Tropicalization of positive Grassmannians

Abstract

We introduce combinatorial objects which are parameterized by the positive part of the tropical Grassmannian $\mathrm{Gr}(k,n)$. Our method is to relate the Grassmannian to configuration spaces of flags. By work of the first author, and of Goncharov and Shen, configuration spaces of flags naturally tropicalize to give configurations of points in the affine building, which we call higher laminations. We use higher laminations to give two dual objects that are parameterized by the positive tropicalization of $\mathrm{Gr}(k,n)$: equivalence classes of higher laminations; or certain restricted subset of higher laminations. This extends results of Speyer and Sturmfels on the tropicalization of $\mathrm{Gr}(2,n)$, and of Speyer and Williams on the tropicalization of $\mathrm{Gr}(3,6)$ and $\mathrm{Gr}(3,7)$. We also analyze the ${\mathcal {X}}$-variety associated to the Grassmannian, and give an interpretation of its positive tropicalization.

Appendix: Cluster algebras

We review here the basic definitions of cluster algebras [4]. Cluster algebras are commutative algebras that come equipped with a collection of distinguished set of algebra generators, called cluster variables or ${\mathcal {A}}$-coordinates. These generators are grouped in specific subsets known as clusters. A given cluster does not generate the algebra, but each cluster is a transcendence basis for the field of fractions. There is a procedure known as mutation which replaces a given cluster with a new one.

Each cluster belongs to a seed, which roughly consists of the cluster together with the extra underlying data of a B-matrix (see below). The B-matrix prescribes how to mutate from one seed to any adjacent seed. Starting from an initial choice of seed, all other seeds (and hence, all clusters, and all cluster variables) can be obtained recursively by applying a sequence of mutations. Because mutation is involutive, the resulting pattern of clusters and cluster variables does not depend on the choice of initial seed.

Each cluster provides a coordinate system on the ${\mathcal {A}}$-space. The same combinatorial data underlying a seed gives rise to a second, related, algebra of functions, the algebra of ${\mathcal {X}}$-coordinates. Just as before, the ${\mathcal {X}}$-coordinates come in specified groupings that mutate when passing from one seed to another. The ${\mathcal {X}}$-coordinates are functions on the ${\mathcal {X}}$ space. The ${\mathcal {A}}$-coordinates and ${\mathcal {X}}$-coordinates are related by a canonical monomial transformation, which gives a map from the ${\mathcal {A}}$-space to the ${\mathcal {X}}$-space. Together, the data of the ${\mathcal {A}}$-space and the ${\mathcal {X}}$-space, along with their distinguished sets of coordinates, is called a cluster ensemble.

Now we give more precise definitions. A seed $\Sigma = (I,I_0,B,d)$ consists of the following data:

• (1) An index set I with a subset $I_0 \subset I$ of “frozen” indices.

• (2) A rational $I \times I$ exchange matrix B. It should have the property that $b_{ij} \in {\mathbb {Z}}$ unless both i and j are frozen.

• (3) A set $d = \{d_i \}_{i \in I}$ of positive integers that skew-symmetrize B; that is, $b_{ij}d_j = -b_{ji}d_i$ for all $i,j \in I$.

For most purposes, the values of $d_i$ are only important up to simultaneous scaling. Also note that the values of $b_{ij}$ where i and j are both frozen will play no role in the cluster algebra, though it is sometimes convenient to assign values to $b_{ij}$ for bookkeeping purposes.

In the simplest case, B is skew-symmetric, i.e. each $d_i = 1$. In this case, the seed is given simply by the data $\Sigma = (I,I_0,B)$. Moreover, we can depict the B-matrix by a quiver (as was done in previous sections). This quiver will have vertices labelled by the set I. When $b_{ij} > 0$, we will have $b_{ij}$ arrows going from j to i.

Let $k \in I {\setminus } I_0$ be an unfrozen index of a seed $\Sigma$. We say another seed $\Sigma ' = \mu _k(\Sigma )$ is obtained from $\Sigma$ by mutation at k if we identify the index sets in such a way that the frozen variables are preserved, and the exchange matrix $B'$ of $\Sigma '$ satisfies

$$\begin{aligned} b'_{ij} = {\left\{ \begin{array}{ll} -b_{ij} &{}\quad i = k \text { or } j=k \\ b_{ij} &{}\quad b_{ik}b_{kj} \le 0 \\ b_{ij} + |b_{ik}|b_{kj} &{}\quad b_{ik}b_{kj} > 0. \end{array}\right. } \end{aligned}$$

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We can view the procedure of mutation of a seed as follows. We consider the arrows involving the vertex k. For every pair of arrows such that one arrow goes into k and one arrow goes out of k, we compose these arrows to get a new arrow. Then we cancel arrows in opposite directions. Finally, we reverse all the arrows going in or out of k. Two seeds $\Sigma$ and $\Sigma '$ are said to be mutation equivalent if they are related by a finite sequence of mutations.

To a seed $\Sigma$ we associate a collection of cluster variables $\{A_i\}_{i \in I}$ and a split algebraic torus ${\mathcal {A}}_\Sigma := {\text {Spec}}{\mathbb {Z}}[A^{\pm 1}_I]$, where ${\mathbb {Z}}[A^{\pm 1}_I]$ denotes the ring of Laurent polynomials in the cluster variables. If $\Sigma '$ is obtained from $\Sigma$ by mutation at $k \in I {\setminus } I_0$, there is a birational cluster transformation $\mu _k: {\mathcal {A}}_\Sigma \rightarrow {\mathcal {A}}_{\Sigma '}$. This is defined by the exchange relation

$$\begin{aligned} \mu _k^*(A'_i) = {\left\{ \begin{array}{ll} A_i &{} i \ne k \\ A_k^{-1}\biggl (\prod \nolimits _{b_{kj}>0}A_j^{b_{kj}} + \prod \nolimits _{b_{kj}<0}A_j^{-b_{kj}}\biggr ) &{} i = k. \end{array}\right. } \end{aligned}$$

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Composing these transformations yields gluing data between any tori ${\mathcal {A}}_\Sigma$ and ${\mathcal {A}}_{\Sigma '}$ of mutation equivalent seeds $\Sigma$ and $\Sigma '$. The ${\mathcal {A}}$-space ${\mathcal {A}}_{|\Sigma |}$ is defined as the scheme obtained from gluing together all such tori of seeds mutation equivalent with an initial seed $\Sigma$.

Given a seed $\Sigma$ we also associate a second algebraic torus ${\mathcal {X}}_\Sigma := {\text {Spec}}{\mathbb {Z}}[X_I^{\pm 1}]$, where ${\mathbb {Z}}[X_I^{\pm 1}]$ again denotes the Laurent polynomial ring in the variables $\{X_i\}_{i \in I}$. If $\Sigma '$ is obtained from $\Sigma$ by mutation at $k \in I {\setminus } I_0$, we again have a birational map $\mu _k: {\mathcal {X}}_\Sigma \rightarrow {\mathcal {X}}_{\Sigma '}$. It is defined by

$$\begin{aligned} \mu _k^*(X'_i) = {\left\{ \begin{array}{ll} X_iX_k^{[b_{ik}]_+}(1+X_k)^{-b_{ik}} &{} i \ne k \\ X_k^{-1} &{} i = k, \end{array}\right. } \end{aligned}$$

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where $[b_{ik}]_+:=\mathrm {max}(0,b_{ik})$. The ${\mathcal {X}}$-space ${\mathcal {X}}_{|\Sigma |}$ is defined as the scheme obtained from gluing together all such tori of seeds mutation equivalent with an initial seed $\Sigma$.

Now we will describe the natural map from ${\mathcal {A}}_{\Sigma }$ to ${\mathcal {X}}_{\Sigma }$. Let us assume that the entries of the B-matrix are all integers. Then we can define $p: {\mathcal {A}}_\Sigma \rightarrow {\mathcal {X}}_\Sigma$ by

$$\begin{aligned} p^*(X_i) = \prod _{j \in I}A_j^{b_{ij}}. \end{aligned}$$

This formula appears to depend on the seed, but it actually intertwines the mutation of both the ${\mathcal {A}}$-coordinates and the ${\mathcal {X}}$-coordinates. In other words, if $\Sigma '$ is obtained from $\Sigma$ by mutation at k, there is a commutative diagram

So the maps ${\mathcal {A}}_{\Sigma }$ to ${\mathcal {X}}_{\Sigma }$ glue to give a map ${\mathcal {A}}_{|\Sigma |}$ to ${\mathcal {X}}_{|\Sigma |}$.