### 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* = *T*(*x*). 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 *M*, *M*^{*}, *M**M*^{*}, *M*^{*}*M*, *M**M*^{*}*M*, *M*^{*}*M**M*^{*}, … along with their intersections and sums. In our notation, *M**M*^{*} means Span{*R**S* ∣ *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.

#### Keywords

DOI: https://doi.org/10.26493/1855-3974.1559.390

ISSN: 1855-3974

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