Spectra of Symmetrized Shuffling Operators
MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY(2014)
摘要
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 noninv(O)(w) on w in W that counts the number of "O-noninversions" of w. This generalizes the classical (non-)inversion statistic for permutations w in the symmetric group G(n). We then study the operator nu(O) of right-multiplication within the group algebra CW by the element that has noninv(O)(w) as its coefficient on w. We reinterpret nu(O) geometrically in terms of the arrangement of reflecting hyperplanes for W, and more generally, for any real arrangement of linear hyperplanes. At this level of generality, one finds that, after appropriate scaling, nu(O) corresponds to a Markov chain on the chambers of the arrangement. We show that nu(O) is self-adjoint and positive semidefinite, via two explicit factorizations into a symmetrized form pi(t)pi. In one such factorization, the matrix pi 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 nu(O) as pi(t)pi, the matrix pi 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 {nu((k, 1n-k))}(k=1,2, ..., n), corresponding to the case where O is the conjugacy classes of Young subgroups in W = G(n) of type (k, 1(n-k)). The k = n-1 special case within this family is the operator nu((n-1,1)) corresponding to random-to-random shuffling, factoring as pi(t)pi where pi corresponds to random-to-top shuffling. We show in a purely enumerative fashion that this family of operators {nu((k, 1n-k))} pairwise commute. We furthermore conjecture that they have integer spectrum, generalizing a conjecture of Uyemura-Reyes for the case k = n-1. 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 CW-module structure on each block. We further use representation theory to show that if O is a conjugacy class of rank one parabolics in W, multiplication by nu(O) has integer spectrum; as a very special case, this holds for the matrix (inv(sigma tau(-1)))(sigma,tau is an element of Gn). The proof uncovers a fact of independent interest. Let W be an irreducible finite reflection group and s any reflection in W, with reflecting hyperplane H. Then the {+/- 1}-valued character chi of the centralizer subgroup Z(W)(s) given by its action on the line H-perpendicular to has the property that chi is multiplicity-free when induced up to W. In other words, (W, Z(W)(s), chi) forms a twisted Gelfand pair. We also closely study the example of the family of operators {nu((2k, 1n-2k))}(k=0,1,2, ... , left perpendicularn/2right perpendicular) corresponding to the case where O is the conjugacy classes of Young subgroups in W = G(n) of type (2(k), 1(n-2k)). Here the construction of a Gelfand model for G(n) shows both that these operators pairwise commute, and that they have integer spectrum. We conjecture that, apart from these two commuting families {nu((k, 1n-k))} and {nu((2k, 1n-2k))} and trivial cases, no other pair of operators of the form nu(O) commutes for W = G(n).
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关键词
representation theory,symmetric group,group algebra,transition matrix,conjugacy class,gelfand pair,linear order,spectrum
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