Polynomial invariants on matrices and partition, Brauer algebras

We identify the dimension of the centralizer of the symmetric group S-d in the partition algebra A(d)(delta) and in the Brauer algebra B-d(delta) with the number of multidigraphs with darrows and the number of disjoint union of directed cycles with darrows, respectively. Using Schur-Weyl duality as a fundamental theory, we conclude that each centralizer is related to the G-invariant space P-d(M-n(k))(G) of degree dhomogeneous polynomials on n xn 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(k))(G) are stable for sufficiently largen. (C) 2021 Elsevier Inc. All rights reserved.