Extended double Dyck path algebra actions

Establish that the rank- polynomial representations \(\mathcal{L}^{(r)}_{\bullet}(q_1,\ldots,q_r)\) admit extended actions by \(\mathbb{B}_{t,q_1}^{\mathrm{ext}}\), and that these actions extend to the universal polynomial representation \(\mathcal{U}_{\bullet}(q_1,q_2,\ldots)\).

Background

The paper uses compatible sequences of DAHA representations to construct modules $\mathcal{L}^{(r)}_{\bullet}(q_1,\ldots,q_r)$ for the algebra $\mathbb{B}_{t,q_1}$. It also constructs maps between these modules for successive values of the rank and defines their direct-limit universal polynomial representation $\mathcal{U}_{\bullet}(q_1,q_2,\ldots)$.

The extended algebra $\mathbb{B}_{t,q_1}^{\mathrm{ext}}$ contains additional operators related to Macdonald operators. The paper does not establish actions of this extended algebra on the higher-rank modules; instead, it explicitly formulates the existence and compatibility of such actions as a conjecture.