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A combinatorial formula for rank 2 cluster variables. (English) Zbl 1266.13017
Let \(r\) be a positive integer and \(x_{1},x_{2}\) be indeterminates. Then we can construct a sequence \(\left\{ x_{n}\right\} \subset\mathbb{Q}\left( x_{1},x_{2}\right) \) recursively by \(x_{n}=\left( x_{n}^{r}+1\right) /x_{n-1}\). The subring \(\mathbb{Q}\left( x_{1},x_{2}\right) \) generated by \(\left\{ x_{n}\right\} \) is called a (rank 2) cluster algebra, and the \(x_{n}\)’s are called cluster variables. In general (rank not necessarily \(2\), more general recurrences), it is an open problem to give explicit computable formulas for the \(x_{n}\)’s in terms of its initial variables; the Positivity Conjecture states that they are Laurent polynomials with positive coefficients.
In the work under review, the authors provide a formula for \(x_{n},\;n\geq4\). The expression is in terms of subpaths of the maximal Dyck path on a rectangle whose size depends on \(r\) and \(n\). Previously, P. Caldero and A. Zelevinsky [Mosc. Math. J. 6, No. 3, 411–429 (2006; Zbl 1133.16012)] obtained an expression for \(x_{n}\) in terms of the Euler-Poincaré characteristic of the variety Gr\(_{\left( e_{1},e_{2}\right) }\left( M\left( n\right) \right) \). Their expression has a similar form to the formula above. Thus, as an application, the formula above is used to give a combinatorial formula for \(\chi\left( \text{Gr}_{\left( e_{1},e_{2}\right) }\left( M\left( n\right) \right) \right)\).

13F60 Cluster algebras
16G20 Representations of quivers and partially ordered sets
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