Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 033, 18 pages      arXiv:2310.10223      https://doi.org/10.3842/SIGMA.2024.033

A Laurent Phenomenon for the Cayley Plane

Oliver Daisey and Tom Ducat
Department of Mathematical Sciences, Durham University, Upper Mountjoy Campus, Stockton Road, Durham DH1 3LE, UK

Received October 22, 2023, in final form April 11, 2024; Published online April 15, 2024

Abstract
We describe a Laurent phenomenon for the Cayley plane, which is the homogeneous variety associated to the cominuscule representation of $E_6$. The corresponding Laurent phenomenon algebra has finite type and appears in a natural sequence of LPAs indexed by the $E_n$ Dynkin diagrams for $n\leq6$. We conjecture the existence of a further finite type LPA, associated to the Freudenthal variety of type $E_7$.

Key words: Laurent phenomenon; cluster structure; mirror symmetry; Cayley plane.

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