Some matrices associated with the split decomposition for a [formula omitted]-polynomial distance-regular graph

Abstract

We consider a Q -polynomial distance-regular graph Γ with vertex set X and diameter D ≥ 3 . For μ , ν ∈ { ↓ , ↑ } we define a direct sum decomposition of the standard module V = C X , called the ( μ , ν ) -split decomposition. For this decomposition we compute the complex conjugate and transpose of the associated primitive idempotents. Now fix b , β ∈ C such that b ≠ 1 and assume Γ has classical parameters ( D , b , α , β ) with α = b − 1 . Under this assumption Ito and Terwilliger displayed an action of the q -tetrahedron algebra ⊠ q on the standard module of Γ . To describe this action they defined eight matrices in Mat X ( C ) , called A , A ∗ , B , B ∗ , K , K ∗ , Φ , Ψ . For each matrix in the above list we compute the transpose and complex conjugate. Using this information we compute the transpose and complex conjugate for each generator of ⊠ q on V .