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Determinants of representations of Coxeter groups

Authors
  • Ghosh, Debarun1
  • Spallone, Steven2
  • 1 Central European University, Budapest, Hungary , Budapest (Hungary)
  • 2 Indian Institute of Science Education and Research, Pune, Pune, Maharashtra, India , Pune (India)
Type
Published Article
Journal
Journal of Algebraic Combinatorics
Publisher
Springer US
Publication Date
Sep 28, 2018
Volume
49
Issue
3
Pages
229–265
Identifiers
DOI: 10.1007/s10801-018-0842-2
Source
Springer Nature
Keywords
License
Yellow

Abstract

In Ayyer et al. (J Comb Theory Ser A 150:208–232, 2017), the authors characterize the partitions of n whose corresponding representations of Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_n$$\end{document} have nontrivial determinant. The present paper extends this work to all irreducible finite Coxeter groups W. Namely, given a nontrivial multiplicative character ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document} of W, we give a closed formula for the number of irreducible representations of W with determinant ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}. For Coxeter groups of type Bn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_n$$\end{document} and Dn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_n$$\end{document}, this is accomplished by characterizing the bipartitions associated to such representations.

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