Cartan Invariants of Symmetric Groups and Iwahori-Hecke Algebras

TL;DR

This paper investigates the invariant factors of Cartan matrices for symmetric groups and Iwahori-Hecke algebras, proposing explicit combinatorial formulas for these invariants and providing evidence supporting their validity.

Contribution

It formulates an explicit combinatorial method to determine Cartan invariants for individual blocks of symmetric groups, extending previous conjectures and results.

Findings

Formulated combinatorial formulas for Cartan invariants.

Predicted the determinants of the Cartan matrices accurately.

Connected Hill's conjecture to K"{u}lshammer, Olsson, and Robinson's conjecture.

Abstract

K\"{u}lshammer, Olsson and Robinson conjectured that a certain set of numbers determined the invariant factors of the $\ell$-Cartan matrix for $S_n$ (equivalently, the invariant factors of the Cartan matrix for the Iwahori-Hecke algebra $\mathcal{H}_n(q)$, where $q$ is a primitive $\ell$th root of unity). We call these invariant factors Cartan invariants. In a previous paper, the second author calculated these Cartan invariants when $\ell=p^r$, $p$ prime, and $r\leq p$ and went on to conjecture that the formulae should hold for all $r$. Another result was obtained, which is surprising and counterintuitive from a block theoretic point of view. Namely, given the prime decomposition $\ell=p_1^{r_1}... p_k^{r_k}$, the Cartan matrix of an $\ell$-block of $S_n$ is a product of Cartan matrices associated to $p_i^{r_i}$-blocks of $S_n$. In particular, the invariant factors of the Cartan matrix associated to an $\ell$-block of $S_n$ can be recovered from the Cartan matrices associated to the $p_i^{r_i}$-blocks. In this paper, we formulate an explicit combinatorial determination of the Cartan invariants of $S_n$--not only for the full Cartan matrix, \emph{but for an individual block}. We collect evidence for this conjecture, by showing that the formulae predict the correct determinant of the $\ell$-Cartan matrix. We then go on to show that Hill's conjecture implies the conjecture of KOR.