Polynomial representations, algebraic monoids, and Schur algebras of classical type

Abstract

First we study Zariski-closed subgroups of general linear groups over an infinite field with the property that their polynomial representation theory is graded in a natural way. There are “Schur algebras” associated with such a group and their representations completely determine the polynomial representations of the original subgroup. Moreover, the polynomial representations of the subgroup are equivalent with the rational representations of a certain algebraic monoid associated with the subgroup, and the aforementioned Schur algebras are the linear duals of the graded components of the coordinate bialgebra on the monoid. Then we study the monoids and Schur algebras associated with the classical groups. We obtain some structural information on the monoids. Then we characterize the associated Schur algebras as centralizer algebras for the Brauer algebras, in characteristic zero.