Characteristic quasi-polynomials of ideals and signed graphs of classical root systems

TL;DR

This paper uses signed graphs to fully describe the characteristic quasi-polynomials of ideals in classical root systems and their associated toric arrangements, providing new combinatorial insights.

Contribution

It introduces a comprehensive description of characteristic quasi-polynomials for ideals in classical root systems using signed graphs, linking algebraic and combinatorial structures.

Findings

Complete characterization of characteristic quasi-polynomials for classical root system ideals

Explicit formulas for characteristic polynomials of associated toric arrangements

Verification of factorization over dual partitions in classical cases

Abstract

With a main tool is signed graphs, we give a full description of the characteristic quasi-polynomials of ideals of classical root systems ($ABCD$) with respect to the integer and root lattices. As a result, we obtain a full description of the characteristic polynomials of the toric arrangements defined by these ideals. As an application, we provide a combinatorial verification to the fact that the characteristic polynomial of every ideal subarrangement factors over the dual partition of the ideal in the classical cases.