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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2405.14915 |
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| _version_ | 1866918278010503168 |
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| author | Ciliberti, Azzurra |
| author_facet | Ciliberti, Azzurra |
| contents | Let $\mathbf{P}_{2n+2}$ be the regular polygon with $2n+2$ vertices, and let $θ$ be the rotation of 180$^\circ$. Fomin and Zelevinsky proved that $θ$-invariant triangulations of $\mathbf{P}_{2n+2}$ are in bijection with the clusters of cluster algebras of type $B_n$ or $C_n$. Furthermore, cluster variables correspond to the orbits of the action of $θ$ on the diagonals of $\mathbf{P}_{2n+2}$. In this paper, we associate a labeled modified snake graph $\mathcal{G}_{ab}$ to each $θ$-orbit $[a,b]$, and we get the cluster variables of type $B_n$ and $C_n$ which correspond to $[a,b]$ as perfect matching Laurent polynomials of $\mathcal{G}_{ab}$. This extends the work of Musiker for cluster algebras of type B and C to every seed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_14915 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cluster expansion formulas and perfect matchings for type B and C Ciliberti, Azzurra Representation Theory Combinatorics Let $\mathbf{P}_{2n+2}$ be the regular polygon with $2n+2$ vertices, and let $θ$ be the rotation of 180$^\circ$. Fomin and Zelevinsky proved that $θ$-invariant triangulations of $\mathbf{P}_{2n+2}$ are in bijection with the clusters of cluster algebras of type $B_n$ or $C_n$. Furthermore, cluster variables correspond to the orbits of the action of $θ$ on the diagonals of $\mathbf{P}_{2n+2}$. In this paper, we associate a labeled modified snake graph $\mathcal{G}_{ab}$ to each $θ$-orbit $[a,b]$, and we get the cluster variables of type $B_n$ and $C_n$ which correspond to $[a,b]$ as perfect matching Laurent polynomials of $\mathcal{G}_{ab}$. This extends the work of Musiker for cluster algebras of type B and C to every seed. |
| title | Cluster expansion formulas and perfect matchings for type B and C |
| topic | Representation Theory Combinatorics |
| url | https://arxiv.org/abs/2405.14915 |