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Instanton partition functions of $\mathcal{N}=1$ 5d Super Yang-Mills reducedon $S^1$ can be engineered in type IIB string theory from the $(p,q)$-branesweb diagram. To this diagram is superimposed a web of representations of theDing-Iohara-Miki (DIM) algebra that acts on the partition function. In thiscorrespondence, each segment is associated to a representation, and the(topological string) vertex is identified with the intertwiner operatorconstructed by Awata, Feigin and Shiraishi. We define a new intertwiner actingon the representation spaces of levels $(1,n)\otimes(0,m)\to(1,n+m)$, therebygeneralizing to higher rank $m$ the original construction. It allows us to usea folded version of the usual $(p,q)$-web diagram, bringing greatsimplifications to actual computations. As a result, the characterization ofGaiotto states and vertical intertwiners, previously obtained by some of theauthors, is uplifted to operator relations acting in the Fock space ofhorizontal representations. We further develop a method to build qq-charactersof linear quivers based on the horizontal action of DIM elements. Whilefundamental qq-characters can be built using the coproduct, higher ones requirethe introduction of a (quantum) Weyl reflection acting on tensor products ofDIM generators.

(p,q)-webs of DIM representations, 5d N=1 instanton partition functions and qq-characters