On polynomials interpolating between the stationary state of a O(n) model and a Q.H.E. ground state

TL;DR

This paper introduces a family of polynomials linked to tangles and an O(n) loop model, exploring their properties, positivity conjectures, and connections to combinatorics and conformal field theory.

Contribution

It constructs new polynomials associated with tangles and O(n) models, and relates them to CFT partition functions via affine Hecke algebra representation theory.

Findings

Polynomials are defined by vanishing conditions and related to tangles.

Conjecture of positivity at specific parameter values.

Connection established between polynomials, combinatorics, and conformal field theory.

Abstract

We obtain a family of polynomials defined by vanishing conditions and associated to tangles. We study more specifically the case where they are related to a O(n) loop model. We conjecture that their specializations at $z_i=1$ are {\it positive} in $n$. At $n=1$, they coincide with the the Razumov-Stroganov integers counting alternating sign matrices. We derive the CFT modular invariant partition functions labelled by Coxeter-Dynkin diagrams using the representation theory of the affine Hecke algebras.