Diagonal operators, $q$-Whittaker functions and rook theory
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2023
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| author | Ram, Samrith Schlosser, Michael J. |
| author_facet | Ram, Samrith Schlosser, Michael J. |
| contents | 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 $b_{μν}(q)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_06401 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Diagonal operators, $q$-Whittaker functions and rook theory Ram, Samrith Schlosser, Michael J. Combinatorics 15B33, 05A15, 05A18, 05A05, 05E05, 11B65 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 $b_{μν}(q)$ 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. |
| title | Diagonal operators, $q$-Whittaker functions and rook theory |
| topic | Combinatorics 15B33, 05A15, 05A18, 05A05, 05E05, 11B65 |
| url | https://arxiv.org/abs/2309.06401 |