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Results tagged with cluster-algebras
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Questions related to cluster algebras, a class of commutative rings introduced around 2000 by Fomin and Zelevinsky, and nearby topics.
17
votes
What do cluster algebras tell us about Grassmannians?
One simple answer is to talk about the totally positive part of $(G_{k,n})_{> 0}$, the part of the Grassmannian where all the maximal minors (=Plücker coordinates) are real and positive. Naively, if …
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16
votes
0
answers
556
views
Catalan objects associated to a univariate polynomial
Given a monic degree $n$ polynomial $f(z)$ with no double roots, and a phase $0\leq \theta < \pi$, there are natural constructions which associate to this data:
a noncrossing matching on $2n$ vertice …
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12
votes
2
answers
323
views
Easy way to understand theta basis for X-cluster algebras of finite type?
For $\mathcal A$-cluster algebras of finite type, it is very easy to describe the theta-basis: it consists of the cluster monomials. Is there any similarly easy way to describe the theta-basis for $\m …
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12
votes
1
answer
1k
views
What is a good introduction to cluster algebras from surfaces?
What is a good reference for cluster algebras from surfaces, with a view to their connection to Teichmuller theory?
In my view, that means it should start off with unpunctured surfaces (and in fact …
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10
votes
Which cluster algebras have been categorified?
Jan's answer includes many excellent references. I will try to give a few quick comments.
First of all, although the original Buan-Marsh-Reineke-Reiten-Todorov paper contained some results which wer …
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10
votes
Accepted
Is it possible that the GHKK canonical basis for cluster algebras is the Lusztig/Kashiwara d...
I think there is good reason to think the answer is "no".
In rank 2, the theta basis agrees with the greedy basis (arXiv:1508.01404). Greedy basis elements are indecomposable positive elements (see …
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6
votes
Quiver representations of type $D_n$ mutation class
The quiver given in the question has five simple modules, six which correspond to a single arrow, and the remaining representations have support as follows:
123, 124, 125, 235, 345, 1235, 12235 (note …
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4
votes
References about tropical cluster algebras and tropical Laurent phenomenon
There are certainly things related to tropical cluster algebras associated to surfaces. See Fomin, Shapiro, Thurston, https://arxiv.org/pdf/math/0608367, especially section 9.5, and the sequel by Fom …
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4
votes
Accepted
Reference request: Associahedron
The original source would be Fomin-Zelevinsky, https://arxiv.org/abs/hep-th/0111053. Note that, for them, the "associahedron" is really just a fan (the normal fan to the simple polytope associahedron …
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3
votes
Accepted
Quiver folding and maximal green sequences
This will certainly work fine in finite type. Folding $Q$ to $Q'$ corresponds to an inclusion of $W'$ into $W$, where the reflections of $W'$ are mapped to products of commuting reflections in $W$. …
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3
votes
Some interesting and elementary topics with connections to the representation theory?
One example of an elementary application of cluster algebras is the proof that the Somos-4 and Somos-5 sequences, which are defined by a simple recursion, are integral. This is so because the entries …
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