Chevalley operations on TNN Grassmannians
Prateek Kumar Vishwakarma
TL;DR
The article introduces Chevalley operations on Grassmannian index sets to classify all homogeneous quadratic determinantal inequalities that hold on the totally nonnegative Grassmannian Gr^{≥0}(m,m+n). By relating these operations to cluster mutations and leveraging a dimension-reduction mechanism, the authors establish a complete reduction-based classification and produce multiple applications: certificates via sums over 321-avoiding permutations, an alternate proof of log-supermodularity of Plücker coordinates with several lattice and positivity consequences, and a new perspective on majorization-immanant relations for totally nonnegative matrices. The work connects to Temperley–Lieb immanants, 321-avoiding combinatorics, and Lam–Postnikov–Pylyavskyy positivity results, while hinting at deeper cluster-algebraic interpretations and future directions in dual canonical-basis phenomena. Overall, the Chevalley-operations framework provides a unifying, reduction-driven approach to quadratic Plücker inequalities with broad combinatorial and algebraic implications.
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: \[\sum_{I,J}c_{I,J}Δ_IΔ_J\ge 0\quad over\quad \mathrm{Gr}^{\ge 0}(m,m+n)\] where each $c_{I,J}$ is real, and $Δ_I,Δ_J$ 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.