the associated projective space. We describe all finite subsets $X\subset V$ such that every permutation on $X$ can be extended to a linear automorphism of $V$ and all finite subsets ${\mathcal X}\subset {\mathcal P}(V)$ such that every permutation on ${\mathcal X}$ can be extended to an element of ${\rm PGL}(V)$. Also, we reformulate the results in terms of linear and projective representations of symmetric groups.

Cluster automorphism groups of cluster algebras of finite type

We study the cluster automorphism group $Aut(\mathcal{A})$ of a coefficient free cluster algebra $\mathcal{A}$ of finite type. A cluster automorphism of $\mathcal{A}$ is a permutation of the cluster variable set $\mathscr{X}$ that is compatible with cluster mutations. We show that, on the one hand, by the well-known correspondence between $\mathscr{X}$ and the almost positive root system $Φ_{\geq -1}$ of the corresponding Dynkin type, the piecewise-linear transformations $τ_+$ and $τ_-$ on $Φ_{\geq -1}$ induce cluster automorphisms $f_+$ and $f_-$ of $\mathcal{A}$ respectively; on the other ha…

关于张量空间自同构的一个注记

A remark on automorphisms of tensor spaces

本文证明一大类置换群可实现为无穷维张量空间的自同构群，并据此揭示张量空间作为其自同构群表示时可能呈现病态行为。

Representations of automorphism groups of finite O-modules of rank two

Let O be a complete discrete valuation domain with finite residue field. In this paper we describe the irreducible representations of the groups Aut(M) for any finite O-module M of rank two. The main emphasis is on the interaction between the different groups and their representations. An induction scheme is developed in order to study the whole family of these groups coherently. The results obtained depend on the ring O in a very weak manner, mainly through the degree of the residue field. In particular, a uniform description of the irreducible representations of GL(2,O/P^k) is obtained, wher…

类型为 $(\ell,1^n)$ 的 $\mathfrak{o}$-模的自同构群的表示

The Representations of Automorphism Groups of $\mathfrak{o}$-modules of type $(\ell,1^n)$

本文针对有限剩余域的非阿基米德局部域的赋值环上的特定模，提出通过归纳计算其自同构群的表示 zeta 多项式，并证明表示维数可由有限多项式在 $q$ 处取值得到。

关于 $p$-群上排列模的识别定理，扩展 Weiss 定理

A recognition theorem for permutation modules over $p$-groups extending Weiss' Theorem

本文提出一个关于 $p$-群上排列模的识别定理，扩展了 Weiss 定理，并在混合特征下对排列模块进行了更一般的刻画。