Major Index for Standard Young Tableaux

Abstract

For a standard tableau \(T\) of shape \(\lambda \vdash n\), \(maj(T)\) is the sum of \(i\)’s such that \(i+1\) appears in a row strictly below that of \(i\) in \(T\). We consider the \(g\)-polynomial \(f^\lambda(q) = \sum_\tau q^{ maj(T)}\), which appears in many contexts: as a dimension of an irreducible representation of finite general linear group, as a special case of Kostka-Foulkes polynomials, and so on. In this article, we try to understand `maj’ on a standard tableau \(T\) in relation to `inv’ on a multiset permutation (or a permutation of type \(\lambda\)). We construct an injective map from the set of standard tableaux to the set of permutations of type \(\lambda\) (increasing in each block) such that the `maj’ of the tableau is the `maj’ of the corresponding permutation when \(\lambda\) is a two-part partition. We believe that this helps to understand irreducible unipotent representations of finite general linear groups.