The Jantzen -Shapovalov form and Cartan invariants of symmetric groups and Hecke algebras

Abstract

The standard lattice Δ inside a highest weight representation of a Kac-Moody algebra of type A can be identified with a direct sum of representation rings of cyclotomic Hecke algebras. Under this identification, weight spaces correspond to blocks and the Jantzen-Shapovalov form corresponds to the Cartan pairing on isomorphism classes of finitely generated projective modules. When Δ is the basic representation of the Kac-Moody algebra of type [special characters omitted], the corresponding Hecke algebra is either the group algebra of the symmetric group over a field of characteristic l, or the Iwahori-Hecke algebra over a field of arbitrary characteristic, but at an lth root of unity. In this case, we extend the work of Brundan and Kleshchev, who calculated the determinant of the blocks of H _n, by giving an algorithm for calculating the invariant factors of the blocks, and proving a formula for these numbers in some cases. When Δ is a higher level representation of a Kac-Moody algebra of type A _∞, the corresponding Hecke algebras are either the degenerate cyclotomic Hecke algebra over a field of characteristic 0, or the (non-degenerate) cyclotomic Hecke algebra over a field of arbitrary characteristic, but with generic parameter q. In this case, we provide a formula for the determinant of the blocks.