Combinatorics and topology of toric arrangements defined by root systems

We introduce and study the notion of the G-Tutte polynomial for a list A of elements in a finitely generated abelian group Γ and an abelian group G, which is defined by counting the number of homomorphisms from associated finite abelian groups to G.The G-Tutte polynomial is a common generalization of the (arithmetic) Tutte polynomial for realizable (arithmetic) matroids, the characteristic quasi-polynomial for integral arrangements, Brändén-Moci's arithmetic version of the partition function of an abelian group-valued Potts model, and the modified Tutte-Krushkal-Renhardy polynomial for a finite CW-complex.As in the classical case, G-Tutte polynomials carry topological and enumerative information (e.g., the Euler characteristic, point counting and the Poincaré polynomial) of abelian Lie group arrangements.We also discuss differences between the arithmetic Tutte and the G-Tutte polynomials related to the axioms for arithmetic matroids and the (non-)positivity of coefficients.

Section: Examples: Root Systemsmentioning

confidence: 99%

G-Tutte Polynomials and Abelian Lie Group Arrangements

Liu

Tran

Yoshinaga

2019

“…The next step is the work of Moci, in particular his papers [9], [10] and [11], developing the theory with a special focus on combinatorics. In particular, Moci introduces a two-variable polynomial that encodes enumerative invariants of many of the different objects populating the landscape outlined by De Concini and Procesi in [4].…”

Section: Introductionmentioning

confidence: 99%

A Salvetti Complex for Toric Arrangements and its Fundamental Group

d'Antonio

Delucchi

2011

“…In fact, the construction of the arithmetic Tutte polynomial was largely motivated by this special case. [11,24] We will pay special attention to the four infinite families of finite root systems, known as the classical root systems:…”

Section: Root Systems and Latticesmentioning

confidence: 99%

The Arithmetic Tutte Polynomials of the Classical Root Systems

Ardila

Castillo

Henley

2014

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 A n , B n , C n , and D n 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.