The direct-sum Askey–Wilson quotient conjecture for the centralizer

Let $j_1,j_2,j_3$ be representation labels and let $\mathcal{J}(j_1,j_2,j_3)$ be the set of admissible intermediate labels. For each $k\in\mathcal{J}(j_1,j_2,j_3)$, let $\overline{AW}^{k}(j_1,j_2,j_3)$ be the quotient of the centrally extended Askey–Wilson algebra obtained by setting $\alpha_i=\chi_{j_i}$ and $K=\chi_k$, and imposing the four annihilating polynomial relations for $A$, $B$, $D$, and $D'$ whose roots are indexed by the corresponding sets $\mathcal{S}^{k}$. Define

$$\widetilde{AW}(j_1,j_2,j_3)=\bigoplus_{k\in\mathcal{J}(j_1,j_2,j_3)}\overline{AW}^{k}(j_1,j_2,j_3).$$

Here $\mathcal{S}^{k}(j_a,j_b,j_c)=\{j\in\mathcal{J}(j_a,j_b)\mid k\in\mathcal{J}(j,j_c)\}$, and the four annihilating polynomials have degree $d_k$. The direct-sum Askey–Wilson quotient conjecture. The algebra $\widetilde{AW}(j_1,j_2,j_3)$ is isomorphic to $\mathcal{C}_{j_1,j_2,j_3}$. This gives a decomposition of the centralizer into summands indexed by the possible intermediate labels $k$; the supplied text gives no resolution status.