Grassmannian tree webs and cluster variables

Abstract

Homogeneous coordinate rings of Grassmannians are among the most fundamental instances of cluster algebras arising in ``nature.'' Despite their importance, however, the cluster variables of these cluster algebras are not well-understood, except in special cases. For the Grassmannian of 3-planes in n-space, Fomin and Pylyavskyy conjectured [FP16] that the cluster and frozen variables are exactly the indecomposable, non-elliptic web invariants which are arborizable. They proved that in particular, if an SL3 tensor diagram is a planar tree, its web invariant is a cluster or frozen variable. In this article, we give a new proof of their result for Gr(3, n) planar tree webs, and we generalize the result to Gr(4, n): if an SL4 tensor diagram for Gr(4, n) is a planar tree web, its web invariant is a cluster or frozen variable. Our proof uses some of the cluster quasihomomorphisms from [EZLP+23] and [EZPT+25] (namely, upper promotion and spurion promotion), as well as a new 6-leg pattern map.

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BibTeX RIS

BibTeX

@misc{li2026grassmannian,
  title = {{Grassmannian tree webs and cluster variables}},
  author = {Jian-Rong Li and Chenglu Wang and Lauren Williams},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.38398},
  eprint = {2609.38398},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.38398},
  note = {CC BY-NC-SA 4.0},
}

RIS

TY  - UNPB
TI  - Grassmannian tree webs and cluster variables
AU  - Jian-Rong Li
AU  - Chenglu Wang
AU  - Lauren Williams
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.38398
UR  - https://arxiv.org/abs/2609.38398
AB  - Homogeneous coordinate rings of Grassmannians are among the most fundamental instances of cluster algebras arising in ``nature.'' Despite their importance, however, the cluster variables of these cluster algebras are not well-understood, except in special cases. For the Grassmannian of 3-planes in n-space, Fomin and Pylyavskyy conjectured [FP16] that the cluster and frozen variables are exactly the indecomposable, non-elliptic web invariants which are arborizable. They proved that in particular, if an SL3 tensor diagram is a planar tree, its web invariant is a cluster or frozen variable. In this article, we give a new proof of their result for Gr(3, n) planar tree webs, and we generalize the result to Gr(4, n): if an SL4 tensor diagram for Gr(4, n) is a planar tree web, its web invariant is a cluster or frozen variable. Our proof uses some of the cluster quasihomomorphisms from [EZLP+23] and [EZPT+25] (namely, upper promotion and spurion promotion), as well as a new 6-leg pattern map.
ER  -

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