Topological quantum field theories from Hecke algebras

TL;DR

This paper constructs new 2D topological quantum field theories based on Hecke algebras, providing explicit formulas and positivity results for invariants associated with ciliated surfaces.

Contribution

It introduces a novel class of TQFTs linked to Hecke algebras for finite Coxeter systems, with explicit computation methods and positivity properties.

Findings

Invariant is a Laurent polynomial for punctured surfaces

Graphical computation method using minimal colored graphs

Positivity of invariants for classical and certain exceptional Coxeter types

Abstract

We construct two-dimensional non-commutative topological quantum field theories (TQFTs), one for each Hecke algebra corresponding to a finite Coxeter system. These TQFTs associate an invariant to each ciliated surface, which is a Laurent polynomial for punctured surfaces. There is a graphical way to compute the invariant using minimal colored graphs. We give explicit formulas in terms of the Schur elements of the Hecke algebra and prove positivity properties for the invariants when the Coxeter group is of classical type, or one of the exceptional types $H_3$, $E_6$ and $E_7$.