On Dyck path expansion formulas for rank 2 cluster variables
Algebraic Combinatorics, Volume 7 (2024) no. 2, pp. 529-553.

In this paper, we simplify and generalize formulas for the expansion of rank 2 cluster variables. In particular, we prove an equivalent, but simpler, description of the colored Dyck subpaths framework introduced by Lee and Schiffler. We then prove the conjectured bijectivity of a map constructed by Feiyang Lin between collections of colored Dyck subpaths and compatible pairs, objects introduced by Lee, Li, and Zelevinsky to study the greedy basis. We use this bijection along with Rupel’s expansion formula for quantum greedy basis elements, which sums over compatible pairs, to provide a quantum generalization of Lee and Schiffler’s colored Dyck subpaths formula.

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DOI: 10.5802/alco.343
Classification: 13F60
Keywords: rank two cluster algebra, maximal Dyck path, compatible pair, quantum cluster algebra
Burcroff, Amanda 1

1 Harvard University Department of Mathematics Cambridge MA 02138 (USA)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Burcroff, Amanda. On Dyck path expansion formulas for rank 2 cluster variables. Algebraic Combinatorics, Volume 7 (2024) no. 2, pp. 529-553. doi : 10.5802/alco.343. https://alco.centre-mersenne.org/articles/10.5802/alco.343/

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