From i-boxes to signed words

TL;DR

This paper connects the combinatorics of i-boxes with signed words to efficiently determine exchange matrices in cluster algebras related to quantum affine algebra representations.

Contribution

It introduces a method to associate signed words to chains of i-boxes, enabling explicit computation of exchange matrices in cluster algebra frameworks.

Findings

Provides a canonical association between i-box chains and signed words

Enables quick determination of exchange matrices for i-box chains

Bridges combinatorics of i-boxes with cluster algebra seeds

Abstract

The combinatorics of i-boxes has recently been introduced by Kashiwara--Kim--Oh--Park in the study of cluster algebras arising from the representation theory of quantum affine algebras. In this article, we associate to each chain of i-boxes a signed word, which canonically determines a cluster seed following Berenstein--Fomin--Zelevinsky. By bridging these two different languages, we are able to provide a quick solution to the problem of explicit determining the exchange matrices associated with chains of i-boxes.