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Spectra of symmetrized shuffling operators

About this Title

Victor Reiner, Franco Saliola and Volkmar Welker

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 228, Number 1072
ISBNs: 978-0-8218-9095-0 (print); 978-1-4704-1484-9 (online)
Published electronically: July 24, 2013

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Defining the operators
  • Chapter 3. The case where $\mathcal {O}$ contains only hyperplanes
  • Chapter 4. Equivariant theory of BHR\xspace random walks
  • Chapter 5. The family $\nu _{(2^k,1^{n-2k})}$
  • Chapter 6. The original family $\nu _{(k,1^{n-k})}$
  • Chapter 7. Acknowledgements
  • Appendix A. $\mathfrak {S}_n$-module decomposition of $\nu _{(k,1^{n-k})}$


For a finite real reflection group and a -orbit of flats in its reflection arrangement – or equivalently a conjugacy class of its parabolic subgroups – we introduce a statistic on in that counts the number of “-noninversions” of . This generalizes the classical (non-)inversion statistic for permutations in the symmetric group . We then study the operator of right-multiplication within the group algebra by the element that has as its coefficient on . We reinterpret geometrically in terms of the arrangement of reflecting hyperplanes for , and more generally, for any real arrangement of linear hyperplanes. At this level of generality, one finds that, after appropriate scaling, corresponds to a Markov chain on the chambers of the arrangement. We show that is self-adjoint and positive semidefinite, via two explicit factorizations into a symmetrized form . In one such factorization, the matrix is a generalization of the projection of a simplex onto the linear ordering polytope from the theory of social choice. In the other factorization of as , the matrix is the transition matrix for one of the well-studied Bidigare-Hanlon-Rockmore random walks on the chambers of an arrangement. We study closely the example of the family of operators , corresponding to the case where is the conjugacy classes of Young subgroups in of type . The special case within this family is the operator corresponding to random-to-random shuffling, factoring as where corresponds to random-to-top shuffling. We show in a purely enumerative fashion that this family of operators pairwise commute. We furthermore conjecture that they have integer spectrum, generalizing a conjecture of Uyemura-Reyes for the case . Although we do not know their complete simultaneous eigenspace decomposition, we give a coarser block-diagonalization of these operators, along with explicit descriptions of the -module structure on each block. We further use representation theory to show that if is a conjugacy class of rank one parabolics in , multiplication by has integer spectrum; as a very special case, this holds for the matrix . The proof uncovers a fact of independent interest. Let be an irreducible finite reflection group and any reflection in , with reflecting hyperplane . Then the -valued character of the centralizer subgroup given by its action on the line has the property that is multiplicity-free when induced up to . In other words, forms a twisted Gelfand pair. We also closely study the example of the family of operators

corresponding to the case where is the conjugacy classes of Young subgroups in of type . Here the construction of a Gelfand model for shows both that these operators pairwise commute, and that they have integer spectrum. We conjecture that, apart from these two commuting families and and trivial cases, no other pair of operators of the form commutes for .

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