All Questions
Tagged with ag.algebraic-geometry cluster-algebras
14 questions
33
votes
2
answers
2k
views
What do cluster algebras tell us about Grassmannians?
One of the first examples of a cluster algebra given in Fomin and Zelevinsky's original paper is the homogeneous coordinate ring $\mathbb{C}[G_{2,n}]$ of the Grassmannian of planes in $\mathbb{C}^n$. ...
- 1,690
10
votes
1
answer
3k
views
Cluster algebras and cluster varieties
I have a really basic question about cluster algebras and cluster varieties. According to the definition of Fomin-Zelevinsky a cluster algebra is generated by a bunch of polynomial rings inside the ...
- 7,037
5
votes
2
answers
533
views
Do we have super Plucker relations for a super Grassmannian?
Super Grassmannians are introduced by Manin, see for example. We have Plucker relation for Grassmannian.
Are there some references about super Plucker relations for super Grassmannian? Thank you ...
- 6,211
4
votes
0
answers
259
views
Road map for learning cluster algebras
I'm a PhD student and I would like learn about cluster algebras. I'm wondering what is a good reference (i.e., has detailed explanations, examples, etc) to learn from the basic and what are some of ...
- 839
4
votes
0
answers
259
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A technical question about a paper by Gross-Hacking-Keel
I have a technical question on the commutativity of diagrams (2.11) and (2.12) in the paper "Birational geometry of cluster algebras" by Gross-Hacking-Keel:
For the leftmost square in (2.11),...
- 61
3
votes
2
answers
362
views
Kahler differentials on cluster varieties
On affine toric varieties there is a classical theorem of Danilov which gives some combinatorial ways to describe the global sections of an appropriate sheaf of Kahler differentials as a vector space. ...
- 1,331
3
votes
0
answers
417
views
Motivic DT-Invariants for the Algebro-Geophobic
I am looking for as gentle of possible of an introduction to Kontsevich-Soibelman's theory of motivic DT-invariants. Specifically I am interested in the algebraic aspects of the theory and the ...
- 2,283
2
votes
1
answer
257
views
Grassmannian $\mathrm{Gr}(k, \pm \infty)$ in infinite dimension
$\DeclareMathOperator\Gr{Gr}$The Grassmnnian variety $\Gr(k,n)$ is the set of $k$-dimensional subspaces of $\mathbb{C}^n$. The coordinate ring $\mathbb{C}[\Gr(k,n)]$ is generated by Plucker ...
- 6,211
2
votes
1
answer
223
views
Cluster algebra structure on the coordinate ring of $Mat_3$
Let $Mat_3$ be the set of all 3 by 3 matrices. I have some questions on the cluster algebra structure on the coordinate ring of $Mat_3$.
We use $\Delta_{j_1\ldots j_n}^{i_1\ldots i_n}$ to denote the ...
- 6,211
2
votes
0
answers
116
views
What algebraic varieties arise as intersections of finitely many cluster charts?
A cluster variety $V$ admits, by definition, many charts of the form $(\mathbb{C}^*)^n \hookrightarrow V$. These charts do not always cover the variety of interest, but when they do, one could e.g. ...
- 8,723
2
votes
0
answers
74
views
Berenstein-Fomin-Zelevinsky's Ininital seeds and initial seeds from Postnikov diagrams
In Cluster algebra III by Berenstein-Fomin-Zelevinsky, Theorem 2.10, for any pair of reduced words $(u,v)$, they constructed an initial seed for the cluster algebra $\mathbb{C}[B^{u,v}]$, where $B^{u,...
- 6,211
2
votes
0
answers
137
views
Questions about cluster $\mathcal{X}$-varieties and amalgamation
I am trying to learn the amalgamation of two cluster seeds and I am reading the paper https://arxiv.org/pdf/math/0508408.pdf written by V.V. Fock and A. B. Goncharov. I am at a loss for the Lemma 2....
- 348
1
vote
1
answer
204
views
Are there some relations between F-polynomials and theta functions?
F-polynomials are certain polynomials appears in the expansion formula of a cluster variable, see for example the formula (6.5) in cluster algebras IV. Theta functions in the paper correspond to ...
- 6,211
0
votes
1
answer
262
views
What are the cluster algebra structures on $Gr(3,5)$?
In the paper, cluster algebra structures on $Gr(2,n)$, $Gr(3,6)$, $Gr(3,7)$, $Gr(3,8)$, $Gr(4,6)$ are described. But what are the cluster algebra structures on $Gr(3,5)$ (and $Gr(3,4)$)? Do we have ...
- 6,211