Melissa Sherman-Bennett (Harvard): Many cluster structures on Schubert varieties in the Grassmannian

Scott (2006) showed that the coordinate ring of (the affine cone over) the Grassmannian is a cluster algebra. Moreover, the seeds for this cluster algebra consisting entirely of Plucker coordinates are combinatorially well-understood: they can be obtained from Postnikov’s plabic graphs for the Grassmannian. In work with K. Serhiyenko and L. Williams, we showed that Postnikov’s plabic graphs give seeds for a cluster structure on (open) Schubert varieties in the Grassmannian. However, the situation for Schubert varieties is a bit more mysterious than for the Grassmannian. In particular, plabic graphs naturally give rise to two different cluster algebras, depending on if one uses “source labeling” or “target labeling”. These two cluster algebras are both equal to the coordinate ring of the (affine cone over the open) Schubert variety, but have different frozen variables and different cluster variables. A priori, they give rise to different positive parts of the Schubert variety. I’ll discuss work with C. Fraser, in which we determine the precise relationship between these two cluster algebras: loosely, the seeds of the target cluster algebra can be rescaled by Laurent monomials in the frozens to obtain seeds of the source cluster algebra. Along the way to proving this result, we find many more cluster structures on Schubert varieties given by “generalized” plabic graphs.