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 at roots of unity. These matrices are $${\mathbb {Z}}[v,v^{-1}]$$ -valued and 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 Kulshammer-Olsson-Robinson conjecture over $${\mathbb {Z}}$$ .