Polynomial invariants on matrices and partition, Brauer algebras

Abstract

We identify the dimension of the centralizer of the symmetric group Sd in the partition algebra Ad(δ) and in the Brauer algebra Bd(δ) 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 to the G-invariant space Pd(Mn(k))G of degree d homogeneous polynomials on n×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 Pd(Mn(k))G are stable for sufficiently large n.