A note on stable toric sheaves of low rank

Kaneyama and Klyachko have shown that any torus equivariant vector bundle of rank $r$ over ${\mathbb{C}\mathbb{P}}^n$ splits if $r<n$. In particular, any such bundle is not slope stable. In contrast, we provide explicit examples of stable equivariant reflexive sheaves of rank $r$ on any polarised toric variety $(X,L)$, for $2\le r<\mathrm{d}\mathrm{i}\mathrm{m}(X)+\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(\mathrm{P}\mathrm{i}\mathrm{c}(X))$, and show that the dimension of their singular locus is strictly bounded by $n-r$.