Positivity for cluster algebras

Lee, Kyungyong; Schiffler, Ralf

doi:10.4007/annals.2015.182.1.2

Cited by 131 publications

(113 citation statements)

References 29 publications

Supporting

Mentioning

109

Contrasting

Order By: Relevance

“…For example, it is a direct consequence of our result that the positivity conjecture for skew-symmetric cluster algebras as proved by Lee and Schiffler in [13] holds for cluster algebras of infinite rank.…”

Section: Theorem 1 Every Rooted Cluster Algebra Of Infinite Rank Can supporting

confidence: 56%

Cluster algebras of infinite rank as colimits

Gratz

2015

Math. Z.

View full text Add to dashboard Cite

We formalize the way in which one can think about cluster algebras of infinite rank by showing that every rooted cluster algebra of infinite rank can be written as a colimit of rooted cluster algebras of finite rank. Relying on the proof of the posivity conjecture for skew-symmetric cluster algebras (of finite rank) by Lee and Schiffler, it follows as a direct consequence that the positivity conjecture holds for cluster algebras of infinite rank. Furthermore, we give a sufficient and necessary condition for a ring homomorphism between cluster algebras to give rise to a rooted cluster morphism without specializations. Assem, Dupont and Schiffler proposed the problem of a classification of ideal rooted cluster morphisms. We provide a partial solution by showing that every rooted cluster morphism without specializations is ideal, but in general rooted cluster morphisms are not ideal.Comment: Included cluster algebras of uncountable rank, fixed some typos. Results on the countable case unchanged, comments appreciate

show abstract

Section: Theorem 1 Every Rooted Cluster Algebra Of Infinite Rank Can supporting

confidence: 56%

Cluster algebras of infinite rank as colimits

Gratz

2015

Math. Z.

View full text Add to dashboard Cite

show abstract

“…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.…”

Section: Ord(v ) ⊂ Mid(v ) ⊂ Up(v )mentioning

confidence: 99%

Canonical bases for cluster algebras

Gross¹,

Hacking²,

Keel³

et al. 2017

J. Amer. Math. Soc.

363

968

View full text Add to dashboard Cite

Abstract. 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 H 0 (Y, O Y ) extending to a basis of H 0 (U, O 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.

show abstract

“…This is called the Laurent phenomenon. The coefficients are now known to be non-negative [25,30], but this had been an open problem for more than ten years. Several different bases have been constructed for cluster algebras of different types; see [4,22,25,29,33].…”

Section: What Is a Cluster Algebra?mentioning

confidence: 99%