© 1998 by London Mathematical Society
© The London Mathematical Society
A Matrix Setting for the q-Schur Algebra
Department of Mathematics and Statistics, Lancaster University Lancaster LA1 4YF. E-mail: r.m.green{at}lancs.ac.uk
Received 17 May 1995. Revision received 27 March 1996.
The Schur algebra S(n, r) has a basis (described in [6, 
2.3]) consisting of certain elements 
i,j, where i, j
I(n, r), the set of all ordered r-tuples of elements from the set n={1, 2, ..., n}. The multiplication of two such basis elements is given by a formula known as Schur's product rule. In recent years, a q-analogue Sq(n, r) of the Schur algebra has been investigated by a number of authors, particularly Dipper and James [3, 4]. The main result of the present paper, Theorem 3.6, shows how to embed the q-Schur algebra in the rth tensor power Tr(Mn) of the nxn matrix ring. This embedding allows products in the q-Schur algebra to be computed in a straightforward manner, and gives a method for generalising results on S(n, r) to Sq(n, r). In particular we shall make use of this embedding in subsequent work to prove a straightening formula in Sq(n, r) which generalises the straightening formula for codeterminants due to Woodcock [12].
We shall be working mainly with three types of algebra: the quantized enveloping algebra U(gln) corresponding to the Lie algebra gln, the q-Schur algebra Sq(n, r), and the Hecke algebra, H(Ar–1). It is often convenient, in the case of the q-Schur algebra and the Hecke algebra, to introduce a square root of the usual parameter q which will be denoted by v, as in [5]. This corresponds to the parameter v in U(gln). We shall denote this ‘extended’ version of the q-Schur algebra by Sv(n, r), and we shall usually refer to it as the v-Schur algebra. All three algebras are associative and have multiplicative identities, and the base field will be the field of rational functions, Q(v), unless otherwise stated. The symbols n and r shall be reserved for the integers given in the definitions of these three algebras.