Rook Theory of the Finite General Linear Group

ABSTRACT

Matrices over a finite field having fixed rank and restricted support are a natural q-analog of rook placements on a board, even though their enumeration is known to yield nonpolynomial answers in some cases. We develop this q-rook theory by defining a corresponding analog of the hit numbers. Using tools from coding theory, we show that these q-hit and q-rook numbers obey a variety of identities analogous to the classical case. We also explore connections to earlier q-analogs of rook theory, as well as settling a polynomiality conjecture and finding experimentally a counterexample of a positivity conjecture of the authors and Klein.

Keywords:

• rook number
• hit number
• q-analogue
• reciprocity
• positivity

2000 AMS SUBJECT CLASSIFICATION:

• MSC 2010: Primary 05A05
• Secondary 05A15
• 11B65

Acknowledgments

We are grateful to Jim Haglund and Igor Pak for helpful conversations. We thank Dennis Stanton for his valuable insights into q-series and Krawtchouk polynomials. Finally, we are indebted to Jeffrey Remmel, from whose crucial suggestions this project initially grew.

The experiments in Sections 5.4 and 6 were done in the servers of the math departments of the University of Minnesota and UCLA using SageMath [The Sage Developers 17].

Notes

1 There are many possible variations on the diagram I_w: different choices of coordinates for w give different correspondences between the set of permutations and the set of their diagrams, or amount to reflecting or rotating the diagrams; recording the pairs (i, j) instead of (i, w_j) produces diagrams with permuted columns; recording inversions instead of coinversions is equivalent to recoordinatizing; and so on. None of these differences materially affect our results. Appropriate variations are known in the literature as inversion diagrams or Rothe diagrams of permutations.

2 Note that the set of matrices supported on a given board is a linear subspace, and thus a code in this sense.

3 Regrettably, there are several families of polynomials that go by this name; see, e.g., [Gasper and Rahman 04, Ex. 7.8, 7.11] and [Stanton 84, Section 4], where the polynomials related to ours are the affine q-Krawtchouk polynomials.

4 When comparing the statement there, note a notational conflict: in [Lewis et al. 11], ${\mathfrak{m}}_q(n,B,r)$ counts matrices with support in the complement $\bar{B}$.

5 We use the definition by Haglund [Haglund 98, (3)] of these q-hit numbers as opposed to the original definition [Garsia and Remmel 86, (2.1)]. The two definitions are equivalent up to dividing by tⁿ and replacing t by 1/t.

Additional information

Funding

JBL was supported in part by NSF grant DMS-1401792. AHM was supported in part by an AMS–Simons Foundation travel grant.