Spectra of Symmetrized Shuffling Operators
Spectra of Symmetrized Shuffling Operators
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对称洗牌算子的谱
DOI：
10.1090/memo/1072
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
V. Welker
中科院分区：
文献类型：
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作者：
V. Reiner;Franco V. Saliola;V. Welker
对于有限实反射群W和其反射排列中平坦的W轨道O-或等价于它的抛物子群的一个共轭类-，我们引入了关于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与其系数由该统计量给出的元素的乘法运算。
我们用W的反射超平面的排列对算子作了几何上的重新解释，证明了它们是自伴的，是半正定的。通过两个显式因式分解成对称形式A^T。在其中一个因式分解中，A是单纯形在线性序多面体上的投影的推广。在另一种因式分解中，A是研究得很好的一种Bidigare-Hanlon-Rockmore随机游走在排列的腔上的转移矩阵。
我们研究了其中O是(k，1^{n-k})型Young子群的共轭类的算子族。这个系列中的一个特例是对应于随机到随机洗牌的运算符。我们用纯列举的方式证明了这些算子是成对交换的。进一步，我们猜想它们具有整数谱，推广了Uyemura-Reyes关于k=n-1的一个猜想。
利用表象理论证明了：如果O是W中一阶抛物线的共轭类，则相应的算子具有整数谱。我们的证明使用了(显然)一族新的双绞吉尔芬德对W。
我们还研究了其中O是(2^k，1^{n-2k})型Young子群的共轭类的算子族。本文构造了对称群的Gelfand模型，证明了这些算子是成对交换的，并且它们具有整数谱。
对于对称群，我们猜想，除了上面的两个交换族外，没有其他对这种形式的算子是交换的。
(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.