Positroid cluster structures from relabeled plabic graphs

Abstract

The Grassmannian is a disjoint union of open positroid varieties <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Π</mml:mi> <mml:mi>μ</mml:mi> <mml:mo>∘</mml:mo> </mml:msubsup> </mml:math> , certain smooth irreducible subvarieties whose definition is motivated by total positivity. The coordinate ring <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ℂ</mml:mi> <mml:mo>[</mml:mo> <mml:msubsup> <mml:mi>Π</mml:mi> <mml:mi>μ</mml:mi> <mml:mo>∘</mml:mo> </mml:msubsup> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> is a cluster algebra, and each reduced plabic graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Π</mml:mi> <mml:mi>μ</mml:mi> <mml:mo>∘</mml:mo> </mml:msubsup> </mml:math> determines a cluster. We study the effect of relabeling the boundary vertices of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> by a permutation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> . Under suitable hypotheses on the permutation, we show that the relabeled graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>G</mml:mi> <mml:mi>ρ</mml:mi> </mml:msup> </mml:math> determines a cluster for a different open positroid variety <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Π</mml:mi> <mml:mi>π</mml:mi> <mml:mo>∘</mml:mo> </mml:msubsup> </mml:math> . As a key step in the proof, we show that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Π</mml:mi> <mml:mi>π</mml:mi> <mml:mo>∘</mml:mo> </mml:msubsup> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Π</mml:mi> <mml:mi>μ</mml:mi> <mml:mo>∘</mml:mo> </mml:msubsup> </mml:math> are isomorphic by a nontrivial twist isomorphism. Our constructions yield a family of cluster structures on each open positroid variety, given by plabic graphs with appropriately permuted boundary labels. We conjecture that the seeds in all of these cluster structures are related by a combination of mutations and rescalings by Laurent monomials in frozen variables. We establish this conjecture for (open) Schubert and opposite Schubert varieties. As an application, we also show that for certain reduced plabic graphs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> , the “source” cluster and the “target” cluster are related by mutation and Laurent monomial rescalings.

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