Total positivity for cominuscule Grassmannians

By Thomas Lam, Lauren K. Williams. 2007

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In this paper we explore the combinatorics of the nonneg- ative part (G/P )≥0 of a cominuscule Grassmannian. For each such Grassmannian we define Γ — certain fillings of generalized Young diagrams which are in bijection with the cells of (G/P )≥0 .I n the classical cases, we describe Γ explicitly in terms of pattern avoidance. We also define a game on diagrams, by which one can reduce an arbitrary diagram to a Γ We give enumerative results and relate our Γ 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.

Subfields: Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4483143796b4f5db98aa8e4a1c8d37b72698edc5