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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

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 $$w$$.

• 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