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

TL;DR

This paper explores the combinatorial structures of graded module categories over skew polynomial algebras at roots of unity, introducing new operations and classifications of matrices that relate to algebraic and topological properties.

Contribution

It introduces the switching operation and modular Eulerian matrices, linking combinatorics with the structure of graded modules over skew polynomial algebras at roots of unity.

Findings

Introduction of switching operation on skew-symmetric matrices

Definition and classification of modular Eulerian matrices

Analysis of point simplicial complexes at cube 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.