Graded Cartan determinants of the symmetric groups
Shunsuke Tsuchioka

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.
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