Tensor products of the defining representations over the Witt algebra in positive characteristic

TL;DR

This paper analyzes the tensor product of natural representations over the Witt algebra in positive characteristic, decomposing it into invariant subspaces and explicitly determining their composition series.

Contribution

It provides a detailed decomposition of tensor products of defining modules over the Witt algebra in prime characteristic, including explicit composition series.

Findings

Decomposition of A(1)⊗A(1) into invariant subspaces

Explicit construction of Jordan-Hölder series for these submodules

Complete description of tensor product factors with highest weight p-1

Abstract

Let $A(1):=k[X]/(X^p)$ be the natural representation of the Witt algebra $W(1)$ over an algebraically closed field of prime characteristic $p>3$. In this note, we decompose the $W(1)$-module $A(1)\otimes A(1)$ into two invariant subspaces, and precisely construct their Jordan-H\"older composition series. As a consequence, we obtain all decomposition factors of the tensor product of the simple restricted $W(1)$-module with "highest" weight $p-1$.