From i-boxes to signed words

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.