Fully Packed Loop Configurations: Polynomiality and Nested Arches

Florian Aigner

doi:10.37236/6963

Abstract

This article proves a conjecture by Zuber about the enumeration of fully packed loops (FPLs). The conjecture states that the number of FPLs whose link pattern consists of two noncrossing matchings which are separated by $m$ nested arches is a polynomial function in $m$ of certain degree and with certain leading coefficient. Contrary to the approach of Caselli, Krattenthaler, Lass and Nadeau (who proved a partial result) we make use of the theory of wheel polynomials developed by Di Francesco, Fonseca and Zinn-Justin. We present a new basis for the vector space of wheel polynomials and a polynomiality theorem in a more general setting. This allows us to finish the proof of Zubers conjecture.

Highlights

Razumov and Stroganov conjectured in [5] a relation between fully packed loop configurations and the ground state vector in the O(1) loop model
Theorem 1.1 ([11, Conjecture 7]) For noncrossing matchings π ∈ N Cn and π ∈ N Cn and an integer m the number of FPLs with link pattern (π)mπ is a polynomial in m of degree |λ(π)| + |λ(π )| with leading coefficient dim(λ(π)) dim(λ(π |λ(π)|!|λ(π )|!
We can assign to every FPL F a noncrossing matching π(F ) by setting π(F )(i) the label of the external edge which is connected to the i-th external edge for 1 ≤ i ≤ 2n, and call π(F ) the link pattern of F

Summary

Introduction

Razumov and Stroganov conjectured in [5] a relation between fully packed loop configurations (short FPLs) and the ground state vector in the O(1) loop model This conjectural connection made it possible to come up with many new conjectures concerning FPLs with certain link patterns, see e. With the proof of the RazumovStroganov Conjecture in [1], it was possible to use new methods – namely the theory of wheel polynomials which was developed in [2], [9] – for solving open problems concerning FPLs. Using integral wheel polynomials, Fonseca and Zinn-Justin reproved in [4] Theorem 1.1 for the special case of one matching. 1365–8050 c 2016 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

Noncrossing matchings

Fully packed loop configurations

Wheel polynomials

The proof