In this paper, we study the correlation functions of the quantum toroidal $\mathfrak{gl}_1$ algebra. The first key properties we establish are similar to those of the correlation functions of quantum affine algebras $U_q\mathfrak{n}_+$ as established by Enriquez in (Eneiquez, 2000), while the proof of the remaining key ``vanishing property" relies on a certain ``Master Equality'' of formal series, which constitutes the main technical result of this paper.
We use the isomorphisms between the R-matrix and Drinfeld presentations of the quantum affine algebras in types B, C and D produced in our previous work to describe finite-dimensional irreducible representations in the R-matrix realization.We also review the isomorphisms for the Yangians of these types and use Gauss decomposition to establish an equivalence of the descriptions of the representations in the R-matrix and Drinfeld presentations of the Yangians.
Dual canonical bases are expected to satisfy a certain (double) triangularity property by Leclerc's conjecture. We propose an analogous conjecture for common triangular bases of quantum cluster algebras. We show that a weaker form of the analogous conjecture is true. Our result applies to the dual canonical bases of quantum unipotent subgroups. It also applies to the t-analogs of q-characters of simple modules of quantum affine algebras.
Categories over quantum affine algebras and monoidal categorification
