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

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

2007-12-14

arxiv.org/abs/0712.2293v1

Mathematics

Representation Theory

9 pages

Scientific paper

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.

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