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Abstract We initiate the study of a class of polytopes, which we coinpolypositroids, defined to be those polytopes that are simultaneously generalized permutohedra (or polymatroids) and alcoved polytopes. Whereas positroids are the matroids arising from the totally nonnegative Grassmannian, polypositroids are “positive” polymatroids. We parametrize polypositroids using Coxeter necklaces and balanced graphs, and describe the cone of polypositroids by extremal rays and facet inequalities. We introduce a notion of$$(W,c)$$-polypositroidfor a finite Weyl groupWand a choice of Coxeter elementc. We connect the theory of$$(W,c)$$-polypositroids to cluster algebras of finite type and to generalized associahedra. We discussmembranes, which are certain triangulated 2-dimensional surfaces inside polypositroids. Membranes extend the notion of plabic graphs from positroids to polypositroids.
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This content will become publicly available on December 20, 2025
Associahedra as moment polytopes
Abstract Generalized associahedra are a well‐studied family of polytopes associated with a finite‐type cluster algebra and a choice of starting cluster. We show that the generalized associahedra constructed by Padrol, Palu, Pilaud, and Plamondon, building on ideas from Arkani‐Hamed, Bai, He, and Yan, can be naturally viewed as moment polytopes for an open patch of the quotient of the cluster ‐variety with universal coefficients by its maximal natural torus action. We prove our result by showing that the construction of Padrol, Palu, Pilaud, and Plamondon can be understood on the basis of the way that moment polytopes behave under symplectic reduction.
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- Award ID(s):
- 2100785
- PAR ID:
- 10644743
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Bulletin of the London Mathematical Society
- Volume:
- 57
- Issue:
- 2
- ISSN:
- 0024-6093
- Format(s):
- Medium: X Size: p. 551-564
- Size(s):
- p. 551-564
- Sponsoring Org:
- National Science Foundation
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