On graded Cartan invariants of symmetric groups and Hecke algebras

Abstract

We consider graded Cartan matrices of the symmetric groups and the Iwahori-Hecke algebras of type A, which have entries in the ring $\mathbb Z[v,v^{-1}]$. These matrices may also be interpreted as Gram matrices of the Shapovalov form on sums of weight spaces of a basic representation of an affine quantum group. We present a conjecture predicting the invariant factors of these matrices and give evidence for the conjecture by proving its implications under a localization and certain specializations of the ring $\mathbb Z[v,v^{-1}]$. This proves and generalizes a conjecture of Ando-Suzuki-Yamada on the invariants of these matrices over $\mathbb Q[v,v^{-1}]$ and also generalizes the first author's recent proof of the K\"ulshammer-Olsson-Robinson conjecture over $\mathbb Z$.