Twice Q-polynomial distance-regular graphs are thin

Let Γ be a distance-regular graph of diameter d ⩾ 3. For each vertex χ of Γ, let T(χ) denote the subconstituent algebra for Γ with respect to χ. An irreducible T(χ)-module W is said to be thin if dim Ei∗(χ) W ⩽ 1 for 0 ⩽ i ⩽ d, where Ei∗(χ) is the projection onto the ith subconstituent for Γ with respect to χ. The graph Γ is said to be thin if, for each vertex χ of Γ, very irreducible T(χ)-module is thin. Our main result is the following Theorem: If Γ has two Q-polynomial structures, then Γ is thin.