对称群上多项式下降矩阵的对角化及多项式反演统计量
Diagonalization of the Matrices of the Multinomial Descent and Multinomial Inversion Statistics on the Symmetric Group
摘要：
在Varchenko，Zagier，Thibon和Reiner，Saliola，Welker的工作中，研究了群代数元素的群代数上的乘法映射的线性代数性质，该乘法映射是以q^{inv}，q^{maj}，inv加权的所有置换的和。这里q是一个变量，inv和maj是经典的统计反演和主索引。我们定义了一个多项式下降统计量des_X和一个多项式反演统计量inv_X。这些新定义的统计量是经典统计量下降、下降和反转的多项式表示。我们确定了群代数上类似定义的乘法映射的谱和谱中每个元素的重数。作为推论，我们导出了群代数上定义的乘法映射的谱和谱中每个元素的重数。
英文摘要：
In the work of Varchenko, Zagier, Thibon, and Reiner, Saliola, Welker, linear algebraic properties of the multiplication map on the group algebra of the group algebra element are studied, which is the sum over all permutations weighted by q^{inv}, q^{maj}, inv. Here q is a variable, and inv and maj are the classical statistics inversion and major index. We define a multinomial descent statistic des_X and a multinomial inversion statistic inv_X. These new defined statistics are the multinomial expressions of the classical statistics descent des and inversion. We determine the spectrum and the multiplicity of each element of the spectrum of the analogously defined multiplication map on the group algebra for both des_X and inv_X. As corollaries we deduce the spectrum and the multiplicity of each element of the spectrum of the defined multiplication map on the group algebra for the statistics des, maj and inv.
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