The tropical totally positive Grassmannians

Speyer, David ; Williams, Lauren

Journal of Algebraic Combinatorics, Tome 22 (2005) no. 2, pp. 189-210.

Résumé

Summary: Tropical algebraic geometry is the geometry of the tropical semiring ($\mathbb{R},min,$ +). The theory of total positivity is a natural generalization of the study of matrices with all minors positive. In this paper we introduce the totally positive part of the tropicalization of an arbitrary affine variety, an object which has the structure of a polyhedral fan. We then investigate the case of the Grassmannian, denoting the resulting fan Trop ${}^{+}G{r}_{k,n}$. We show that Trop ${}^{+}G{r}_{2,n}$ is the Stanley-Pitman fan, which is combinatorially the fan dual to the (type $A_{n-3}$) associahedron, and that Trop ${}^{+}G{r}_{3,6}$ and Trop ${}^{+}G{r}_{3,7}$ are closely related to the fans dual to the types $D_4$ and $E_6$ associahedra. These results are strikingly reminiscent of the results of Fomin and Zelevinsky, and Scott, who showed that the Grassmannian has a natural cluster algebra structure which is of types $A_{n-3},D_4$, and $E_6$ for $G{r}_{2,n},G{r}_{3,6}$, and $G{r}_{3,7}$. We suggest a general conjecture about the positive part of the tropicalization of a cluster algebra.

Détail

Mots-clés : keywords tropical geometry, total positivity, cluster algebras, generalized associahedra, Grassmannian