Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: hep-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: mtrl-th
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The categories of this paper are: nucl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: eess
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The categories of this paper are: acc-phys
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: solv-int
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The categories of this paper are: q-fin