The Pieri Rule for GLn Over Finite Fields

TL;DR

This paper extends the Pieri rule, a fundamental tool for decomposing tensor products of representations, to the context of the finite field GL(n,F_q), using a two-step reduction involving symmetric groups and Schur duality.

Contribution

It derives the Pieri rule for GL(n,F_q) by connecting representation theory over finite fields with symmetric group theory and Schur duality, providing a new perspective and methodology.

Findings

Derived the Pieri rule for GL(n,F_q).

Reduced the problem to symmetric group S_n using natural relations.

Highlighted the importance of Young diagram dominance order.

Abstract

The Pieri rule gives an explicit formula for the decomposition of the tensor product of irreducible representation of the complex general linear group GL(n,C) with a symmetric power of the standard representation on C^n. It is an important and long understood special case of the Littlewood-Richardson rule for decomposing general tensor products of representations of GL(n,C). In our recent work [Gurevich-Howe17, Gurevich-Howe19] on the organization of representations of the general linear group over a finite field F_q using small representations, we used a generalization of the Pieri rule to the context of this latter group. In this note, we demonstrate how to derive the Pieri rule for GL(n,Fq). This is done in two steps; the first, reduces the task to the case of the symmetric group S_n, using the natural relation between the representations of S_n and the spherical principal series representations of GL(n,F_q); while in the second step, inspired by a remark of Nolan Wallach, the rule is obtained for S_n invoking the S_\ell-GL_(n,C)) Schur duality. Along the way, we advertise an approach to the representation theory of the symmetric group which emphasizes the central role played by the dominance order on Young diagrams. The ideas leading to this approach seem to appear first, without proofs, in [Howe-Moy86].