A note on tensor categories of Lie type $E_9$

We consider the problem of decomposing tensor powers of the fundamental level 1 highest weight representation $V$ of the affine Kac-Moody algebra $\text{g}(E_9)$. We describe an elementary algorithm for determining the decomposition of the submodule of $\text{Vn}$ whose irreducible direct summands have highest weights which are maximal with respect to the null-root. This decomposition is based on Littelmann’s path algorithm and conforms with the uniform combinatorial behavior recently discovered by H. Wenzl for the series $E_N$, $N≠9$.