Cluster algebras were conceived by Fomin and Zelevinsky (1) in the spring of 2000 as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. However, the theory of cluster algebras has since taken on a life of its own, as connections and applications have been discovered in diverse areas of mathematics, including representation theory of quivers and finite dimensional algebras, cf., for example, refs. 2⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–15; Poisson geometry (16⇓⇓–19); Teichmuller theory (20⇓⇓⇓–24); string theory (25⇓⇓⇓⇓⇓–31); discrete dynamical systems and integrability (6, 32⇓⇓⇓⇓⇓–38); and combinatorics (39⇓⇓⇓⇓⇓⇓⇓–47).

Quite remarkably, cluster algebras provide a unifying algebraic and combinatorial framework for a wide variety of phenomena in these and other settings. We refer the reader to the survey papers (36, 48⇓⇓⇓⇓–53) and to the cluster algebras portal (www.math.lsa.umich.edu/~fomin/cluster.html) for various introductions to cluster algebras and their links with other subjects in mathematics (and physics).

In brief, a cluster algebra A of rank k is a subring of an ambient field ℱ of rational functions in k variables, say x 1, …, x k . Unlike most commutative rings, a cluster algebra is not presented at the outset via a complete set of generators and relations. Instead, from the data of the initial seed — which includes the k initial cluster variables x 1, …, x k , plus an exchange matrix — one uses an iterative procedure called “mutation” to produce the rest of the cluster variables. In particular, each new cluster … 

[↵][1]1To whom correspondence should be addressed. Email: williams{at}math.berkeley.edu.

 [1]: #xref-corresp-1-1

https://www.pnas.org/content/pnas/111/27/9676.full.pdf

Cluster algebras

In [GHK11], Conjecture 0.6, 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 tropicalisations 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, [FG06], Conjecture 4.3. In particular, under suitable hypotheses, for each Y the partial compactification of an affine cluster variety U given by allowing some frozen variables to vanish, we obtain canonical bases for H0(Y,OY ) extending to a basis of H0(U,OU ). 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 U in the basic affine space Y we obtain a canonical basis of each irreducible representation of SLr, parameterized by a set which each choice of seed identifies with 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.

/pdf/canonical-bases-for-cluster-algebras-f4gz0rr2up.pdf

Canonical bases for cluster algebras

In previous work, 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 tropicalisations 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. In particular, under suitable hypotheses, for each Y the partial compactification of an affine cluster variety U given by allowing some frozen variables to vanish, we obtain canonical bases for the ring of functions on Y extending to a basis for functions on U. 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 U in the basic affine space Y we obtain a canonical basis of each irreducible representation of SL_r, parameterized by a set which each choice of seed identifies with 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.

We prove the positivity conjecture for all skew-symmetric cluster algebras.

https://annals.math.princeton.edu/wp-content/uploads/Lee_Schiffler.pdf

Positivity for cluster algebras

We give a geometric interpretation of cluster varieties in terms of blowups of toric varieties. This enables us to provide, among other results, an elementary geometric proof of the Laurent phenomenon for cluster algebras (of geometric type), extend Speyer’s example [Spe13] of upper cluster algebras which are not nitely generated, and show that the Fock-Goncharov dual basis conjecture is usually false. Contents

/pdf/birational-geometry-of-cluster-algebras-20j297acc9.pdf

Birational Geometry of Cluster Algebras

A lot of recent activity in the theory of cluster algebras has been directed toward various constructions of “natural” bases in them. One of the approaches to this problem was developed several years ago by Sherman and Zelevinsky who have shown that the indecomposable positive elements form an integer basis in any rank 2 cluster algebra of finite or affine type. It is strongly suspected (but not proved) that this property does not extend beyond affine types. Here, we go around this difficulty by constructing a new basis in any rank 2 cluster algebra that we call the greedy basis. It consists of a special family of indecomposable positive elements that we call greedy elements. Inspired by a recent work of Lee and Schiffler; Rupel, we give explicit combinatorial expressions for greedy elements using the language of Dyck paths.

Greedy elements in rank 2 cluster algebras

Some degenerations of Kazhdan–Lusztig ideals and multiplicities of Schubert varieties

This article concerns properties of mixed $$\ell $$
 -adic complexes on varieties over finite fields, related to the action of the Frobenius automorphism. We establish a fiberwise criterion for the semisimplicity and Frobenius semisimplicity of the direct image complex under a proper morphism of varieties over a finite field. We conjecture that the direct image of the intersection complex on the domain is always semisimple and Frobenius semisimple; this conjecture would imply that a strong form of the decomposition theorem of Beilinson–Bernstein–Deligne–Gabber is valid over finite fields. We prove our conjecture for (generalized) convolution morphisms associated with partial affine flag varieties for split connected reductive groups over finite fields. As a crucial tool, we develop a new schematic theory of big cells for loop groups. With suitable reformulations, the main results are valid over any algebraically closed ground field.

Frobenius semisimplicity for convolution morphisms

The higher q,t-Catalan polynomial $C^{(m)}_{n}(q,t)$ can be defined combinatorially as a weighted sum of lattice paths contained in certain triangles, or algebraically as a complicated sum of rational functions indexed by partitions of n. This paper proves the equivalence of the two definitions for all m?1 and all n≤4. We also give a bijective proof of the joint symmetry property $C^{(m)}_{n}(q,t)=C^{(m)}_{n}(t,q)$ for all m?1 and all n≤4. The proof is based on a general approach for proving joint symmetry that dissects a collection of objects into chains, and then passes from a joint symmetry property of initial points and terminal points to joint symmetry of the full set of objects. Further consequences include unimodality results and specific formulas for the coefficients in $C^{(m)}_{n}(q,t)$ for all m?1 and all n≤4. We give analogous results for certain rational-slope q,t-Catalan polynomials.

https://link.springer.com/content/pdf/10.1007%2Fs10801-013-0466-5.pdf

Combinatorics of certain higher q,t-Catalan polynomials: chains, joint symmetry, and the Garsia---Haiman formula

We identify a quantum lift of the greedy basis for rank 2 coefficient-free cluster algebras. Our main result is that our construction does not depend on the choice of initial cluster, that it builds all cluster monomials, and that it produces bar-invariant elements. We also present several conjectures related to this quantum greedy basis and the triangular basis of Berenstein and Zelevinsky.

https://www.pnas.org/content/pnas/111/27/9712.full.pdf

Li Li

Papers

Greedy elements in rank 2 cluster algebras

Some degenerations of Kazhdan–Lusztig ideals and multiplicities of Schubert varieties

Frobenius semisimplicity for convolution morphisms

Combinatorics of certain higher q,t-Catalan polynomials: chains, joint symmetry, and the Garsia---Haiman formula

Greedy bases in rank 2 quantum cluster algebras.