# Questions tagged [cluster-algebra]

Use this tag for questions about constructively deﬁned commutative rings equipped with a distinguished set of generators grouped into overlapping subsets of the same ﬁnite cardinality.

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### Dimers from Postnikov diagrams

I am reading several papers on Postnikov diagrams, dimer models, and quivers. From the Postnikov diagram, we can draw a dimer model and an ice quiver. My question is as follows. Does every dimer model(...
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### How to describe the partition of $GL_3(\mathbb{C})$ by Bruhat decomposition accurately?

We know that $GL_n(\mathbb{C})$ can be decomposed as $GL_n(\mathbb{C}) = \bigsqcup_{w \in W}BwB$ where $B$ is the subgroup of upper triangular invertible matrices and $W$ is the Weyl group isomorphic ...
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### What are the generating elements of a cluster algebra?

I'm trying to get into cluster algebras, and so have been looking at the basic stuff to get the foundations settled. A problem I've run into is that I've come across two (seemingly contradictory) ...
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### What are the good reading books to learn cluster algebra?

What are the good reading books to learn cluster algebra? I need a basic introductory books or notes in particular. I do not have any physics background and I want a book which starts with graph ...
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### projective basis of a $P^2$ projective space

I'm working on Grassmanians, specifically on the Plucker embeding $p:G_{d,V}\rightarrow P(\wedge^2(V))$ where $V$ is a $k$-vector space 3-dimensional. Say {v_1,v_2,v_3} is a basis for $V$. Then we can ...
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### Cluster algebra of finite type

It is proved in the paper that a cluster algebra is of finite type if its Cartan counter part of the principal part of its seeds is a Cartan matrix of finite type. If the initial quiver of a cluster ...
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### Reference request: bounded derived categories and their Auslander-Reiten quivers

I have some knowledge of Auslander-Reiten theory, tilting theory, derived categories and triangulated categories though I still find most proofs using derived categories in "Tilting Theory and Cluster ...
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### Mutations of an $A_n$ quiver to reach $A_n$ straight orientation

(1)For a quiver $A_n$ of arbitrary orientation does there exist a finite sequence of quiver mutations that can mutate it to $A_n$ straight orientation $1\rightarrow 2\rightarrow\cdots\rightarrow n$ or ...
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### How to understand that minors are matrix elements in fundamental representations of $SL_n$?

In the video, Lecture 3 of June 14, 49:00-53:00, it is said that "minors are matrix elements in fundamental representations of $SL_n$". What are fundamental representations of $SL_n$? How to ...
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I have a question about the definition of a seed of a cluster algebra. It is said that a seed is a pair $(R, u)$, where $R$ is a quiver with $n$ vertices, $u = \{u_1, \ldots, u_n\}$ is a free ...