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Journal of Discrete Mathematics
Volume 2013 (2013), Article ID 140537, 4 pages
(0, 2)-Graphs and Root Systems
1Department of Mathematics, Technical University Eindhoven, Eindhoven, The Netherlands
2The University of Georgia, Athens, GA, USA
Received 7 August 2012; Accepted 18 October 2012
Academic Editor: Zhan Zhou
Copyright © 2013 Andries E. Brouwer and Leonard Chastkofsky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We construct (0, 2)-graphs from root systems with simply laced diagram and study their properties.
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. ) 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, .
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 .
Given a nonbipartite -graph, its bipartite double (the unique bipartite 2-cover, cf. ) 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 ).
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 () 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, it is a bipartite -graph.
Proof. That is bipartite follows since adjacent vertices are sets of roots of which the sizes have different parity.
We show that is connected. Since has simply laced diagram (so that all connected components of the diagram are of type , or , ), we may assume that all roots have the same length . Now the roots are precisely the vectors of squared norm 2 in the (integral) lattice spanned by and for we have when and when . Let and be two vertices of . We want to join them by a path. Use induction on the size of the symmetric difference. Let . Suppose and with . Now is a root, and either or is positive. Say . If , then is a vertex adjacent to and we are done by induction. If , then is a vertex adjacent to and we are done by induction. This means that in the remaining case whenever and with we have either or . Since or , and the pair can be retrieved from the pair or , there is a contribution of for each contribution of 1 in the expanded inner product , so that and hence , so that . This proves connectedness.
We show that is a -graph. Let be two vertices with at least one common neighbour, and put . Since neighbours have sizes that differ by 1, the sizes of and differ by 0 or 2. Suppose first that . If and , then the common neighbours are and . If and and , then the common neighbours are (i) if and otherwise, and (ii) if and otherwise. Note that since , so that , precisely two of the inner products are (and the third is 0). Now suppose that . If the symmetric difference of and has size 6, so that and , then the common neighbours are and . If their symmetric difference has size 4, then and , where . If , then one common neighbour is if and otherwise. If , say , then one common neighbour is if and otherwise. In this way, we find one common neighbour for each of and that is in , that is, for each inproduct , that equals 1. Now , and . If , then and precisely one of , is 1, and we find two common neighbours. If , then so that , again two common neighbours. Finally, if , then , impossible.
Different choices for may yield isomorphic graphs .
The map sending each set of positive roots to its complement induces an isomorphism from the graph onto the graph , where is the sum of the positive roots.
Let be the Weyl group (generated by the reflections in the elements of ), let , and let be the set of positive roots made negative by . The graph is mapped isomorphically onto the graph , where , by the map that sends to the union of and with pairs of opposite elements removed. If we parametrize the graphs using instead of , this means that is mapped isomorphically onto . (Indeed, , so ). It follows that we may choose , where is a dominant weight.
5. The Number of Vertices
Let be half the sum of the positive roots.
Proposition 2. The number of vertices of the graph , where and is a dominant weight, equals the multiplicity of the weight in the Verma module .
Proof. This is Lemma 5.9 in . Or, from Weyl’s character formula: the formal character of equals , so that the multiplicity of equals the number of ways to write as sum of positive roots.
Now Freudenthal's formula gives a straightforward way to compute the number of vertices for any given .
6. The Valency
Let be the set of fundamental roots.
Proposition 3. The graph has valency . For , this becomes , where the last sum is over all edges in the Coxeter diagram.
Proof. Let be a vertex, and its complement. Compute (which equals ). If are distinct positive roots with , then and determine a root system of type with a unique third positive root , and w.l.o.g. , so that , . The pairs taken from only contribute to when , or , (and then the contribution is 2). This means that we find a nonzero contribution (of 2) for each neighbour or of . This proves the formula for the valency . On the basis of fundamental roots, inner products are given by the Cartan matrix, and , and .
Lemma 4. Let , where is a dominant weight. Then, the graph has valency .
Proof. Let be the Cartan matrix. A vector is converted from root coordinates to weight coordinates by multiplication by . In weight coordinates and , so that , and .
7. Direct Products
Let us write things like for the graph where is a root system of type and on the root basis, where fundamental roots are numbered as in Bourbaki .
If the Coxeter diagram is disconnected (or the support of is), then clearly the corresponding graph is the direct product of the graphs for the components. For example, is the direct product .
If the target vector contains a 1 on a nonterminal position , then the graph is the direct product of the two or three graphs that arise by splitting the Coxeter diagram (and target vector) into components at , preserving a copy of in each component. For example, is the direct product . And is the direct product . (Proof: Since , each vertex contains a unique root involving . Let the projection on one of the components consist of the roots in with support in that component, together with the projection of on that component. This establishes an isomorphism).
In particular, is the -cube.
Theorem 5. The graphs are signable.
Proof. Let be a semisimple complex Lie algebra with Cartan subalgebra and root system , where the structure constants are chosen such that for (this is possible because is simply laced—see, e.g., [6, Theorem ]).
Let be a field, and let be the -vector space with basis . Let be an arbitrary fixed total order on and identify each vertex , where , with the exterior product . Give the edge joining and , where , the sign (with ) if and , where . We show that this assignment of signs has the property that the product of the signs on the four edges of a quadrangle is always . Note that if is a quadrangle, we may choose the ordering of the roots at these four vertices independently (since each of these vertices is on two edges of the quadrangle), and invariant tails in the exterior products can be ignored.
Examine the possible shapes of a 4-gon , as found in the proof of the -property. Define by .
If , , , and , then the edges , and have signs , , , and with product , as desired.
If , , , and , then the signs are , , , and with product , since the Jacobi identity reduces to (note that ).
That covers the case where (or ). Remains the case where and and, say, .
If , , , and , with , then the signs are , , , and with product since the Jacobi identity reduces to .
If , , , and (or ), with , then the signs are , , (or ), and (or ) with product by the Jacobi identity on (or ).
That covers all cases.
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