A Matrix Setting for the q-Schur Algebra

The Schur algebra S(n, r) has a basis (described in 6, 2.3])consisting of certain elements _i,j, where i, jI(n, r), the setof all ordered r-tuples of elements from the set n={1, 2, ...,n}. The multiplication of two such basis elements is given bya formula known as Schur's product rule. In recent years, aq-analogue S_q(n, r) of the Schur algebra has been investigatedby a number of authors, particularly Dipper and James 3, 4].The main result of the present paper, Theorem 3.6, shows howto embed the q-Schur algebra in the rth tensor power T^r(M_n)of the nxn matrix ring. This embedding allows products in theq-Schur algebra to be computed in a straightforward manner,and gives a method for generalising results on S(n, r) to S_q(n,r). In particular we shall make use of this embedding in subsequentwork to prove a straightening formula in S_q(n, r) which generalisesthe straightening formula for codeterminants due to Woodcock12]. We shall be working mainly with three types of algebra: thequantized enveloping algebra U(gl_n) corresponding to the Liealgebra gl_n, the q-Schur algebra S_q(n, r), and the Hecke algebra,H(A_r–1). It is often convenient, in the case of the q-Schuralgebra and the Hecke algebra, to introduce a square root ofthe usual parameter q which will be denoted by v, as in 5].This corresponds to the parameter v in U(gl_n). We shall denotethis ‘extended’ version of the q-Schur algebra byS_v(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 bereserved for the integers given in the definitions of thesethree algebras.