Samrith Ram and Michael J. Schlosser

Diagonal operators, q-Whittaker functions and rook theory

(41 pages)

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 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_μν(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_μν(q); each such statistic arises from a Mahonian statistic on multiset permutations. The polynomials b_μν(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_μν(q) and present connections with q-rook theory in the spirit of Garsia and Remmel.

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