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Spectra of Symmetrized Shuffling Operators
 
Victor Reiner University of Minnesota, Minneapolis, Minnesota
Franco Saliola Université du Québec à Montréal, Montréal, Canada
Volkmar Welker Philipps-Universitaet Marburg, Marburg, Germany
Spectra of Symmetrized Shuffling Operators
eBook ISBN:  978-1-4704-1484-9
Product Code:  MEMO/228/1072.E
List Price: $76.00
MAA Member Price: $68.40
AMS Member Price: $45.60
Spectra of Symmetrized Shuffling Operators
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Spectra of Symmetrized Shuffling Operators
Victor Reiner University of Minnesota, Minneapolis, Minnesota
Franco Saliola Université du Québec à Montréal, Montréal, Canada
Volkmar Welker Philipps-Universitaet Marburg, Marburg, Germany
eBook ISBN:  978-1-4704-1484-9
Product Code:  MEMO/228/1072.E
List Price: $76.00
MAA Member Price: $68.40
AMS Member Price: $45.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2282014; 109 pp
    MSC: Primary 05; 20; 60;

    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
     
     
    • Chapters
    • 1. Introduction
    • 2. Defining the operators
    • 3. The case where $\mathcal {O}$ contains only hyperplanes
    • 4. Equivariant theory of BHR\xspace random walks
    • 5. The family $\nu _{(2^k,1^{n-2k})}$
    • 6. The original family $\nu _{(k,1^{n-k})}$
    • 7. Acknowledgements
    • A. $\mathfrak {S}_n$-module decomposition of $\nu _{(k,1^{n-k})}$
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2282014; 109 pp
MSC: Primary 05; 20; 60;

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 \(w\).

  • Chapters
  • 1. Introduction
  • 2. Defining the operators
  • 3. The case where $\mathcal {O}$ contains only hyperplanes
  • 4. Equivariant theory of BHR\xspace random walks
  • 5. The family $\nu _{(2^k,1^{n-2k})}$
  • 6. The original family $\nu _{(k,1^{n-k})}$
  • 7. Acknowledgements
  • A. $\mathfrak {S}_n$-module decomposition of $\nu _{(k,1^{n-k})}$
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.