Abstract
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.
Highlights
1.1 Diagram varieties A diagram is a finite subset D of Z2
One can associate a representation SpD of the symmetric group SD to a diagram D, called the Specht module of D
These generalize the usual irreducible Specht modules, which occur when D is the Young diagram of a partition; the definition for general diagrams is due to James and Peel [8]
Summary
1.1 Diagram varieties A diagram is a finite subset D of Z2. Write [n] for {1, 2, . . . , n}. One can associate a (complex) representation SpD of the symmetric group SD to a diagram D, called the Specht module of D These generalize the usual irreducible Specht modules, which occur when D is the Young diagram of a partition; the definition for general diagrams is due to James and Peel [8]. A positroid variety is defined by imposing some rank conditions on cyclic intervals of columns of matrices representing points in Grk(n), and any irreducible variety defined by such rank conditions is a positroid variety They show that the positroid variety Πf has cohomology class φ(Ff ), where Ff is the affine Stanley symmetric function of f , as defined in [12]. Conjecture 5 can fail for permutation diagrams D = D(w), and in general [Πfw] and [XD(w)] need not be equal. The cohomology class φ(Fw) − [XD(w)] is a nonnegative integer combination of Schubert classes
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