The arithmetic Tutte polynomials of the classical root systems

TL;DR

This paper computes the arithmetic Tutte polynomials for classical root systems A, B, C, D using finite field methods and signed graph enumeration, linking combinatorial invariants to algebraic structures.

Contribution

It introduces a finite field approach and signed graph enumeration to explicitly compute arithmetic Tutte polynomials for classical root systems.

Findings

Derived explicit formulas for arithmetic Tutte polynomials of types A, B, C, D

Established connections between signed graphs and arithmetic Tutte polynomials

Provided new computational methods for invariants of hypertoric arrangements

Abstract

Many combinatorial and topological invariants of a hyperplane arrangement can be computed in terms of its Tutte polynomial. Similarly, many invariants of a hypertoric arrangement can be computed in terms of its arithmetic Tutte polynomial. We compute the arithmetic Tutte polynomials of the classical root systems of types A,B,C, and D, with respect to their integer, root, and weight lattices. We do it in two ways: by introducing a finite field method for arithmetic Tutte polynomials, and by enumerating signed graphs with respect to six parameters.