Combinatorics of graded module categories over skew polynomial algebras at roots of unity

We introduce an operation on skew-symmetric matrices over $\mathbb{Z}\mathrm{/}\mathrm{\ell }\mathbb{Z}$ called switching, and also define a class of skew-symmetric matrices over $\mathbb{Z}\mathrm{/}\mathrm{\ell }\mathbb{Z}$ referred to as modular Eulerian matrices. We then show that these are closely related to the graded module categories over skew polynomial algebras at $\mathrm{\ell }$-th roots of unity. As an application, we study the point simplicial complexes of skew polynomial algebras at cube roots of unity.