Counting integral matrices with a given characteristic polynomial

计数矩阵的生成空间

Counting generating spaces of matrices

本文证明有限域上矩阵环的生成子空间由多项式计数，并据此定义和研究了自由代数绝对不可约表示同构类计数的双变量多项式。

Counting Quasi-Idempotent Irreducible Integral Matrices

Given any polynomial $p$ in $C[X]$, we show that the set of irreducible matrices satisfying $p(A)=0$ is finite. In the specific case $p(X)=X^2-nX$, we count the number of irreducible matrices in this set and analyze the arising sequences and their asymptotics. Such matrices turn out to be related to generalized compositions and generalized partitions.

Computing character tables and Cartan matrices of finite monoids with fixed point counting

In this paper we present an algorithm for efficiently counting fixed points in a finite monoid $M$ under a conjugacy-like action. We then prove a formula for the character table of $M$ in terms of fixed points and radical, which allows for the effective computation of the character table of $M$ over a field of null characteristic, as well as its Cartan matrix, using a formula from [Thiéry '12], again in terms of fixed points. We discuss the implementation details of the resulting algorithms and provide benchmarks of their performances.

通过交换自同态研究交换矩阵

Commuting matrices via commuting endomorphisms

本文提出一个一般框架，将交换矩阵计数问题归结为有限阿贝尔p-群上的自同态问题，并应用于非约化曲线上的有限模计数，同时将交换矩阵三元组难题与Onn猜想联系起来。