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arXiv · 0712.2293

Representation theory of the $α$-determinant and zonal spherical functions

Kazufumi Kimoto
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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 $\det^{(α)}(X)^l$ of the $α$-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 $α$-determinants.

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Kazufumi Kimoto. 2007-12-14. Representation theory of the $α$-determinant and zonal spherical functions. https://arxiv.org/abs/0712.2293

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