Chevalley operations on TNN Grassmannians

TL;DR

This paper introduces Chevalley operations on TNN Grassmannians, classifies quadratic inequalities in Plucker coordinates, and applies these operations to prove positivity properties, inequalities, and order relations in TNN matrices and Grassmannians.

Contribution

It develops Chevalley operations on TNN Grassmannians, classifies quadratic Plucker inequalities, and applies these to prove positivity, lattice structures, and order relations in TNN matrices.

Findings

Complete classification of quadratic inequalities in Plucker coordinates.

Chevalley operations motivate cluster mutation perspective.

Proof of Lam's log-supermodularity and related positivity properties.

Abstract

Lusztig showed that invertible totally nonnegative (TNN) matrices form a semigroup generated by positive diagonal matrices and Chevalley generators. From its Grassmann analogue, we introduce Chevalley operations on index sets, which we show have a rich variety of applications. We first completely classify all inequalities that are quadratic in Plucker coordinates over the TNN part of the Grassmannian: \[\sum_{I,J}c_{I,J}\Delta_I\Delta_J\ge 0\quad over\quad \mathrm{Gr}^{\ge 0}(m,m+n)\] where each $c_{I,J}$ is real, and $\Delta_I,\Delta_J$ are Plucker coordinates with a homogeneity condition. Using an idea of Gekhtman-Shapiro-Vainshtein, we also explain how our Chevalley operations can be motivated from cluster mutations, and lead to working in Grassmannians of smaller dimension, akin to cluster algebras. We then present several applications of Chevalley operations. First, we obtain certificates for the above inequalities via sums of coefficients $c_{I,J}$ over 321-avoiding permutations and involutions; we believe this refined results of Rhoades-Skandera for TNN-matrix inequalities via their Temperley-Lieb immanant idea. Second, we provide a novel proof via Chevalley operations of Lam's log-supermodularity of Plucker coordinates. This has several consequences: (a) Each positroid, corresponding to the positroid cells in Postnikov's decomposition of the TNN Grassmannian, is a distributive lattice. (b) It also yields numerical positivity in the main result of Lam-Postnikov-Pylyavskyy. (c) We show the coordinatewise monotonicity of ratios of Schur polynomials, first proved by Khare-Tao and which is the key result they use to obtain quantitative estimates for positivity preservers. Third, we employ Chevalley operations to show that the majorization order over partitions implicates a partial order for induced character immanants over TNN matrices, proved originally by Skandera-Soskin.