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

Abstract

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

arXiv commentv1: 16 pages; v2: 17 pages, added Example 4.10; v3: minor updates; v4: 18 pages