TY - JOUR

T1 - G-Tutte polynomials and Abelian Lie group arrangements

AU - Liu, Ye

AU - Tran, Tan Nhat

AU - Yoshinaga, Masahiko

N1 - Publisher Copyright:
© The Author(s) 2019.

PY - 2021

Y1 - 2021

N2 - For a list A of elements in a finitely generated abelian group Г and an abelian group G, we introduce and study an associated G-Tutte polynomial, 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 quasipolynomial 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.

AB - For a list A of elements in a finitely generated abelian group Г and an abelian group G, we introduce and study an associated G-Tutte polynomial, 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 quasipolynomial 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.

UR - http://www.scopus.com/inward/record.url?scp=85107848414&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnz092

DO - 10.1093/imrn/rnz092

M3 - Article

AN - SCOPUS:85107848414

SN - 1073-7928

VL - 2021

SP - 152

EP - 190

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 1

ER -