This chapter introduces the Schur algebra Sr = Sr(Γ). The definition is motivated by consideration of the matrix representation afforded by the KΓ-module E⊗r, the tensor power of the natural module. Theorem 2.1.3 unearths a characterisation of the Schur algebra as a Hecke algebra, namely as the centralising algebra of the action of G = Σr on the r-tensor space. This shadows the method of Schur [1927]. The rest of the chapter is devoted to an examination of the basic properties of Sr. We exhibit a basis, and a rule for multiplying two basis elements together that generalises the rule of (1.3). At this stage though, the most important fact is the statement that the module category for Sr is equivalent to the category of homogeneous polynomial representations, PK(n, r). There is a simple rule (2.6) allowing one to switch easily from one category to another.

The Schur algebra was defined rather differently in Green's monograph [1980]. There it was defined as the vector space dual of the coalgebra Ar. In 2.3 we shall show how to identify the two algebras. This identification is very important, and will be generalised to q-Schur algebras later.

In part of the next chapter we shall be concerned with the problem of constructing a complete set of irreducible modules for Sr. There are now several approaches available in the literature, most of which involve the coalgebra Ar to some degree.