Quantum Weyl Reciprocity and Tilting Modules

Abstract:

Quantum Weyl reciprocity relates the representation theory of Hecke algebras of type A with that of q-Schur algebras. This paper establishes that Weyl reciprocity holds integrally (i. e., over the ring ℤ[q, q ^{− 1}] of Laurent polynomials) and that it behaves well under base-change. A key ingredient in our approach involves the theory of tilting modules for q-Schur algebras. New results obtained in that direction include an explicit determination of the Ringel dual algebra of a q-Schur algebra in all cases. In particular, in the most interesting situation, the Ringel dual identifies with a natural quotient algebra of the Hecke algebra.