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Results tagged with cluster-algebras
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user 468
Questions related to cluster algebras, a class of commutative rings introduced around 2000 by Fomin and Zelevinsky, and nearby topics.
12
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2
answers
323
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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 …
- 306
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 …
- 10.5k
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 …
- 6,327
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 …
- 6,327
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 …
- 6,327
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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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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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 $W$. …
- 6,327
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 …
- 6,327
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 …
- 6,327
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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