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Total positivity for cominuscule Grassmannians

Abstract : In this paper we explore the combinatorics of the non-negative part $(G/P)_{\geq 0}$ of a cominuscule Grassmannian. For each such Grassmannian we define Le-diagrams ― certain fillings of generalized Young diagrams which are in bijection with the cells of $(G/P)_{\geq 0}$. In the classical cases, we describe Le-diagrams explicitly in terms of pattern avoidance. We also define a game on diagrams, by which one can reduce an arbitrary diagram to a Le-diagram. We give enumerative results and relate our Le-diagrams to other combinatorial objects. Surprisingly, the totally non-negative 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.
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Thomas Lam, Lauren Williams. Total positivity for cominuscule Grassmannians. 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), 2008, Viña del Mar, Chile. pp.39-51. ⟨hal-01185168⟩



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