Title: (p,q)-webs of DIM representations, 5d N=1 instanton partition functions and qq-characters,
Abstract: Instanton partition functions of $\mathcal{N}=1$ 5d Super Yang-Mills reduced
on $S^1$ can be engineered in type IIB string theory from the $(p,q)$-branes
web diagram. To this diagram is superimposed a web of representations of the
Ding-Iohara-Miki (DIM) algebra that acts on the partition function. In this
correspondence, each segment is associated to a representation, and the
(topological string) vertex is identified with the intertwiner operator
constructed by Awata, Feigin and Shiraishi. We define a new intertwiner acting
on the representation spaces of levels $(1,n)\otimes(0,m)\to(1,n+m)$, thereby
generalizing to higher rank $m$ the original construction. It allows us to use
a folded version of the usual $(p,q)$-web diagram, bringing great
simplifications to actual computations. As a result, the characterization of
Gaiotto states and vertical intertwiners, previously obtained by some of the
authors, is uplifted to operator relations acting in the Fock space of
horizontal representations. We further develop a method to build qq-characters
of linear quivers based on the horizontal action of DIM elements. While
fundamental qq-characters can be built using the coproduct, higher ones require
the introduction of a (quantum) Weyl reflection acting on tensor products of
DIM generators.