对称重排算子的谱
Spectra of Symmetrized Shuffling Operators
摘要：
对于有限实反射群W和它的反射排列中的平坦的W-轨道O--或等价地它的抛物子群的共轭类--我们引入关于W的元素的一个统计量。然后，我们研究了W的群代数内的右乘算子，其系数由该统计量给出。我们根据W的反射超平面的排列重新几何地解释了算子。我们证明了它们是自伴的和半正定的。通过两个显式分解成对称形式A^t A。在一个这样的因式分解中，A是单形在线性序多面体上的投影的推广。在另一种分解中，A是一种在排列的腔室上充分研究的Bidigare-Hanlon-Rockmore随机游动的转移阵。研究了（k，1 ^{n-k}）型Young子群的共轭类为O的算子族.这个家族中的一个特例是对应于随机到随机洗牌的算子。我们表明，在一个纯粹的枚举方式，这些运营商成对交换。我们进一步猜想它们有整数谱，推广了Uyemura-Reyes的猜想，当k=n-1时，我们得到了它们的整数谱.利用表示论证明了：若O是W中一秩抛物的共轭类，则相应的算子具有整数谱.我们的证明使用了W的（显然）新的双绞Gelfand对族。我们还研究了（2^k，1 ^{n-2k}）型Young子群的共轭类为O的算子族.在这里，一个对称群的Gelfand模型的建设表明，这些运营商两两交换，他们有整数谱。对于对称群，我们猜想，除了上面的两个交换族，没有其他对这种形式的算子交换。
英文摘要：
(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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