Diagonal operators, \(q\)-Whittaker functions and rook theory

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.Mathematics Subject Classifications: 15B33, 05A15, 05A18, 05A05, 05E05, 11B65Keywords: Diagonal matrix, finite field, semistandard tableau, Mahonian statistic, \(q\)-Whittaker function, chord diagram, Touchard-Riordan formula, \(q\)-Stirling number, \(q\)-rook theory