 
Summary: CYCLICITY OF LUSZTIG'S STRATIFICATION OF
GRASSMANNIANS AND POISSON GEOMETRY
MILEN YAKIMOV
Abstract. We prove that the standard Poisson structure on the Grass
mannian 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 Knut
son, Lam, and Speyer that the Coxeter element interchanges the Lusztig strata
of Gr(k, n). We also relate the main result to known antiinvariance properties
of the standard Poisson structures on cominuscule flag varieties.
1. Introduction
For the purpose of the study of canonical bases, Lusztig defined [4] the totally
nonnegative part (G/P ) #0 of an arbitrary complex flag variety G/P . He also
constructed an algebrogeometric stratification of G/P and conjectured that in
tersecting this stratification with (G/P ) #0 is producing a cell decomposition of
(G/P ) #0 . This was latter proved by Rietsch in [5]. Both the nonnegative part
(G/P ) #0 and the Lusztig stratification of a flag variety were studied in recent
years from many di#erent combinatorial and Lie theoretic points of view.
In a recent work Knutson, Lam, and Speyer proved that the Lusztig strat
ification of the Grassmannian Gr(k, n) has a remarkable cyclicity property. If
