W -representations for multi-character partition functions and their β -deformations

In this letter we continue the development of W -representations. We propose several generalizations of the known models, such as the hypergeometric Hurwitz τ -functions. We construct W -representations for multi-character expansions, which involve a generic number of sets of time variables. We propose integral representations for such kind of partition functions which are given by tensor models and multi-matrix models with multi-trace couplings. We further propose the β -deformation of the discussed W -representation for the Hurwitz case for two sets of times as well as for the multi-character case.