3 - Representation theory of the Schur algebra

Summary

The aim of this chapter is to discuss the representation–theoretic aspects of the Schur algebra, S_r. Fulfilling a promise made in 1.6, we construct explicitly a complete set of irreducible S_r-modules indexed by the dominant weights Λ⁺(n, r). After our study of bideterminants and codeterminants in previous chapter, most of the work has already been done. We define left and right Schur and Weyl modules as certain S_r-modules having bases consisting of standard bideterminants or codeterminants with shapes in Λ⁺(n, r).

In 3.3 we prove the fundamental fact that S_r is a quasi-hereditary algebra, which means roughly that S_r can be filtered by certain (S_r, S_r)-bimodules V = S(λ) indexed by dominant weights and that for each λ ∈ Λ⁺(n, r) there is a factorisation V ≅ V^λ ⊗ ^λV as (S_r, S_r)-bimodules into right and left weight spaces of V. The modules V^λ and ^λV are precisely the left and right Weyl modules, and constitute the standard modules in the highest weight category P_K(n, r) (see CPS [1988b]). Taking duals we re-discover the fundamental filtration appearing in 2.5, which had been obtained by DeConcini et al. some dozen years before anyone had thought about such categories. The sections of this filtration are then seen to be the Schur modules i.e. the costandard modules in the highest weight category P_K(n, r).