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Generalized characters of the generalized symmetric group

Omar Tout
Dec 2022
摘要
We prove that $(\mathbb{Z}_k \wr \mathcal{S}_n \times \mathbb{Z}_k \wr\mathcal{S}_{n-1}, \text{diag} (\mathbb{Z}_k \wr \mathcal{S}_{n-1}) )$ is asymmetric Gelfand pair, where $\mathbb{Z}_k \wr \mathcal{S}_n$ is the wreathproduct of the cyclic group $\mathbb{Z}_k$ with the symmetric group$\mathcal{S}_n.$ The proof is based on the study of the $\mathbb{Z}_k \wr\mathcal{S}_{n-1}$-conjugacy classes of $\mathbb{Z}_k \wr \mathcal{S}_n.$ Wedefine the generalized characters of $\mathbb{Z}_k \wr \mathcal{S}_n$ using thezonal spherical functions of $(\mathbb{Z}_k \wr \mathcal{S}_n \times\mathbb{Z}_k \wr \mathcal{S}_{n-1}, \text{diag} (\mathbb{Z}_k \wr\mathcal{S}_{n-1}) ).$ We show that these generalized characters haveproperties similar to usual characters. A Murnaghan-Nakayama rule for thegeneralized characters of the hyperoctahedral group is presented. Thegeneralized characters of the symmetric group were first studied by Strahov in[7].
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