# A Clebsch-Gordan decomposition in positive characteristic

## TL;DR

This paper investigates the decomposition of tensor products of symmetric powers of the natural module for SL(2) over fields of positive characteristic, revealing multiplicity-one indecomposables and identifying the modules involved.

## Contribution

It extends the Clebsch-Gordan decomposition to positive characteristic, detailing the structure and modules involved in the tensor product decomposition.

## Key findings

- Each indecomposable component occurs with multiplicity one.
- Explicit identification of modules involved in the decomposition.
- Generalization of Clebsch-Gordan formula to positive characteristic.

## Abstract

Let $G$ be the special linear group of degree $2$ over an algebraically closed field $K$. Let $E$ be the natural module and $S^rE$ the $r$th symmetric power. We consider here, for $r,s\geq 0$, the tensor product of $S^rE$ and the dual of $S^sE$. In characteristic zero this tensor product decomposes according to the Clebsch-Gordan formula. We consider here the situation when $K$ is a field of positive characteristic. We show that each indecomposable component occurs with multiplicity one and identify which modules occur for given $r$ and $s$.