The nucleus of a $Q$-polynomial distance-regular graph

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

💡 Research Summary

The paper studies Q‑polynomial distance‑regular graphs Γ with diameter D ≥ 1, focusing on the subconstituent algebra T(x)=⟨A,A*⟩ attached to a fixed vertex x. Here A is the adjacency matrix of Γ and A* is the dual adjacency matrix defined with respect to x; their primitive idempotents {E_i} and {E*i} generate the algebras h_A and h{A*}. The authors introduce a new T‑module, denoted 𝒩(x) and called the nucleus of Γ at x.

First, the authors recall basic notions of distance‑regular graphs, Q‑polynomial property, and the subconstituent algebra. They then develop the theory of tridiagonal pairs (A,A*), which are two diagonalizable linear maps satisfying a three‑term recurrence on each other’s eigenspaces. Standard and dual standard orderings of the eigenspaces are introduced, together with the raising map R and lowering map L. The shape sequence {ρ_i}=dim E_iV=dim E*_iV is defined; when all ρ_i=1 the system is a Leonard system.

A key new invariant, the displacement of an irreducible T‑module W, is defined as r + t − D + d, where r and t are the endpoint and dual endpoint of W, and d its diameter. The authors prove that displacement zero is equivalent to r = t and d = D − 2r, and that any irreducible T‑module with zero displacement is thin (i.e., each E_iW and each E*_iW has dimension at most one).

The nucleus 𝒩(x) is defined as the direct sum of all irreducible T‑submodules of V having displacement zero. Equivalently, for each i (0 ≤ i ≤ D) one sets
N_i = (E*0+…+E*i)V ∩ (E_0+…+E{D−i})V,
and then 𝒩(x)=⊕{i=0}^D N_i. The sum is direct, and each N_i is invariant under both A and A*. The authors show that every irreducible component of 𝒩(x) is thin, and that the endpoint r uniquely determines the isomorphism class of each component.

When Γ is a non‑bipartite dual polar graph (the graphs arising from the geometry of totally isotropic subspaces of a symplectic, orthogonal, or unitary space over GF(q)), the paper gives a concrete description of the nucleus. Define an equivalence relation ∼ on the vertex set X by y ∼ z iff (i) they lie at the same distance from x, and (ii) they belong to the same connected component of the subgraph induced by vertices at that distance. The ∼‑equivalence classes are precisely the fibers of the map that sends a vertex to the subspace of the fixed D‑dimensional space x that it represents. The characteristic vectors of these classes form a basis of 𝒩(x).

The set of all subspaces of x, ordered by inclusion, is the projective geometry L_D(q). The authors construct an explicit bijection between L_D(q) and the basis of 𝒩(x). Under this bijection, the action of A on 𝒩(x) becomes a weighted adjacency operator on L_D(q); the weights coincide with those introduced by Bernard, Crampé, and Vinet for the symplectic dual polar graph when q is prime. The action of A* simply records the distance from x, acting diagonally on the basis. Consequently, dim 𝒩(x) equals the total number of subspaces of a D‑dimensional vector space over GF(q), i.e. the sum of q‑binomial coefficients ∑_{i=0}^D \binom{D}{i}_q.

The paper also discusses several elementary examples (Hamming, Johnson, Grassmann graphs) to illustrate the construction, and it outlines open problems: (1) description of the nucleus for Q‑polynomial graphs that are not dual polar; (2) classification of irreducible T‑modules with non‑zero displacement; (3) connections between the weighted adjacency maps on projective geometries and cohomological or topological invariants.

Overall, the work blends the algebraic theory of tridiagonal pairs, the combinatorial structure of Q‑polynomial distance‑regular graphs, and finite geometry, providing a unified framework for understanding a distinguished T‑module—the nucleus—and revealing its deep links with projective geometries in the dual polar case.