Reflection factorizations of Singer cycles

J. B. Lewis, V. Reiner, D. Stanton

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

The number of shortest factorizations into reflections for a Singer cycle in GLn(Fq) is shown to be (qn-1)n-1. Formulas counting factorizations of any length, and counting those with reflections of fixed conjugacy classes are also given. The method is a standard character-theory technique, requiring the compilation of irreducible character values for Singer cycles, semisimple reflections, and transvections. The results suggest several open problems and questions, which are discussed at the end.

Original languageEnglish (US)
Pages (from-to)663-691
Number of pages29
JournalJournal of Algebraic Combinatorics
Volume40
Issue number3
DOIs
StatePublished - Oct 2 2014

Bibliographical note

Funding Information:
Acknowledgments The authors thank A. Ram and P. Diaconis for pointing them to this work of Hildebrand [12] used in Sect. 4.4. They also thank A. Henderson and E. Letellier for pointing them to [11,21]. This work is partially supported by NSF grants DMS-1148634 and DMS-1001933.

Publisher Copyright:
© Springer Science+Business Media New York 2014.

Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

Keywords

  • Factorization
  • Reflection
  • Singer cycle
  • Transvection

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