Hypercubes With Another Q–polynomial Structure And Associated Leonard Triples

Let D denote an integer at least 3 and let H(D,2)denote the graph of the D-dimensional hypercube.It is known that if D is even,then H(D,2)has two P-polynomial structures and two Q-polynomial structures.Let A0,A1,…,AD denote the original P-polynomial structure of H(D,2),where for 0 ≤ i ≤D,Ai denotes ith distance matrix of H(D,2).Let E0,E1,…,ED denote the original Q-polynomial structure of H(D,2),where for 0 ≤ i≤D,Ei denotes ith primitive idempotent of H(D,2).H(D,2)also has an-other P-polynomial structure A0,AD-1,A2…,AD and another Q-polynomial structure E,ED-1,E2,…,ED.In this paper，we consider H(D,2)with the original P-polynomial structure A0,A1,…,AD and another Q-polynomial structure E0,ED-1-E2,…,ED,this hypercube is denoted by H(D,2)’.Let X denote the vertex set of H(D,2)’.Obviously,Ai(0 ≤ i ≤ D)is the ith distance matrix of H(D,2)’.And its adjacency matrix A is equal to A1.Fix x∈ X,for 0 ≤ i ≤ D let Bi = Bi(x)denote the ith dual distance matrix of H(D,2)’.And its dual adjacency matrix B is equal to B1.We observe that if i is odd,then Bi =AD-i*;otherwise Bi = Ai*where Ai*= Ai*(x)denotes the diagonal matrix in Matx(C)with(y,y)-entry(Ai*)yy = |X|(Ei)xy(y ∈ X).For 0 ≤ i ≤ D let Fi denote the ith primitive idempotent of H(D,2)’.We observe that if i is odd,then Fi = ED-i;otherwise Fi = Ei.For ＝0 ≤ i ≤ D let Fi*= Fi*(x)denote the ith dual idempotent of H(D,2)’.Obviously,Fi*= Ei*.Let T denote the subalgebra of MatX(C)generated by A,B.We refer to T as the Terwilliger algebra of H(D,2)’ with respect to x.In this paper we construct the imaginary matrix of H(D,2)’ B~ε=(AB + BA)/2 with respect to x,we show that the matrices A and B are related by the fact thatBB~ε+ B~εB = 2A and AB~ε + B~ε = 2B.We display the matrices which represent the action of triple(A,B,B~ε)on W with respect to the six bases.Especially,we show that the triple(A,B,B~ε)acts on each irreducible T-module as a Leonard triple.Finally,we give inner products for the vectors in these bases of irreducible T-module W.