论文标题
广义对称组的广义字符
Generalized characters of the generalized symmetric group
论文作者
论文摘要
我们证明$（\ Mathbb {z} _k \ wr \ Mathcal {s} _n \ times \ times \ Mathbb {z} _k \ wr \ wr \ wr \ Mathcal {s} _ {n-1} ）$是一对对称的gelfand对，其中$ \ mathbb {z} _k \ wr \ mathcal {s} _n $是环环$ \ mathbb {z} _k $ symmetric Group $ \ Mathcal $ \ natcal {s} _n。 \ Mathcal {s} _ {n-1} $ - $ \ Mathbb {Z} _K \ wr \ Mathcal {s} _n。$我们定义$ \ Mathbb {z} _k} _k \ wr \ nathcal { $）证明这些广义字符具有类似于通常字符的属性。介绍了穆纳汉 - 纳卡亚山（Murnaghan-Nakayama）规则，该规则针对超巨大群体的广义角色。对称组的广义字符首先由[7]中的Strahov研究。
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 a symmetric Gelfand pair, where $\mathbb{Z}_k \wr \mathcal{S}_n$ is the wreath product 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 $\mathbb{Z}_k \wr \mathcal{S}_n.$ We define the generalized characters of $\mathbb{Z}_k \wr \mathcal{S}_n$ using the zonal 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 have properties similar to usual characters. A Murnaghan-Nakayama rule for the generalized characters of the hyperoctahedral group is presented. The generalized characters of the symmetric group were first studied by Strahov in [7].
