an:02105790
Zbl 1071.20041
Dipper, Richard; James, Gordon
On Specht modules for general linear groups
EN
J. Algebra 275, No. 1, 106-142 (2004).
00107123
2004
j
20G05 20G40 20C33 20C05 05E10
Specht modules; standard tableaux
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 \(m\leq 11\).
Wilberd van der Kallen (Utrecht)