eBook ISBN:  9781470414849 
Product Code:  MEMO/228/1072.E 
List Price:  $76.00 
MAA Member Price:  $68.40 
AMS Member Price:  $45.60 
eBook ISBN:  9781470414849 
Product Code:  MEMO/228/1072.E 
List Price:  $76.00 
MAA Member Price:  $68.40 
AMS Member Price:  $45.60 

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 rightmultiplication 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 $\nu _{(2^k,1^{n2k})}$

6. The original family $\nu _{(k,1^{nk})}$

7. Acknowledgements

A. $\mathfrak {S}_n$module decomposition of $\nu _{(k,1^{nk})}$


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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 rightmultiplication 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 $\nu _{(2^k,1^{n2k})}$

6. The original family $\nu _{(k,1^{nk})}$

7. Acknowledgements

A. $\mathfrak {S}_n$module decomposition of $\nu _{(k,1^{nk})}$