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

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

Collections: Mathematics