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Some reducible Specht modules for Iwahori-Hecke algebras of type \(A\) with \(q=-1\). (English) Zbl 1176.20004

The authors consider the question of classifying the irreducible Specht modules \(S_{\mathbb{F},q}^\lambda\) for the Iwahori-Hecke algebra \(\mathcal H=\mathcal H_{\mathbb{F},q}(\mathfrak S_n)\) of type \(A\), where \(\mathbb{F}\) is a field with \(q\in\mathbb{F}\). They start by giving a historic overview of this problem: let \(e\) be the multiplicative order of \(q\) in \(\mathbb{F}\) if \(q\neq 1\), or the characteristic of \(\mathbb{F}\) if \(q=1\). Then the classification of the simple Specht modules is complete in the cases where \(e>2\) (including \(\infty\)). For a regular partition \(\lambda\) this follows from the Carter Criterion, whereas the general case was done by the authors and others over the course of several papers. They hence go on to consider the only remaining case, that of \(q=-1\) and \(\mathbb{F}\) of characteristic different from 2.
Their main Theorem 2.1 states that \(S_{\mathbb{F},-1}^\lambda\) is reducible if \(\lambda\) contains a disconnected ladder.
The proof of the theorem involves a number of different techniques and ideas, several of which are drawn from previous works of the authors. The most powerful tool is the Fock space approach to the representation theory of \(\mathcal H\). Indeed Ariki’s proof of the Lascoux-Leclerc-Thibon conjecture in principle solves the question of reduciblity of \(S_{\mathbb{F},-1}^\lambda\) since it gives the decomposition numbers for \(\mathcal H\). Still, it does not seem easy to apply the algorithm concretely for the question of reduciblity and the authors’ application is therefore much less direct. Other important ingredients in the proof are Rouquier blocks, James’s regularisation theorem and the kernel intersection theorem.
The main Theorem does not solve the classification problem but constitutes a step forward. The authors present a conjectural solution to the classification problem, based on calculations with the Lascoux-Leclerc-Thibon algorithm that were done in collaboration with A. Mathas.

MSC:

20C08 Hecke algebras and their representations
05E10 Combinatorial aspects of representation theory
20C30 Representations of finite symmetric groups
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References:

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