Gentle algebras arising from surfaces with orbifold points, Part II: Locally free Caldero-Chapoton functions
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| author | Labardini-Fragoso, Daniel Mou, Lang |
| author_facet | Labardini-Fragoso, Daniel Mou, Lang |
| contents | We prove that in the skew-symmetrizable cluster algebras associated by Felikson-Shapiro-Tumarkin to unpunctured surfaces with orbifold points of order $2$ and a specific choice of weights, the Laurent expansion of any cluster variable with respect to any cluster coincides with the locally free Caldero-Chapoton function of a $τ$-rigid representation of a gentle algebra. These cluster algebras are typically non-acyclic and of infinite type, whereas for polygons with one orbifold point one recovers cluster algebras of finite type $C$; so, our result is an ample extension of a seminal result established by Geiss-Leclerc-Schröer for skew-symmetrizable cluster algebras of finite type and acyclic initial seeds. As the main means to achieve the result, we provide a generalization of Derksen-Weyman-Zelevinsky's mutation theory of loop-free quivers with potential to the quivers-with-loops with potential we associate to the triangulations of unpunctured surfaces with orbifold points, and study the relation with $τ$-tilting theory.
As a result of independent interest, we compute the aforementioned $τ$-rigid representations explicitly. To this end, we show that the indecomposable $τ$-rigid string modules arising from arcs on the surface, and the quasi-simple band modules arising from simple closed curves, are well-behaved under the mutations of representations we define in the paper, thus extending results of the first author's Ph.D. thesis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_16061 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Gentle algebras arising from surfaces with orbifold points, Part II: Locally free Caldero-Chapoton functions Labardini-Fragoso, Daniel Mou, Lang Representation Theory 13F60, 16G20 We prove that in the skew-symmetrizable cluster algebras associated by Felikson-Shapiro-Tumarkin to unpunctured surfaces with orbifold points of order $2$ and a specific choice of weights, the Laurent expansion of any cluster variable with respect to any cluster coincides with the locally free Caldero-Chapoton function of a $τ$-rigid representation of a gentle algebra. These cluster algebras are typically non-acyclic and of infinite type, whereas for polygons with one orbifold point one recovers cluster algebras of finite type $C$; so, our result is an ample extension of a seminal result established by Geiss-Leclerc-Schröer for skew-symmetrizable cluster algebras of finite type and acyclic initial seeds. As the main means to achieve the result, we provide a generalization of Derksen-Weyman-Zelevinsky's mutation theory of loop-free quivers with potential to the quivers-with-loops with potential we associate to the triangulations of unpunctured surfaces with orbifold points, and study the relation with $τ$-tilting theory. As a result of independent interest, we compute the aforementioned $τ$-rigid representations explicitly. To this end, we show that the indecomposable $τ$-rigid string modules arising from arcs on the surface, and the quasi-simple band modules arising from simple closed curves, are well-behaved under the mutations of representations we define in the paper, thus extending results of the first author's Ph.D. thesis. |
| title | Gentle algebras arising from surfaces with orbifold points, Part II: Locally free Caldero-Chapoton functions |
| topic | Representation Theory 13F60, 16G20 |
| url | https://arxiv.org/abs/2309.16061 |