The Folded (2D + 1)-cube and Its Uniform Posets

Let Γ denote the folded (2 D + 1)-cube with vertex set X and diameter D ≥ 3. Fix x ∈ X . We first define a partial order ≤ on X as follows. For y , z ∈ X let y ≤ z whenever ∂ ( x , y ) + ∂ ( y , z ) = ∂ ( x , z ). Let R (resp. L ) denote the raising matrix (resp. lowering matrix) of Γ. Next we show that there exists a certain linear dependency among RL 2 , LRL , L 2 R and L for each given Q -polynomial structure of Γ. Finally, we determine whether the above linear dependency structure gives this poset a uniform structure or strongly uniform structure.