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On Specht modules for general linear groups. (English) Zbl 1071.20041
Let \(q\) be a power of the prime \(p\). Let \(F\) be a field of characteristic different from \(p\) and containing \(p\)-th roots of unity. Fix \(n\geq 2\). For each partition \(\lambda\) of \(n\) one has the Specht module \(S^\lambda\), which is a representation over \(F\) of the finite group \(\text{GL}_n(q)\).
As the authors explain, it is important to understand these Specht modules, even if one cares only about irreducible representations of symmetric groups in arbitrary characteristic, or irreducible rational representations of \(\text{GL}_n(\overline{\mathbb{F}_q})\) in the defining characteristic \(p\).
They conjecture that there is a natural basis of \(S^\lambda\) with the following properties. To each basis element is associated a standard \(\lambda\)-tableau \(t\) and the number of basis elements that is associated with the same \(t\) is given by a polynomial \(g_t(\lambda)\) in \(q\). The aim of the paper is to provide evidence for this phenomenon, in particular for two part partitions \(\lambda=(n,n-m)\) with \(m\leq 11\).

20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
20C33 Representations of finite groups of Lie type
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
05E10 Combinatorial aspects of representation theory
Full Text: DOI
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