The Nucleus of a Q-Polynomial Distance-Regular Graph

Source abstract

Abstract Let $$\Gamma$$ Γ denote a Q -polynomial distance-regular graph with diameter $$D\ge 1$$ D ≥ 1 . For a vertex x of $$\Gamma$$ Γ the corresponding subconstituent algebra $$T=T(x)$$ T = T ( x ) is generated by the adjacency matrix A of $$\Gamma$$ Γ and the dual adjacency matrix $$A^*=A^*(x)$$ A ∗ = A ∗ ( x ) of $$\Gamma$$ Γ with respect to x . We introduce a T -module $${\mathcal {N}} = {\mathcal {N}}(x)$$ N = N ( x ) called the nucleus of $$\Gamma$$ Γ with respect to x . We describe $${\mathcal {N}}$$ N from various points of view. We show that all the irreducible T -submodules of $${\mathcal {N}}$$ N are thin. Under the assumption that $$\Gamma$$ Γ is a nonbipartite dual polar graph, we give an explicit basis for $${\mathcal {N}}$$ N and the action of $$A, A^*$$ A , A ∗ on this basis. The basis is in bijection with the set of elements for the projective geometry $$L_D(q)$$ L D ( q ) , where GF ( q ) is the finite field used to define $$\Gamma$$ Γ .