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

TL;DR

This paper introduces a novel Robinson-Schensted correspondence for specific group algebras involving $p$-Young tableaux, extending classical combinatorial and representation theory concepts with new algebraic and identity results.

Contribution

It develops a new Robinson-Schensted correspondence for groups $G_r$ and $SG_r$ using matrix units from primitive idempotents, and extends the Cauchy identity to this setting.

Findings

Established a new correspondence between group elements and $p$-Young tableaux.

Extended the classical Cauchy identity to the lacunary Cauchy identity.

Provided insights into the representation theory of $G_r$ and $SG_r$ groups.

Abstract

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.