# From partition identities to a combinatorial approach to explicit Satake inversion

## TL;DR

This paper develops a combinatorial framework to prove partition identities related to Langlands' beyond endoscopy, enabling explicit calculations of plethysm expansions and basic functions for symmetric power L-functions of GL2.

## Contribution

It introduces combinatorial proofs for partition identities and applies them to explicitly compute plethysm expansions and basic functions for specific symmetric powers of GL2.

## Key findings

- Explicit plethysm expansions for k=3, 4
- Combinatorial proofs of partition identities
- Explicit computation of basic functions for symmetric powers

## Abstract

In this paper, we provide combinatorial proofs for certain partition identities which arise naturally in the context of Langlands' beyond endoscopy proposal. These partition identities motivate an explicit plethysm expansion of $\mathrm{Sym}^j(\mathrm{Sym}^kV)$ for $\mathrm{GL}_2$ in the case $k=3$. We compute the plethysm explicitly for the cases $k=3, 4$. Moreover, we use these expansions to explicitly compute the basic function attached to the symmetric power $L$-function of $\mathrm{GL}_2$ for these two cases.