Robert Shwartz

Weighted Coxeter graphs and generalized geometric representations of Coxeter groups

We introduce the notion of weighted Coxeter graph and associate to it a certain generalization of the standard geometric representation of a Coxeter group. We prove sufficient conditions for faithfulness and non-faithfulness of such a representation. In the case when the weighted Coxeter graph is balanced we discuss how the generalized geometric representation is related to the numbers game played on the Coxeter graph.

COXETER COVERS OF THE CLASSICAL COXETER GROUPS

Let C(T) be a generalized Coxeter group, which has a natural map onto one of the classical Coxeter groups, either Bn or Dn. Let CY(T) be a natural quotient of C(T), and if C(T) is simply-laced (which means all the relations between the generators has order 2 or 3), CY(T) is a generalized Coxeter group, too. Let At,n be a group which contains t Abelian groups generated by n elements. The main result in this paper is that CY(T) is isomorphic to At,n ⋊ Bn or At,n ⋊ Dn, depends on whether the signed graph T contains loops or not, or in other words C(T) is simply-laced or not, and t is the number of the cycles in T. This result extends the results of Rowen, Teicher and Vishne to generalized Coxeter groups which have a natural map onto one of the classical Coxeter groups.

Matchings in graphs and groups

Abstract Berge’s Lemma says that for each independent set S and maximum independent set X , there is a matching from S − X into X − S , namely, a function of S − X into X − S such that ( s , f ( s ) ) is an edge for each s ∈ S − X . Levit and Mandrescu prove A Set and Collection Lemma. It is a strengthening of Berge’s Lemma, by which one can obtain a matching M : S − ⋂ Γ → ⋃ Γ − S , where S is an independent set and Γ is a collection of maximum independent sets. Jarden, Levit and Mandrescu invoke new inequalities from A Set and Collection Lemma and yield a new characterization of Konig–Egervary graphs. In the current paper, we study Berge systems (collections of vertex sets, where the conclusion of Berge’s Lemma hold). The work is divided into two parts: the general theory of Berge systems and Berge systems associated with groups (or more precisely, in non-commuting graphs). We first show that there exists many Berge systems. The notion of an ideal is central in several branches of mathematics. We define ‘a graph ideal’, which is a natural generalization of ‘an ideal’ in the context of graph theory. We show that every graph ideal carries a Berge system. We prove a sufficient condition for a Berge system to be a multi-matching system (collections where the conclusion of a Set and Collection Lemma holds). We get new inequalities in multi-matching systems. Here, we do a turn about from the general theory of Berge systems into the study of Berge systems in non-commuting graphs. We prove that for every group G , if A and B are maximal abelian subsets of 〈 A , B 〉 G and | A | ≤ | B | , then there is a non-commutative matching of A − B into B − A . It means that every non-commuting graph carries a Berge system. Moreover, we apply the sufficient condition to get a multi-matching system in the context of non-commuting graphs and invoke new inequalities in group theory. Surprisingly, the new results concerning groups, do not hold for rings.

Enumeration of balanced finite group valued functions on directed graphs

A group valued function on a graph is called balanced if the product of its values along any cycle is equal to the identity element of the group. We compute the number of balanced functions from the set of edges and vertices of a directed graph to a finite group considering two cases: when we are allowed to walk against the direction of an edge and when we are not allowed to walk against the edge direction. In the first case it appears that the number of balanced functions on edges and vertices depends on whether or not the graph is bipartite, while in the second case this number depends on the number of strong connected components of the graph. We study functions vanishing on each cycle of a directed graph.The number of such functions depends on whether or not the underlying undirected graph is bipartite.If every edge is one-way, then the number of such functions depends on the number of strongly connected components.