Cylindric Reverse Plane Partitions and 2D TQFT

Christian Korff ; David Palazzo

Séminaire lotharingien de combinatoire, 80B (2018)

Résumé

The ring of symmetric functions carries the structure of a Hopf algebra. When computing the coproduct of complete symmetric functions h_λ one arrives at weighted sums over reverse plane partitions (RPP) involving binomial coefficients. Employing the action of the extended affine symmetric group at fixed level n we generalise these weighted sums to cylindric RPP and define cylindric complete symmetric functions. The latter are shown to be h-positive, that is, their expansions coefficients in the basis of complete symmetric functions are non-negative integers. We state an explicit formula in terms of tensor multiplicities for irreducible representations of the generalised symmetric group. Moreover, we relate the complete symmetric functions to a 2D topological quantum field theory (TQFT) that is a generalisation of the celebrated sl^~_n-Verlinde algebra or Wess-Zumino-Witten fusion ring, which plays a prominent role in the context of vertex operator algebras and algebraic geometry.