Representation theory of the $\alpha$-determinant and zonal spherical functions

TL;DR

This paper explores the representation theory of the $ ext{alpha}$-determinant, linking multiplicities of irreducible components to matrix ranks derived from spherical Fourier transforms, and provides explicit calculations for the case when n=2.

Contribution

It introduces a novel method to determine multiplicities in $ ext{alpha}$-determinant modules using spherical Fourier transforms and explicitly computes these matrices for n=2.

Findings

Multiplicity of irreducible components is given by matrix rank.

Explicit matrix calculation for the case n=2.

Provides a new proof and perspective on $ ext{alpha}$-determinant representation theory.

Abstract

We prove that the multiplicity of each irreducible component in the $\mathcal{U}(\mathfrak{gl}_n)$-cyclic module generated by the $l$-th power $\det^{(\alpha)}(X)^l$ of the $\alpha$-determinant is given by the rank of a matrix whose entries are given by a variation of the spherical Fourier transformation for $(\mathfrak{S}_{nl},\mathfrak{S}_l^n)$. Further, we calculate the matrix explicitly when $n=2$. This gives not only another proof of the result by Kimoto-Matsumoto-Wakayama (2007) but also a new aspect of the representation theory of the $\alpha$-determinants.