A diagram associated with the subconstituent algebra of a distance-regular graph - Sumalroj, Supalak (Ars mathematica contemporanea, 2019)
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 ?$\mathbb{C}$?-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?$\{RS \mid R \in M, S \in M^*\}$?, 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 ?$U \subseteq W$? 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: distance-regular graph; podkonstitutivna algebra; razdaljno-regularen graf; subconstituent algebra; Terwilliger algebra; Terwilligerjeva algebra