Vector representations for quantum toroidal algebras in general types

Construct explicit vector representations of quantum toroidal algebras U_{q,tor}(X_n^{(r)}) in types beyond A_n^{(1)}, D_n^{(1)}, E_6^{(1)}, and E_7^{(1)}, and systematize their realization (e.g., via Young diagram/column models or vertex operators) to enable exterior power/Fock space constructions across types.

Background

The paper cites existing constructions of vector representations for certain untwisted types and notes obstacles to deriving exterior power and Fock space representations using the Drinfeld coproduct Δ_u. The new Δ^ψ may overcome some of these issues.

Extending vector representations to other types would support broader combinatorial and representation-theoretic constructions, including potential Macmahon-type modules via semi-infinite limits.