Centralizer matrix algebras and symmetric polynomials of partitions

Abstract

For a field R of characteristic p≥0 and a matrix c in the full n×n matrix algebra Mn(R) over R, let Sn(c,R) be the centralizer algebra of c in Mn(R). We show that Sn(c,R) is a Frobenius-finite, 1-Auslander-Gorenstein, and gendo-symmetric algebra, and that the extension Sn(c,R)⊆Mn(R) is separable and Frobenius. Further, we study the isomorphism problem of invariant matrix algebras. Let σ be a permutation in the symmetric group Σn and cσ the corresponding permutation matrix in Mn(R). We give sufficient and necessary conditions for the invariant algebra Sn(cσ,R) to be semisimple. If R is an algebraically closed field, we establish a combinatoric characterization of when two semisimple invariant R-algebras are isomorphic in terms of the cycle types of permutations.