Mixing
Cluster algebra associated to a d-gon
Recently I have been doing some reading on cluster algebras, for example this. When defining the cluster algebras associated to a $d$-gon, they claim that the cluster and coefficient variables of $A_{d−3}$ are in bijection with the diagonals and sides of the $d$-gon, and the clusters are in bijection with triangulations of the $d$-gon, but I'm not really sure how to prove it and I don't think this is simply just the matter of counting.
I'd appreciate any ideas or references, or both.
abstract-algebra cluster-algebra
asked Nov 22 '18 at 11:34
amator2357amator2357
76
add a comment |
Recently I have been doing some reading on cluster algebras, for example this. When defining the cluster algebras associated to a $d$-gon, they claim that the cluster and coefficient variables of $A_{d−3}$ are in bijection with the diagonals and sides of the $d$-gon, and the clusters are in bijection with triangulations of the $d$-gon, but I'm not really sure how to prove it and I don't think this is simply just the matter of counting.
I'd appreciate any ideas or references, or both.
abstract-algebra cluster-algebra
asked Nov 22 '18 at 11:34
amator2357amator2357
76
1
Section 2.2 of the linked paper is a proof of the property you describe, if you're having problem with an aspect of that proof you should describe more clearly which step you're having trouble with.
– Christopher
Nov 23 '18 at 16:53
By definition, the initial cluster comes from a chosen triangulation. The fact that clusters are in bijection with triangulations come from the fact that "flipping" a diagonal corresponds to mutation, and the fact that all triangulations are connected by a sequence of "flips".
– Nick
Dec 27 '18 at 19:56
add a comment |
Recently I have been doing some reading on cluster algebras, for example this. When defining the cluster algebras associated to a $d$-gon, they claim that the cluster and coefficient variables of $A_{d−3}$ are in bijection with the diagonals and sides of the $d$-gon, and the clusters are in bijection with triangulations of the $d$-gon, but I'm not really sure how to prove it and I don't think this is simply just the matter of counting.
I'd appreciate any ideas or references, or both.
abstract-algebra cluster-algebra
asked Nov 22 '18 at 11:34
amator2357amator2357
76
Recently I have been doing some reading on cluster algebras, for example this. When defining the cluster algebras associated to a $d$-gon, they claim that the cluster and coefficient variables of $A_{d−3}$ are in bijection with the diagonals and sides of the $d$-gon, and the clusters are in bijection with triangulations of the $d$-gon, but I'm not really sure how to prove it and I don't think this is simply just the matter of counting.
I'd appreciate any ideas or references, or both.
abstract-algebra cluster-algebra
abstract-algebra cluster-algebra
asked Nov 22 '18 at 11:34
amator2357amator2357
76
asked Nov 22 '18 at 11:34
amator2357amator2357
76
asked Nov 22 '18 at 11:34
amator2357amator2357
76
asked Nov 22 '18 at 11:34
amator2357amator2357
76
asked Nov 22 '18 at 11:34
amator2357amator2357
76
76
1
Section 2.2 of the linked paper is a proof of the property you describe, if you're having problem with an aspect of that proof you should describe more clearly which step you're having trouble with.
– Christopher
Nov 23 '18 at 16:53
By definition, the initial cluster comes from a chosen triangulation. The fact that clusters are in bijection with triangulations come from the fact that "flipping" a diagonal corresponds to mutation, and the fact that all triangulations are connected by a sequence of "flips".
– Nick
Dec 27 '18 at 19:56
add a comment |
1
Section 2.2 of the linked paper is a proof of the property you describe, if you're having problem with an aspect of that proof you should describe more clearly which step you're having trouble with.
– Christopher
Nov 23 '18 at 16:53
By definition, the initial cluster comes from a chosen triangulation. The fact that clusters are in bijection with triangulations come from the fact that "flipping" a diagonal corresponds to mutation, and the fact that all triangulations are connected by a sequence of "flips".
– Nick
Dec 27 '18 at 19:56
1
1
Section 2.2 of the linked paper is a proof of the property you describe, if you're having problem with an aspect of that proof you should describe more clearly which step you're having trouble with.
– Christopher
Nov 23 '18 at 16:53
Section 2.2 of the linked paper is a proof of the property you describe, if you're having problem with an aspect of that proof you should describe more clearly which step you're having trouble with.
– Christopher
Nov 23 '18 at 16:53
By definition, the initial cluster comes from a chosen triangulation. The fact that clusters are in bijection with triangulations come from the fact that "flipping" a diagonal corresponds to mutation, and the fact that all triangulations are connected by a sequence of "flips".
– Nick
Dec 27 '18 at 19:56
By definition, the initial cluster comes from a chosen triangulation. The fact that clusters are in bijection with triangulations come from the fact that "flipping" a diagonal corresponds to mutation, and the fact that all triangulations are connected by a sequence of "flips".
– Nick
Dec 27 '18 at 19:56
add a comment |
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Section 2.2 of the linked paper is a proof of the property you describe, if you're having problem with an aspect of that proof you should describe more clearly which step you're having trouble with.
– Christopher
Nov 23 '18 at 16:53
By definition, the initial cluster comes from a chosen triangulation. The fact that clusters are in bijection with triangulations come from the fact that "flipping" a diagonal corresponds to mutation, and the fact that all triangulations are connected by a sequence of "flips".
– Nick
Dec 27 '18 at 19:56