Absolute order in general linear groups
| Autor: | Jia Huang, Joel Brewster Lewis, Victor Reiner |
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| Rok vydání: | 2017 |
| Předmět: |
Pure mathematics
Group (mathematics) General Mathematics 010102 general mathematics General linear group 0102 computer and information sciences Codimension 01 natural sciences 20G40 05E10 20C33 Finite field Character (mathematics) 010201 computation theory & mathematics Additive function FOS: Mathematics Mathematics - Combinatorics Order (group theory) Combinatorics (math.CO) Representation Theory (math.RT) 0101 mathematics Partially ordered set Mathematics - Representation Theory Mathematics |
| Zdroj: | Journal of the London Mathematical Society. 95:223-247 |
| ISSN: | 1469-7750 0024-6107 |
| DOI: | 10.1112/jlms.12013 |
| Popis: | This paper studies a partial order on the general linear group GL(V) called the absolute order, derived from viewing GL(V) as a group generated by reflections, that is, elements whose fixed space has codimension one. The absolute order on GL(V) is shown to have two equivalent descriptions, one via additivity of length for factorizations into reflections, the other via additivity of fixed space codimensions. Other general properties of the order are derived, including self-duality of its intervals. Working over a finite field F_q, it is shown via a complex character computation that the poset interval from the identity to a Singer cycle (or any regular elliptic element) in GL_n(F_q) has a strikingly simple formula for the number of chains passing through a prescribed set of ranks. Comment: 26 pages. v2: Minor edits; Question 6.3 and some references added |
| Databáze: | OpenAIRE |
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