The Schur algebra S(n, r) has a basis
(described in [6, §2.3]) consisting of certain
elements ζi,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 n×n 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, [Hscr ](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.