Rook Theory of the Finite General Linear Group
| Autor: | Alejandro H. Morales, Joel Brewster Lewis |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
FOS: Computer and information sciences
Mathematics::Combinatorics Rank (linear algebra) Computer Science - Information Theory Information Theory (cs.IT) General Mathematics 010102 general mathematics General linear group 0102 computer and information sciences 01 natural sciences Combinatorics Mathematics - Algebraic Geometry Finite field 010201 computation theory & mathematics Reciprocity (electromagnetism) FOS: Mathematics Enumeration Mathematics - Combinatorics Combinatorics (math.CO) 0101 mathematics Mathematics::Representation Theory Algebraic Geometry (math.AG) Mathematics |
| Zdroj: | Experimental Mathematics. 29:328-346 |
| ISSN: | 1944-950X 1058-6458 |
| DOI: | 10.1080/10586458.2018.1470045 |
| Popis: | Matrices over a finite field having fixed rank and restricted support are a natural $q$-analogue of rook placements on a board. We develop this $q$-rook theory by defining a corresponding analogue 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 $q$-analogues of rook theory, as well as settling a polynomiality conjecture and finding a counterexample of a positivity conjecture of the authors and Klein. Comment: 25 pages, 10 figure files. Minor change in definition of q-hit numbers changes notation but doesn't substantively affect results |
| Databáze: | OpenAIRE |
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