# Representation theory of the $α$-determinant and zonal spherical functions - Mathematics > Representation Theory

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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 amatrix whose entries are given by a variation of the spherical Fouriertransformation for $\mathfrak{S} {nl},\mathfrak{S} l^n$. Further, wecalculate the matrix explicitly when $n=2$. This gives not only another proofof the result by Kimoto-Matsumoto-Wakayama 2007 but also a new aspect of therepresentation theory of the $\alpha$-determinants.

Autor: Kazufumi Kimoto

Fuente: https://arxiv.org/