标题： $p$-Young表，Robinson-Schensted对应和群代数的空缺Cauchy恒等式$KG_{r}$和$KSG_{r}$ 显示英文标题

标题： Study of $p$-Young tableaux, Robinson-Schensted correspondence and the lacunary Cauchy identity of group algebras $KG_{r}$ and $KSG_{r}$

摘要： 在本文中，我们发展了群$G_{r}$ $(\mathbb{Z}_{p^{r}}\rtimes \mathbb{Z}^{*}_{p^{r}})$ 和$SG_{r}$ $(\mathbb{Z}_{p^{r-1}}\rtimes \mathbb{Z}^{*}_{p^{r}})$ 元素之间的 Robinson-Schensted 对应关系，以及一对标准$p$-Young 表。 这种方法不同于经典方法，我们的方法基于为每个群代数计算出的正交原始幂等元所产生的矩阵单位。 讨论了 Robinson-Schensted 对应关系的一些经典性质。 作为副产品，我们还将 Cauchy 恒等式扩展到我们的设置中，我们称之为间隙 Cauchy 恒等式。 这项研究为这些群及其组合结构的表示理论提供了新的见解。 显示英文摘要

摘要： In this paper, we develop the Robinson-Schensted correspondence between the elements of the groups $G_{r}$ $(\mathbb{Z}_{p^{r}}\rtimes \mathbb{Z}^{*}_{p^{r}})$ and $SG_{r}$ $(\mathbb{Z}_{p^{r-1}}\rtimes \mathbb{Z}^{*}_{p^{r}})$, along with a pair of the standard $p$-Young tableaux. This approach differs from the classical method, and ours is based on matrix units arising from orthogonal primitive idempotents computed for every group algebra. Some classical properties of the Robinson-Schensted correspondence are discussed. As a by-product, we also extend the Cauchy identity to our setup, which we refer to as the lacunary Cauchy identity. This study offers new insights into the representation theory of these groups and their combinatorial structures.