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PreprintCC BY 4.0arXiv

Linear Laurent phenomenon algebras are of finite type

  • Oliver Daisey
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Abstract

Laurent phenomenon algebras generalise cluster algebras by allowing irreducible exchange polynomials in place of binomials. Lam and Pylyavskyy proved finite type for those arising from graphs and conjectured that the same holds for every Laurent phenomenon algebra admitting a linear seed. We prove this conjecture over an arbitrary unique factorisation domain. Our proof constructs a nonzero element of the upper algebra divisible by every cluster variable. Comparing noninitial cluster variables with the prime factors of this element in the initial Laurent ring bounds the number of cluster variable classes and yields a finite normalised mutation pattern.

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Cite

BibTeXRIS
BibTeX
@misc{daisey2026linear,
  title = {{Linear Laurent phenomenon algebras are of finite type}},
  author = {Oliver Daisey},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17441},
  eprint = {2610.17441},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17441},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Linear Laurent phenomenon algebras are of finite type
AU  - Oliver Daisey
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.17441
UR  - https://arxiv.org/abs/2610.17441
AB  - Laurent phenomenon algebras generalise cluster algebras by allowing irreducible exchange polynomials in place of binomials. Lam and Pylyavskyy proved finite type for those arising from graphs and conjectured that the same holds for every Laurent phenomenon algebra admitting a linear seed. We prove this conjecture over an arbitrary unique factorisation domain. Our proof constructs a nonzero element of the upper algebra divisible by every cluster variable. Comparing noninitial cluster variables with the prime factors of this element in the initial Laurent ring bounds the number of cluster variable classes and yields a finite normalised mutation pattern.
ER  -