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

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 $p-1$.