
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 |

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 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 228; 2014; 109 ppMSC: 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 \(w\).
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Table of Contents
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Chapters
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1. Introduction
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2. Defining the operators
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3. The case where $\mathcal {O}$ contains only hyperplanes
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4. Equivariant theory of BHR\xspace random walks
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5. The family $\nu _{(2^k,1^{n-2k})}$
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6. The original family $\nu _{(k,1^{n-k})}$
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7. Acknowledgements
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A. $\mathfrak {S}_n$-module decomposition of $\nu _{(k,1^{n-k})}$
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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 \(w\).
-
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 $\nu _{(k,1^{n-k})}$