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Cohomology classes of interval positroid...
Pawlowski, Brendan...

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Cohomology classes of interval positroid varieties and a conjecture of Liu by Pawlowski, Brendan ( Author )
Lemonjar Open Access
Australian National University
06-09-2023
To each finite subset of $\mathbb{Z}^2$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture. However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class $\sigma$ is at least an upper bound on the actual class $\tau$, in the sense that $\sigma - \tau$ is a nonnegative linear combination of Schubert classes. To do this, we exhibit the appropriate diagram variety as a component in a degeneration of one of Knutson's interval positroid varieties (up to Grassmann duality). A priori, the cohomology classes of these interval positroid varieties are represented by affine Stanley symmetric functions. We give a different formula for these classes as ordinary Stanley symmetric functions, one with the advantage of being Schur-positive and compatible with inclusions between Grassmannians.
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Article
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29.34 KB
English
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MYR 0.01
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http://arxiv.org/abs/1410.7419
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