Spectra of Symmetrized Shuffling Operators

(Abridged abstract) For a finite real reflection group W and a W-orbit O of flats in its reflection arrangement---or equivalently a conjugacy class of its parabolic subgroups---we introduce a statistic on elements of W. We then study the operator of right-multiplication within the group algebra of W by the element whose coefficients are given by this statistic. We reinterpret the operators geometrically in terms of the arrangement of reflecting hyperplanes for W. We show that they are self-adjoint and positive semidefinite. via two explicit factorizations into a symmetrized form A^t A. In one such factorization, A is a generalization of the projection of a simplex onto the linear ordering polytope. In the other factorization, A is the transition matrix for one of the well-studied Bidigare-Hanlon-Rockmore random walks on the chambers of an arrangement. We study the family of operators in which O is the conjugacy classes of Young subgroups of type (k,1^{n-k}). A special case within this family is the operator corresponding to random-to-random shuffling. We show in a purely enumerative fashion that these operators pairwise commute. We furthermore conjecture that they have integer spectrum, generalizing a conjecture of Uyemura-Reyes for the case k=n-1. We use representation theory to show that if O is a conjugacy class of rank one parabolics in W, the corresponding operator has integer spectrum. Our proof makes use of an (apparently) new family of twisted Gelfand pairs for W. We also study the family of operators in which O is the conjugacy classes of Young subgroups of type (2^k,1^{n-2k}). Here the construction of a Gelfand model for the symmetric group shows that these operators pairwise commute and that they have integer spectrum. For the symmetric group, we conjecture that apart from the two commuting families above, no other pair of operators of this form commutes.