Centrally symmetric configurations of integer matrices

TL;DR

This paper introduces the concept of centrally symmetric configurations of integer matrices and investigates their algebraic properties, including normality, Gorenstein condition, and Gr"obner bases, with applications to unimodular and graph incidence matrices.

Contribution

It defines centrally symmetric configurations and analyzes their toric rings, providing new insights into their algebraic structure and specific cases like unimodular and graph incidence matrices.

Findings

Conditions for normality and Gorenstein property of the toric ring.

Development of Gr"obner bases for these configurations.

Special focus on unimodular and graph incidence matrices.

Abstract

The concept of centrally symmetric configurations of integer matrices is introduced. We study the problem when the toric ring of a centrally symmetric configuration is normal as well as is Gorenstein. In addition, Gr\"obner bases of toric ideals of centrally symmetric configurations will be discussed. Special attentions will be given to centrally symmetric configurations of unimodular matrices and those of incidence matrices of finite graphs.