(0, 2)-Graphs and Root Systems

We construct (0, 2)-graphs from root systems with simply laced diagram and study their properties. 1. Introduction In the study of the mod cohomology of the Lie algebra of the unipotent radical of groups of Lie type with simply laced diagram, it was found that the connected components of the Hasse diagram of the Koszul complex are -graphs. This note is the result of an attempt to understand these (0, 2)-graphs. 2. (0, 2)-Graphs A -graph is a connected graph with the property that any two vertices have either 0 or 2 common neighbours. The first thing one shows (cf. [10]) is that two adjacent vertices have the same number of neighbours, so that a -graph is regular of some valency (finite or infinite). For a classification of the -graphs of valency at most 8, see [2, 5]. A -graph without triangles is known as a rectagraph. Rectagraphs play a role in diagram geometry, cf., for example, [11]. A semibiplane is a connected incidence structure with points and blocks, where any two points are together in 0 or 2 blocks, and any two blocks meet in 0 or 2 points. Thus, the incidence graph of a semibiplane is a bipartite -graph, and conversely any bipartite -graph defines a semibiplane, up to duality (i.e., up to the choice which part of the bipartition is the set of points and which part is the set of blocks). Semibiplanes were first introduced in order to study projective planes with involution, see [8]. Given a nonbipartite -graph, its bipartite double (the unique bipartite 2-cover, cf. [4]) is a bipartite -graph. A -graph of finite valency has at most vertices, and the -cube is the unique -graph for which equality holds (see [11]). A -graph is called signable if it is possible to label its edges with 1 in such a way that the product of the signs of the four edges of a quadrangle is always ？1. Clearly, a -graph with more than one edge is signable if and only if it has a 2-cover without quadrangles. It is known ([7]) that hypercubes are signable, and ([4, page 372]) that the Gewirtz graph is not. 3. (0,？2)-Graphs from Root Systems Let be a finite root system with simply laced diagram, and let be the collection of positive roots (for some choice of fundamental roots). For any vector (the target vector) in the span of , we define the graph as follows: the vertices of are the subsets of such that . Two vertices and are adjacent when their symmetric difference has size 3 (Smaller is impossible: if the symmetric difference has size 0, then ; it cannot have size 1 since ; it cannot have size 2 since are sets of positive roots). Theorem 1. If has a nonempty vertex set,