Simple operators and $q$-Whittaker coefficients of power sum symmetric functions

TL;DR

This paper presents a new proof connecting subspace counting with $q$-Whittaker coefficients of power sum symmetric functions, leading to solutions for problems in combinatorics and finite field theory.

Contribution

It offers a novel proof of a theorem relating subspace enumeration to symmetric function coefficients, and applies this to solve a problem on splitting subspaces.

Findings

New proof of subspace counting theorem

Explicit calculation of $q$-Whittaker coefficients

Resolution of Niederreiter's splitting subspace problem

Abstract

We give a new proof of a theorem of Bender, Coley, Robbins and Rumsey on counting subspaces with a given profile with respect to a simple operator. Counting such subspaces is equivalent to the problem of determining the $q$-Whittaker coefficients in the expansion of the power sum symmetric function. As a consequence we obtain a result of Chen and Tseng which answers a problem of Niederreiter on splitting subspaces.