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.

In this paper, we compute the second mod 2 homology of an arbitrary Artin group, without assuming the K(π, 1) conjecture. The key ingredients are (A) Hopf's formula for the second integral homology of a group and (B) Howlett's result on the second integral homology of Coxeter groups. 20F36,20J06; 20F55

We obtain vanishing ranges for the mod p cohomology of alternating subgroups of finite p-free Coxeter groups. Here a Coxeter group W is p-free if the order of the product st is prime to p for every pair of Coxeter generators s,t of W . Our result generalizes those for alternating groups formerly proved by Kleshchev-Nakano and Burichenko. As a byproduct, we obtain vanishing ranges for the twisted cohomology of finite p-free Coxeter groups with coefficients in the sign representations. In addition, a weak version of the main result is proved for a certain class of infinite Coxeter groups.

Abstract. e chromatic functor of a simple graph is a functorization of the chromatic polynomial. M. Yoshinaga showed that two nite graphs have isomorphic chromatic functors if and only if they have the same chromatic polynomial. e key ingredient in the proof is the use of stable partitions of graphs. e latter is shown to be closely related to chromatic functors. In this note, we further investigate some interesting properties of chromatic functors associated to simple graphs using stable partitions. Our rst result is the determination of the group of natural automorphisms of the chromatic functor, which is in general a larger group than the automorphism group of the graph. e second result is that the composition of the chromatic functor associated to a nite graph restricted to the category FI of nite sets and injections with the free functor into the category of complex vector spaces yields a consistent sequence of representations of symmetric groups which is representation stable in the sense of Church-Farb.

Kozlov has studied the topological properties of the moduli space of tropical curves of genus 1 with marked points, such as its mod 2 homology, while the integral homology remained a conjecture. In this paper, we present a complete proof of Kozlov's conjecture concerning the integral homology of this moduli space.