Rook Theory of the Finite General Linear Group. Issue 3 (1st September 2020)

• Record Type: Journal Article
• Authors: Lewis, Joel Brewster; Morales, Alejandro H.
• Is Part Of: Experimental mathematics. Volume 29:Issue 3(2020)
• Page Start: 328
• Page End: 346
• Publication Date: 2020-09-01
• Subjects: rook number -- hit number -- q-analogue -- reciprocity -- positivity; MSC 2010: Primary 05A05 -- Secondary 05A15 -- 11B65; Mathematics -- Periodicals; Mathematics -- Research -- Periodicals; 510.724
• DOI: 10.1080/10586458.2018.1470045
• Languages: English
• ISSNs: 1058-6458

Abstract

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.