A generalization of $q$-deformation of graphic arrangements to simplicial complexes

TL;DR

This thesis extends the concept of q-deformation from graphic arrangements to simplicial complexes, exploring new hyperplane arrangements and their algebraic properties, including characteristic polynomials and freeness.

Contribution

Introduces two new hyperplane arrangements inspired by q-deformations and extends q-deformation to simplicial complexes, addressing a conjecture by Nian, Tsujie, Uchiumi, and Yoshinaga.

Findings

Extended q-deformation to simplicial complexes.

Analyzed characteristic polynomials of graphic monomial arrangements.

Investigated freeness over fields with primitive roots.

Abstract

The purpose of this thesis is to introduce two new kinds of hyperplane arrangements, inspired by the graphic arrangements and $q$-deformations of graphic arrangements. In this thesis, the author extends the definition of $q$-deformation to simplicial complexes, with the conjecture by Nian, Tsujie, Uchiumi and Yoshinaga. The author also investigates a special case called graphic monomial arrangement, including the characteristic polynomials and freeness with a further extension to fields with primitive roots.