On graded Cartan invariants of symmetric groups and Hecke algebras

TL;DR

This paper investigates the structure of graded Cartan matrices associated with symmetric groups and Hecke algebras, proposing a conjecture on their invariant factors and providing evidence through algebraic specializations.

Contribution

It introduces a conjecture on the invariant factors of graded Cartan matrices and proves its implications, extending previous conjectures and results in the field.

Findings

Proves implications of the conjecture under localization and specialization.

Generalizes the Ando-Suzuki-Yamada conjecture over rational functions.

Extends the Külshammer-Olsson-Robinson conjecture to a broader setting.

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 $\mathbb Z$.