Generalised Cartan invariants of symmetric groups

Abstract

Külshammer, Olsson, and Robinson developed an ℓ-analogue of modular representation theory of symmetric groups where ℓ is not necessarily a prime. They gave a conjectural combinatorial description for invariant factors of the Cartan matrix in this context. We confirm their conjecture by proving a more precise blockwise version of the conjecture due to Bessenrodt and Hill.

On graded Cartan invariants of symmetric groups and Hecke algebras

“…Case a 2 − b 2 ∈ pZ. This is a generalization of the case v = 1, and we generalize the proof in [Evs,§5], Proposition 2.12 being an extra needed ingredient.…”

“…Proof. Apply [Evs,Lemma 5.6] with α i = v(t i )/2 and β i = −v(t i )/2. Verifying the hypotheses is straightforward.…”

“…Put M n = ν∈CRPp(n) M n | Parp(n,ν)×Parp(n,ν) ∈ Mat Par(n) (Z). Then M n and M n are row equivalent over Z (p) by [Evs,Lemma 4.6]. Thus, by Proposition 2.10,…”

“…Proof. By [Evs,Lemma 4.8], the matrices M n and L (p) n are row equivalent over Z (p) and hence over R. Thus, by (ii) we have…”

“…As in [Evs,§4], we consider a submatrix N (p) n of M n and use it to construct a certain block-diagonal matrix L (p) n , which is row equivalent over Z (p) to M n , for any fixed prime p.…”