A geometric realization of socle-projective categories for posets of type 𝔻

Abstract

We introduce posets of type 𝔻, a family of posets described by an admissible Dynkin quiver of type D_n together with a compatible set of extra arrows (alien arrows), and show that their socle-projective representation category is of finite representation type. Continuing, in the same spirit, a program initiated for posets of type 𝔸 [R. Schiffler, R-J. Serna, A geometric realization of socle-projective categories for posets of type 𝔸, J. Pure Appl. Algebra 224 (2020), no. 12, 106436], we build on an existing geometric model for cluster-tilted algebras of type D_n, whose geometric combinatorics is more intricate than that of type 𝔸. Our main result establishes a Bbbk-linear categorical equivalence Θ between a full subcategory (𝒞/T)_F of the arc category of a punctured (n+3)-gon, whose objects are certain sp-arcs, and the category of finitely generated socle-projective Bbbk𝒫-modules, for 𝒫 a poset of type 𝔻. As a consequence, when the set of alien arrows is empty, we conclude that the cluster subalgebra generated by the sp-arcs coincides with the full cluster algebra.

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BibTeX

@misc{rios2026geometric,
  title = {{A geometric realization of socle-projective categories for posets of type $\mathbbD$}},
  author = {Gabriel Bravo Rios and Ralf Schiffler and Robinson-Julian Serna},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25174},
  eprint = {2609.25174},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25174},
}

RIS

TY  - UNPB
TI  - A geometric realization of socle-projective categories for posets of type 𝔻
AU  - Gabriel Bravo Rios
AU  - Ralf Schiffler
AU  - Robinson-Julian Serna
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.25174
UR  - https://arxiv.org/abs/2609.25174
AB  - We introduce posets of type 𝔻, a family of posets described by an admissible Dynkin quiver of type D_n together with a compatible set of extra arrows (alien arrows), and show that their socle-projective representation category is of finite representation type. Continuing, in the same spirit, a program initiated for posets of type 𝔸 [R. Schiffler, R-J. Serna, A geometric realization of socle-projective categories for posets of type 𝔸, J. Pure Appl. Algebra 224 (2020), no. 12, 106436], we build on an existing geometric model for cluster-tilted algebras of type D_n, whose geometric combinatorics is more intricate than that of type 𝔸. Our main result establishes a Bbbk-linear categorical equivalence Θ between a full subcategory (𝒞/T)_F of the arc category of a punctured (n+3)-gon, whose objects are certain sp-arcs, and the category of finitely generated socle-projective Bbbk𝒫-modules, for 𝒫 a poset of type 𝔻. As a consequence, when the set of alien arrows is empty, we conclude that the cluster subalgebra generated by the sp-arcs coincides with the full cluster algebra.
ER  -

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