Quasi-coincidence of cluster structures on positroid varieties
By work of a number of authors, beginning with Scott and culminating with Galashin and Lam, the coordinate rings of positroid varieties in the Grassmannian carry cluster algebra structures. In fact, they typically carry many such structures, the two best understood being the source-labelled and targ...
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| Main Author | |
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| Format | Journal Article |
| Language | English |
| Published |
25.08.2023
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| Subjects | |
| Online Access | Get full text |
| DOI | 10.48550/arxiv.2307.13369 |
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| Summary: | By work of a number of authors, beginning with Scott and culminating with
Galashin and Lam, the coordinate rings of positroid varieties in the
Grassmannian carry cluster algebra structures. In fact, they typically carry
many such structures, the two best understood being the source-labelled and
target-labelled structures, referring to how the initial cluster is computed
from a Postnikov diagram or plabic graph. In this article we show that these
two cluster algebra structures quasi-coincide, meaning in particular that a
cluster variable in one structure may be expressed in the other structure as
the product of a cluster variable and a Laurent monomial in the frozen
variables. This resolves a conjecture attributed to Muller and Speyer from
2017. The proof depends critically on categorification: of the relevant cluster
algebra structures by the author, of perfect matchings and twists by the author
with \c{C}anakç{ı} and King, and of quasi-equivalences of cluster algebras
by Fraser-Keller. By similar techniques, we also show that Muller-Speyer's left
twist map is a quasi-cluster equivalence from the target-labelled structure to
the source-labelled structure. |
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| DOI: | 10.48550/arxiv.2307.13369 |