Cyclicity of Lusztig’s Stratification of Grassmannians and Poisson Geometry

We prove that the standard Poisson structure on the Grassmannian Gr(k, n) is invariant under the action of the Coxeter element c = (12 . . . n). In particular, its symplectic foliation is invariant under c. As a corollary, we obtain a second, Poisson geometric proof of the result of Knutson, Lam, and Speyer that the Coxeter element interchanges the Lusztig strata of Gr(k, n). We also relate the main result to known anti-invariance properties of the standard Poisson structures on cominuscule flag varieties.