Generalizations of Total Positivity

Abstract

The theory of total positivity was classically concerned with totally nonnegative matrices (matrices with all nonnegative minors). These matrices appear in many varied areas of mathematics including probability, asymptotic representation theory, algebraic and enumerative combinatorics, and linear algebra. However, motivated by surprising positivity properties of Lusztig's canonical bases for quantum groups, the field of total positivity has more recently grown to include other totally nonnegative varieties. We first discuss results regarding immanants on the space of k-positive matrices (matrices where all minors of size up to k are positive). Immanants are functions on square matrices generalizing the determinant and permanent. Kazhdan--Lustzig immanants, defined by Rhoades and Skandera, are of particular interest, as elements of the dual canonical basis of the coordinate ring of GL_n(C) can be expressed as Kazhdan--Lustzig immanants. Results of Stembridge, Haiman, and Rhoades--Skandera show that Kazhdan--Lustzig immanants are nonnegative on totally nonnegative matrices. Here, we give conditions on v in S_n so that the Kazhdan-Lusztig immanant corresponding to v is positive on k-positive matrices. We then consider a space that arises from the study of totally nonnegative Grassmannians. Postnikov's plabic graphs in a disk are used to parametrize these spaces. In recent years plabic graphs have found numerous applications in math and physics. One of the key features of the theory is the fact that if a plabic graph is reduced, the face weights can be uniquely recovered from boundary measurements. On surfaces more complicated than a disk this property is lost. In this thesis, we undertake a comprehensive study of a certain semi-local transformation of weights for plabic networks on a cylinder that preserve boundary measurements. We call this a plabic R-matrix. We show that plabic R-matrices have underlying cluster algebra structure, generalizing work of Inoue--Lam--Pylyavskyy. Special cases of transformations we consider include geometric R-matrices appearing in Berenstein--Kazhdan theory of geometric crystals, and also certain transformations appearing in a recent work of Goncharov--Shen.