A diagram associated with the subconstituent algebra of a distance-regular graph

Abstract

In this paper we consider a distance-regular graph Γ. Fix a vertex x of Γ and consider the corresponding subconstituent algebra T . The algebra T is the C-algebra generated by the Bose-Mesner algebra M of Γ and the dual Bose-Mesner algebra M * of Γ with respect to x. We consider the subspaces along with their intersections and sums. In our notation, M M * means Span{RS|R ∈ M, S ∈ M * }, and so on. We introduce a diagram that describes how these subspaces are related. We describe in detail that part of the diagram up to M M * + M * M . For each subspace U shown in this part of the diagram, we display an orthogonal basis for U along with the dimension of U . For an edge U ⊆ W from this part of the diagram, we display an orthogonal basis for the orthogonal complement of U in W along with the dimension of this orthogonal complement.

Distance-regular graphs, the subconstituent algebra, and the $Q$-polynomial property

“…In addition, there are papers about the thin condition [10,21,24,64,69,82], irreducible T -modules with endpoint one [30], Γ being bipartite [6,13,14,41], Γ being almost-bipartite [9,42], Γ being dual bipartite [22], Γ being almost dual bipartite [23], Γ being 2-homogeneous [15,17,18,53], Γ being tight [56], Γ being a hypercube [27], Γ being a Doob graph [63], Γ being a Johnson graph [49,62], Γ being a Grassmann graph [48], Γ being a dual polar graph [84], Γ having a spin model in the Bose-Mesner algebra [16,52]. Some miscellaneous topics about irreducible T -modules can be found in [26,34,35,40,43,44,55,59,60,75,83].…”

“…(See[60, Section 7].) With the above notation, (i) the vector space M * MM * has an orthogonal basis…”