Polynomial invariants on matrices and partition, Brauer algebra

TL;DR

This paper explores the dimensions of centralizers of symmetric groups within partition and Brauer algebras, linking them to combinatorial objects and invariant polynomial spaces, and demonstrates stability for large matrix sizes.

Contribution

It establishes a connection between algebraic centralizers and combinatorial graph counts, providing a uniform approach to their dimension stability in invariant polynomial spaces.

Findings

Centralizer dimensions correspond to counts of specific multidigraphs and cycle unions.

Dimensions of invariant polynomial spaces are stable for large matrix sizes.

Provides a unified framework using Schur-Weyl duality for these results.

Abstract

We identify the dimension of the centralizer of the symmetric group $\mathfrak{S}_d$ in the partition algebra $\mathcal{A}_d(\delta)$ and in the Brauer algebra $\mathcal{B}_d(\delta)$ with the number of multidigraphs with $d$ arrows and the number of disjoint union of directed cycles with $d$ arrows, respectively. Using Schur-Weyl duality as a fundamental theory, we conclude that each centralizer is related with the $G$-invariant space $P^d(M_n(\mathbf{k}))^G$ of degree $d$ homogeneous polynomials on $n \times n$ matrices, where $G$ is the orthogonal group and the group of permutation matrices, respectively. Our approach gives a uniform way to show that the dimensions of $P^d(M_n(\mathbf{k}))^G$ are stable for sufficiently large $n$.