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arXiv · 0710.2932

Total positivity for cominuscule Grassmannians

Thomas Lam·Lauren Williams
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Abstract

In this paper we explore the combinatorics of the non-negative part (G/P)+ 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)+. 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. 2007-10-16. Total positivity for cominuscule Grassmannians. https://arxiv.org/abs/0710.2932

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