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Spectra of Symmetrized Shuffling Operators
Victor Reiner, University of Minnesota, Minneapolis, Minnesota, Franco Saliola, Université du Québec à Montréal, Canada, and Volkmar Welker, Philipps-Universitaet Marburg, Germany
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Memoirs of the American Mathematical Society
2014; 109 pp; softcover
Volume: 228
ISBN-10: 0-8218-9095-6
ISBN-13: 978-0-8218-9095-0
List Price: US$76
Individual Members: US$45.60
Institutional Members: US$60.80
Order Code: MEMO/228/1072
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For a finite real reflection group \(W\) and a \(W\)-orbit \(\mathcal{O}\) of flats in its reflection arrangement--or equivalently a conjugacy class of its parabolic subgroups--the authors introduce a statistic \(\operatorname{noninv}_\mathcal{O}(w)\) on \(w\) in \(W\) that counts the number of "\(\mathcal{O}\)-noninversions" of \(w\). This generalizes the classical (non-)inversion statistic for permutations \(w\) in the symmetric group \(\mathfrak{S}_n\). The authors then study the operator \(\nu_\mathcal{O}\) of right-multiplication within the group algebra \(\mathbb{C} W\) by the element that has \(\operatorname{noninv}_\mathcal{O}(w)\) as its coefficient on \(w\).

Table of Contents

  • Introduction
  • Defining the operators
  • The case where \(\mathcal O\) contains only hyperplanes
  • Equivariant theory of \(\mathsf{BHR}\) random walks
  • The family \(\nu_{(2^k,1^{n-2k})}\)
  • The original family \(\nu_{(k,1^{n-k})}\)
  • Acknowledgements
  • Appendix A. \(\mathfrak{G}_n\)-module decomposition of \(\nu_{(k,1^{n-k})}\)
  • Bibliography
  • List of Symbols
  • Index
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