# 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.

## Key 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.