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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 $$\chi\left( \text{Gr}_{\left( e_{1},e_{2}\right) }\left( M\left( n\right) \right) \right)$$.

##### MSC:
 13F60 Cluster algebras 16G20 Representations of quivers and partially ordered sets
##### Keywords:
cluster algebras; Laurent polynomials
Macaulay2
Full Text:
##### References:
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