Diagonal operators, $q$-Whittaker functions and rook theory

Abstract: We discuss the problem posed by Bender, Coley, Robbins and Rumsey of enumerating the number of subspaces which have a given profile with respect to a linear operator over the finite field $\mathbb{F}q$. We solve this problem in the case where the operator is diagonalizable. The solution leads us to a new class of polynomials $b{\mu\nu}(q)$ indexed by pairs of integer partitions. These polynomials have several interesting specializations and can be expressed as positive sums over semistandard tableaux. We present a new correspondence between set partitions and semistandard tableaux. A close analysis of this correspondence reveals the existence of several new set partition statistics which generate the polynomials $b_{\mu\nu}(q)$; each such statistic arises from a Mahonian statistic on multiset permutations. The polynomials $b_{\mu\nu}(q)$ are also given a description in terms of coefficients in the monomial expansion of $q$-Whittaker symmetric functions which are specializations of Macdonald polynomials. We express the Touchard--Riordan generating polynomial for chord diagrams by number of crossings in terms of $q$-Whittaker functions. We also introduce a class of $q$-Stirling numbers defined in terms of the polynomials $b_{\mu\nu}(q)$ and present connections with $q$-rook theory in the spirit of Garsia and Remmel.