Bipartite Q-polynomial distance-regular graphs and uniform posets

Abstract

Let Γ denote a bipartite distance-regular graph with vertex set X and diameter D ≥ 3. Fix x ∈ X and let L (resp. R) denote the corresponding lowering (resp. raising) matrix. We show that each Q-polynomial structure for Γ yields a certain linear dependency among RL 2 , LRL, L 2 R, L. Define a partial order ≤ on X as follows. For y, z ∈ X let y ≤ z whenever ∂(x, y) + ∂(y, z) = ∂(x, z), where ∂ denotes path-length distance. We determine whether the above linear dependency gives this poset a uniform or strongly uniform structure. We show that except for one special case a uniform structure is attained, and except for three special cases a strongly uniform structure is attained.

The attenuated space poset Aq(N,M)

“…Equation (18) follows from this along with (7), (8) and Definition 4.2. Combining (9), (10), (18) we obtain (19). In (19), apply the transpose to each side and use S t = S, L t = R to get (20).…”

“…In [36], Worawannotai found another family of uniform posets using the polar spaces. For each bipartite Q-polynomial distance-regular graph, Miklavič and Terwilliger [18] considered a poset on its vertex set. They found necessary and sufficient conditions for this poset to be uniform.…”

Q -polynomial distance-regular graphs and a double affine Hecke algebra of rank one

“…In (38), evaluate θ * i+1 − θ * i using (14) and evaluate φ i+1 using (16). Simplify the result to get (35).…”

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

“…So it is natural to compute the irreducible T -modules. These modules are important in the study of hypercubes [14,26], dual polar graphs [20,38], spin models [6,10], codes [13,28], the bipartite property [3,4,9,16,21,22,23,25,27], the almost-bipartite property [5,8,17], the Q-polynomial property [3,7,11,12,18,19,27,33], and the thin property [15,24,30,31,32,34,36,37].…”