Cluster algebras and binary subwords

TL;DR

This paper explores the relationship between binary subwords, posets, and snake graphs within cluster algebra theory, introducing new combinatorial structures and bijections to deepen understanding of these connections.

Contribution

It introduces the antichain trie and establishes bijections linking binary subwords, posets, and perfect matchings in snake graphs, advancing combinatorial methods in cluster algebras.

Findings

Antichain trie is isomorphic to the subword trie by Leroy, Rigo, and Stipulanti

Bijections are constructed between subwords, antichains, and perfect matchings

New combinatorial structures enhance understanding of cluster algebra connections

Abstract

This paper establishes a connection between binary subwords and perfect matchings of a snake graph, an important tool in the theory of cluster algebras. Every binary expansion w can be associated to a piecewise-linear poset P and a snake graph G. We construct a tree structure called the antichain trie which is isomorphic to the trie of subwords introduced by Leroy, Rigo, and Stipulanti. We then present bijections from the subwords of w to the antichains of P and to the perfect matchings of G.