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

Supalak Sumalroj

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*, MM*, M*M, MM*M, M*MM*, … along with their intersections and sums. In our notation, MM* 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 MM* + 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


Subconstituent algebra, Terwilliger algebra, distance-regular graph

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

ISSN: 1855-3974

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