Cylindric Reverse Plane Partitions and 2D TQFT

TL;DR

This paper introduces cylindric reverse plane partitions and cylindric complete symmetric functions, connecting them to 2D TQFTs and generalising classical symmetric function theory with new algebraic and geometric insights.

Contribution

It generalises weighted sums over reverse plane partitions to cylindric RPPs, defines cylindric complete symmetric functions, and relates them to a 2D TQFT generalising the $ ext{sl}_n$-Verlinde algebra.

Findings

Cylindric complete symmetric functions are $h$-positive with non-negative integer coefficients.

Explicit formula for tensor multiplicities of irreducible representations of the generalized symmetric group.

Connection established between cylindric symmetric functions and a 2D TQFT generalising the Verlinde algebra.

Abstract

The ring of symmetric functions carries the structure of a Hopf algebra. When computing the coproduct of complete symmetric functions $h_\lambda$ 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 cylindric complete symmetric functions to a 2D topological quantum field theory (TQFT) that is a generalisation of the celebrated $\mathfrak{\widehat{sl}}_n$-Verlinde algebra or Wess-Zumino-Witten fusion ring, which plays a prominent role in the context of vertex operator algebras and algebraic geometry.