 

Thomas Lam and
Lauren Williams
Total positivity for cominuscule Grassmannians


Published: 
February 6, 2008 
Keywords: 
Total positivity, Grassmannian, CW complexes 
Subject: 
Primary 05Exx; Secondary 20G20, 14Pxx 


Abstract
In this paper we explore the combinatorics of the nonnegative part
(G/P)_{≧ 0} of a cominuscule Grassmannian. For each such
Grassmannian we define \Lediagrams  certain fillings of
generalized Young diagrams which are in bijection with the cells of
(G/P)_{≧ 0}. In the classical cases, we describe \Lediagrams
explicitly in terms of pattern avoidance. We also define a game on
diagrams, by which one can reduce an arbitrary diagram to a
\Lediagram. We give enumerative results and relate our
\Lediagrams to other combinatorial objects. Surprisingly, the
totally nonnegative cells in the open Schubert cell of the odd and
even orthogonal Grassmannians are (essentially) in bijection with
preference functions and atomic preference functions respectively.


Acknowledgements
T. L. was partially supported by NSF grant DMS0600677. L. W. was partially supported by an NSF postdoctoral fellowship.


Author information
Department of Mathematics, Harvard University, Cambridge, MA 02138 USA
tfylam@math.harvard.edu lauren@math.harvard.edu

