On Specht modules for general linear groups.
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 \(m\leq 11\).
scientific article; zbMATH DE number 2105790