On Schur algebras and little Schur algebras

Abstract

Let ζ r be the natural algebra homomorphism from the enveloping algebra U Q ( η ) of gl η to the Schur algebra S Q ( η , r ) . We will present a BLM-realization of U Q ( η ) and the restricted enveloping algebra of gl η . This result will be used to construct various bases for U Q ( η , r ) : = ζ r ( U Q ( η ) ) and little Schur algebras. Finally we will discuss the relation between the little q-Schur algebra and the little Schur algebra through the construction of a B -basis for the little q-Schur algebra over B . Here B = Z [ ε , ε − 1 ] and ε ∈ C is an lth primitive root of 1 with l ⩾ 3 being odd.