Polynomial invariants on matrices and partition, Brauer algebra

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 $n$.

20 pages, changes of wrong conditions, typos, and grammar. Brauer algebras. Journal of Algebra (2021)