Graded Cartan determinants of the symmetric groups

TL;DR

This paper computes the graded Cartan determinants of symmetric groups and proposes a gradation of Hill's conjecture, linking it to the generalized Cartan invariants conjecture by Külshammer-Olsson-Robinson.

Contribution

It introduces a gradation of Hill's conjecture and connects it to existing conjectures on Cartan invariants, providing new insights into symmetric group representations.

Findings

Computed graded Cartan determinants for symmetric groups

Proposed a gradation of Hill's conjecture

Linked the gradation to Külshammer-Olsson-Robinson's conjecture

Abstract

We give the graded Cartan determinants of the symmetric groups. Based on it, we propose a gradation of Hill's conjecture which is equivalent to K\"ulshammer-Olsson-Robinson's conjecture on the generalized Cartan invariants of the symmetric groups.