  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].
