# Canonical bases for cluster algebras

## Abstract

In an earlier work (Publ. Inst. Hautes Études Sci., 122 (2015), 65–168) the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi–Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral tropical points of the mirror. Further, the structure constants for the multiplication rule in this basis should be given by counting broken lines (certain combinatorial objects, morally the tropicalizations of holomorphic discs).

Here we prove the conjecture in the case of cluster varieties, where the statement is a more precise form of the Fock–Goncharov dual basis conjecture (Publ. Inst. Hautes Études Sci., 103 (2006), 1–211). In particular, under suitable hypotheses, for each the partial compactification of an affine cluster variety given by allowing some frozen variables to vanish, we obtain canonical bases for extending to a basis of Each choice of seed canonically identifies the parameterizing sets of these bases with integral points in a polyhedral cone. These results specialize to basis results of combinatorial representation theory. For example, by considering the open double Bruhat cell . in the basic affine space we obtain a canonical basis of each irreducible representation of parameterized by a set which each choice of seed identifies with the integral points of a lattice polytope. These bases and polytopes are all constructed essentially without representation-theoretic considerations. ,

Along the way, our methods prove a number of conjectures in cluster theory, including positivity of the Laurent phenomenon for cluster algebras of geometric type.

## Introduction

### 0.1. Statement of the main results

Fock and Goncharov conjectured that the algebra of functions on a cluster variety has a canonical vector space basis parameterized by the tropical points of the mirror cluster variety. Unfortunately, as shown in Reference GHK13 by the first three authors of this paper, this conjecture is usually false: in general the cluster variety may have far too few global functions. One can only expect a power series version of the conjecture, holding in the “large complex structure limit”, and honest global functions parameterized by a subset of the mirror tropical points. For the conjecture to hold as stated, one needs further affineness assumptions. Here we apply methods developed in the study of mirror symmetry, in particular *scattering diagrams*, introduced by Kontsevich and Soibelman in Reference KS06 for two dimensions and by Gross and Siebert in Reference GS11 for all dimensions, *broken lines*, introduced by Gross in Reference G09 and developed further by Carl, Pumperla, and Siebert in Reference CPS, and *theta functions*, introduced by Gross, Hacking, Keel, and Siebert, see Reference GHK11, Reference CPS, Reference GS12, and Reference GHKS, to prove the conjecture in this corrected form. We give in addition a formula for the structure constants in this basis, nonnegative integers given by counts of broken lines. Definitions of all these objects, essentially combinatorial in nature, in the context of cluster algebras will be given in later sections. Here are more precise statements of our results.

For basic cluster variety notions we follow the notation of Reference GHK13, §2, for convenience, as we have collected there a number of definitions across the literature; nothing there is original. We recall some of this notation in Appendices A and B. The various flavors of cluster varieties are all varieties of the form where , is a copy of the algebraic torus

over a field of characteristic and , is a lattice, indexed by running over a set of *seeds* (a seed being roughly an ordered basis for The birational transformations induced by the inclusions of two different copies of the torus are compositions of ).*mutations*. Fock and Goncharov introduced a simple way to dualize the mutations, and using this define the *Fock–Goncharov dual*,Footnote^{1} We write . for the tropical semifield of integers under There is a notion of the set of . points of -valued written as , This can also be viewed as being canonically in bijection with . the set of divisorial discrete valuations on the field of rational functions of , where the canonical volume form has a pole; see §2. Each choice of seed determines an identification .

^{1}

Roughly one can view the Fock–Goncharov dual as the mirror variety, but this is not always precisely the case. With some additional effort, one can make this precise “at the boundary”, but we shall not do so here.

Our main object of study is the * cluster variety with principal coefficients*, see Appendices ;A and B for notation. This comes with a canonical fibration over a torus and a canonical free action by a torus We let . The fiber . ( the identity) is the Fock–Goncharov variety (whose algebra of regular functions is the Fomin–Zelevinsky upper cluster algebra). The quotient is the Fock–Goncharov variety.

For example, is the original cluster algebra defined by Fomin and Zelevinsky in Reference FZ02a, and is the corresponding upper cluster algebra as defined in Reference BFZ05.

Given a global monomial on there is a seed , such that is a character , Because the seed . gives an identification of with we obtain an element , which we show is well-defined (independent of the open set , see Lemma );7.10. This is the * -vector* of the global monomial We show this notion of . coincides with the notion of -vector from -vectorReference FZ07 in the case; see Corollary 5.9. Let be the set of of all global monomials on -vectors Finally, we write . for the space with basis -vector i.e., ,

(where for the moment indicates the abstract basis element corresponding to ).

Fock and Goncharov’s dual basis conjecture says that is canonically identified with the vector space and so in particular , should have a canonical structure. Note that such an algebra structure is determined by its structure constants, a function -algebra

such that for fixed , for all but finitely many and

With this in mind, we have:

There is an analogue to Theorem 0.3 for (the main difference is that the *theta functions*, i.e., the canonical basis for are only defined up to scaling each individual element, and the structure constants will not in general be integers). Injectivity in (7) holds for very general , see Theorem ;7.16.

Note that (5)–(6) immediately imply:

This was conjectured by Fomin and Zelevinsky in their original paper Reference FZ02a. Positivity was obtained independently in the skew-symmetric case by [LS13] by an entirely different argument. In our proof the positivity in (1) and (6) both come from positivity in the scattering diagram, a powerful tool fundamental to the entire paper; see Theorem 1.13.

We conjecture that injectivity in (7) holds for all (without the convexity assumption). Note (7) includes the linear independence of cluster monomials, which has already been established (without convexity assumptions) for skew-symmetric cluster algebras in Reference CKLP by a very different argument. The linear independence of cluster monomials in the principal case also follows easily from our scattering diagram technology, as pointed out to us by Greg Muller; see Theorem 7.20.

When there are frozen variables, one obtains a partial compactification (where the frozen variables are allowed to take the value

Of course if

The tools necessary for the proof of Theorem 0.3 are developed in the first six sections of the paper, with the proof given in §7. This material is summarized in more detail in §0.2.

The second part of the paper turns to criteria for the full Fock–Goncharov conjecture to hold. Precisely:

We prove a number of criteria which guarantee the full Fock–Goncharov conjecture holds. One such condition, which seems to be very natural in our setup and is implied, say, by the existence of a maximal green sequence, is:

Many of the results in the second part of the paper are proved using a generalized notion of convex function or convex polytope; see §§0.3 and 0.4 for more details.

In §8.5, we turn to results on partial compactifications. We first explain how convex polytopes in our sense give rise, under suitable hypotheses, to compactifications of

We now turn to a more detailed summary of the contents of the paper.

### 0.2. Toward the main theorem

Section 1 is devoted to the construction of the fundamental tool of the paper, *scattering diagrams*. While Reference GS11 defined these in much greater generality, here they are collections of walls living in a vector space with attached functions constructed canonically from a choice of seed data. A precise definition can be found in §1.1. Here we simply highlight the main new result, Theorem 1.13, whose proof, being fairly technical, is deferred to Appendix C. This says that the functions attached to walls of a scattering diagram associated to seed data have positive coefficients. All positivity results in this paper flow from this fundamental observation, and indeed many of our arguments use this in an essential way. For the reader’s convenience, we give in §1.2 an elementary construction of the relevant scattering diagrams, drawing on the method given in Reference KS13. Since a scattering diagram depends on a choice of seed, §1.3 shows how scattering diagrams associated to mutation equivalent seeds are related. This shows that a scattering diagram has a chamber structure indexed by seeds mutation equivalent to the initial choice of seed.

In §2 we review some notions of tropicalizations of cluster varieties, showing that scattering diagrams naturally live in such tropicalizations. Indeed, the scattering diagram which is associated to a cluster variety

The collection of cones

Section 3 gives the definition of broken line, the second principal combinatorial tool of the paper. These were originally introduced in Reference G09 and developed further in Reference CPS as tropical replacements for Maslov index two disks. In Reference GHK11, they were used to define *theta functions*, which are, in principle, formal sums over all broken lines with fixed boundary conditions. The relevance of theta functions for us comes in §4. Here we show the direct relationship between scattering diagrams and the

In §5 we begin with what is another essential observation for our approach. A choice of initial seed

Though immediate from our scattering diagram methods, the result is not obvious from the original definitions; indeed, it is equivalent to the sign-coherence of

The last major ingredient in the proof of Theorem 0.3 is a formal version of the Fock–Goncharov conjecture. As mentioned above, this conjecture does not hold in general, but in §6, we show that the Fock–Goncharov conjecture holds in a formal neighborhood of the torus fiber of *large complex structure limit*. This is all one should expect from log Calabi–Yau mirror symmetry in the absence of further affineness assumptions. A crucial point, shown in the proof of Theorem 6.8, is that the expansion of

In §7 we introduce the middle cluster algebra

### 0.3. Convexity conditions

We now turn to the use of convexity conditions to prove the Fock–Goncharov conjecture in a number of different situations, as covered in §8. To motivate the concepts, let us define a *partial minimal model* of a log Calabi–Yau variety

The generalization of the cocharacter lattice

The idea for generalizing the notion of convexity is to instead make use of broken lines, which are piecewise linear paths in *convex* piecewise linear *convex* polytope *convex* cone

The examples of Reference GHK13, §7, show that for the full Fock–Goncharov conjecture to hold, we need to assume *convex* polytope. As we are unable to prove Conjecture 8.11 except in the monomial case, we use a restricted version (which happily still has wide application):

The condition that

The following theorem demonstrates the value of the EGM condition:

We note that

Returning to the role of convexity notions, we note that our formula for the structure constants

A polytope

The Fock–Goncharov conjecture is the cluster special case of Reference GHK11, Conjecture 0.6, which says (roughly) that affine log Calabi–Yau varieties with maximal boundary come in canonical dual pairs with the tropical set of one parameterizing a canonical basis of functions on the other. We can view the conjecture as having two parts: First, the vector space,

For the proof see Theorem 8.32.

### 0.4. Representation-theoretic applications

We turn to §9. Here we study features of partial compactifications coming from frozen variables. As explained in Example 0.5, these partial compactifications are often the relevant ones in representation-theoretic examples. In particular, for a partial minimal model

We shall now describe in more detail what can be proved for partial compactifications of cluster varieties coming from frozen variables. A key point is a technical but combinatorial hypothesis that *each variable has an optimized seed*; see Definition 9.1 and Lemmas 9.2 and 9.3. The main need for this hypothesis is Proposition 9.7, which states that if a linear combination of theta functions extends across a boundary divisor, then each theta function in the sum extends across the divisor. Thus the middle cluster algebra, in this case, behaves well with respect to boundary divisors. Happily, this condition holds for the cluster structures on the Grassmannian, and, for

Let us now work with the principal cluster variety