An affine almost positive roots model
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2017
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| _version_ | 1866914558546804736 |
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| author | Reading, Nathan Stella, Salvatore |
| author_facet | Reading, Nathan Stella, Salvatore |
| contents | We generalize the almost positive roots model for cluster algebras from finite type to a uniform finite/affine type model. We define the almost positive Schur roots $Φ_c$ and a compatibility degree, given by a formula that is new even in finite type. The clusters define a complete fan $\operatorname{Fan}_c(Φ)$. Equivalently, every vector has a unique cluster expansion. We give a piecewise linear isomorphism from the subfan of $\operatorname{Fan}_c(Φ)$ induced by real roots to the ${\mathbf g}$-vector fan of the associated cluster algebra. We show that $Φ_c$ is the set of denominator vectors of the associated acyclic cluster algebra and conjecture that the compatibility degree also describes denominator vectors for non-acyclic initial seeds. We extend results on exchangeability of roots to the affine case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1707_00340 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | An affine almost positive roots model Reading, Nathan Stella, Salvatore Combinatorics Rings and Algebras Representation Theory 13F60 We generalize the almost positive roots model for cluster algebras from finite type to a uniform finite/affine type model. We define the almost positive Schur roots $Φ_c$ and a compatibility degree, given by a formula that is new even in finite type. The clusters define a complete fan $\operatorname{Fan}_c(Φ)$. Equivalently, every vector has a unique cluster expansion. We give a piecewise linear isomorphism from the subfan of $\operatorname{Fan}_c(Φ)$ induced by real roots to the ${\mathbf g}$-vector fan of the associated cluster algebra. We show that $Φ_c$ is the set of denominator vectors of the associated acyclic cluster algebra and conjecture that the compatibility degree also describes denominator vectors for non-acyclic initial seeds. We extend results on exchangeability of roots to the affine case. |
| title | An affine almost positive roots model |
| topic | Combinatorics Rings and Algebras Representation Theory 13F60 |
| url | https://arxiv.org/abs/1707.00340 |