Fusion Hall algebra and shuffle conjectures

          @misc{ pirsa_PIRSA:19030095,
            doi = {10.48660/19030095},
            url = {https://pirsa.org/19030095},
            author = {Mellit, Anton},
            keywords = {Mathematical physics},
            language = {en},
            title = {Fusion Hall algebra and shuffle conjectures},
            publisher = {Perimeter Institute},
            year = {2019},
            month = {mar},
            note = {PIRSA:19030095 see, \url{https://pirsa.org}}
          }
          

Abstract

The classical Hall algebra of the category of representations of one-loop quiver is isomorphic to the ring of symmetric functions, and Hall-Littlewood polynomials arise naturally as the images of objects. I will talk about a second "fusion" product on this algebra, whose structure constants are given by counting of bundles with nilpotent endomorphisms on P^1 with restrictions at 0, 1 and infinity. The two products together make up a structure closely related to the elliptic Hall algebra. In the situations when bundles can be explicitly enumerated, I will explain how this leads to q,t-identities conjectured by combinatorists, such as the shuffle conjecture and its generalizations. This is a joint project with Erik Carlsson.