Averages of alpha-determinants over permutations
Kazufumi Kimoto

TL;DR
This paper establishes a connection between weighted averages of alpha-determinants of specific matrices and k-wreath determinants, providing new formulas for symmetric group functions and generalizing Stanley's character formula.
Contribution
It introduces a novel reduction of alpha-determinant averages to k-wreath determinants and derives a determinantal formula for symmetric group functions, extending Stanley's results.
Findings
Weighted averages of alpha-determinants relate to k-wreath determinants.
Provides a determinantal formula for symmetric group functions invariant under Young subgroups.
Generalizes Stanley's formula for irreducible characters of symmetric groups.
Abstract
We show that certain weighted average of the alpha-determinant of a by matrix of the form , the Kronecker product of a by matrix and by all one matrix , over permutations of letters is reduced to the -wreath determinant of up to constant. The constant is exactly given by the modified content polynomial for the Young diagram . As a corollary, we give a `determinantal' formula for certain functions on the symmetric groups which are invariant under the left and right translation by a Young subgroup, especially the values of the Kostka numbers for rectangular shapes with arbitrary weight. This corollary gives a generalization of the formula of irreducible characters of the symmetric group for rectangular shapes due to Stanley.
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Taxonomy
TopicsBayesian Methods and Mixture Models · Advanced Combinatorial Mathematics · Random Matrices and Applications
