  We prove that the standard Poisson structure on the Grassmannian Gr(k, n) is
invariant under the action of the Coxeter element c =(1 2 ... 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 c 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.
