Cohomology classes of rank varieties and a counterexample to a conjecture of Liu

Pawlowski, Brendan

Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015).

Résumé

To each finite subset of a discrete grid $\mathbb{N}×\mathbb{N}$ (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 consider a degeneration of Coskun's rank varieties which contains the appropriate diagram variety as a component. Rank varieties are instances of Knutson-Lam-Speyer's positroid varieties, whose cohomology classes are represented by affine Stanley symmetric functions. We show that the cohomology class of a rank variety is in fact represented by an ordinary Stanley symmetric function.

Détail

DOI : 10.46298/dmtcs.2462