Schur algebras of classical groups

Abstract

Throughout this paper the base field will be C . By Doty's definition [S. Doty, Polynomial representations, algebraic monoids, and Schur algebras of classical type, J. Pure Appl. Algebra 123 (1998) 165–199], a Schur algebra of a classical group G is the image of the representation map C G → End C ( ( C n ) ⊗ r ) , where C n is the natural representation and r any natural number. These Schur algebras are semisimple over C . Firstly we determine when the Schur algebras are generalized Schur algebras in Donkin's sense (see [S. Donkin, On Schur algebras and related algebras, I, J. Algebra 104 (1986) 310–328]). The main step is to decompose the tensor space ( C n ) ⊗ r , using path model by Littelmann [P. Littelmann, A Littlewood–Richardson rule for symmetrizable Kac–Moody algebras, Invent. Math. 116 (1994) 329–346]. Secondly we relate Schur algebras with different parameters and form inverse systems from Schur algebras in the same type. We find the inverse limit naturally contains the universal enveloping algebra of the corresponding Lie algebra.