Reflection factorizations of Singer cycles

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

doi:10.1007/s10801-014-0502-0

Reflection factorizations of Singer cycles

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

School of Mathematics

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

The number of shortest factorizations into reflections for a Singer cycle in GL_n(F_q) 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 language	English (US)
Pages (from-to)	663-691
Number of pages	29
Journal	Journal of Algebraic Combinatorics
Volume	40
Issue number	3
DOIs	https://doi.org/10.1007/s10801-014-0502-0
State	Published - 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.

Keywords

Factorization
Reflection
Singer cycle
Transvection

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10.1007/s10801-014-0502-0

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title = "Reflection factorizations of Singer cycles",

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.",

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AU - Reiner, V.

AU - Stanton, D.

N1 - 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.

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N2 - 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.

AB - 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.

KW - Factorization

KW - Reflection

KW - Singer cycle

KW - Transvection

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Reflection factorizations of Singer cycles

Abstract

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