POLYNOMIAL REPRESENTATIONS, ALGEBRAIC MONOIDS, AND SCHUR ALGEBRAS OF CLASSICAL TYPE

First we study Zariski-closed subgroups of general linear groups over an infinite field with the property that their polynomial representati on theory is graded in a natural way. There are ''Schur algebras'' ass ociated with such a group and their representations completely determi ne 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 wit h the subgroup, and the aforementioned Schur algebras are the linear d uals of the graded components of the coordinate bialgebra on the monoi d. Then we study the monoids and Schur algebras associated with the cl assical groups. We obtain some structural information on the monoids. Then we characterize the associated Schur algebras as centralizer alge bras for the Brauer algebras, in characteristic zero. (C) 1998 Elsevie r Science B.V.