From $\boldsymbol{i}$-boxes to signed words

Alessandro Contu, Fan Qin and Qiaoling Wei

Vol. 342 (2026), No. 1, 63–78

DOI: 10.2140/pjm.2026.342.63

Abstract

The combinatorics of $i$-boxes has recently been introduced by Kashiwara, Kim, Oh and 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 and Zelevinsky. By bridging these two different languages, we are able to provide a quick solution to the problem of explicitly determining the exchange matrices associated with chains of $i$-boxes.

Keywords

cluster algebra, quantum affine algebra, monoidal categorification

Mathematical Subject Classification

Primary: 13F60

Secondary: 17B37

Milestones

Received: 17 August 2025

Accepted: 21 January 2026

Published: 23 March 2026