Averages of alpha-determinants over permutations

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 $kn$ by $kn$ matrix of the form $A\otimes1_{1,k}$, the Kronecker product of a $kn$ by $n$ matrix $A$ and $1$ by $k$ all one matrix $1_{1,k}$, over permutations of $kn$ letters is reduced to the $k$-wreath determinant of $A$ up to constant. The constant is exactly given by the modified content polynomial for the Young diagram $(k^n)$. 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.