Total positivity for Grassmannians and amplituhedra

Author(s): Karp, Steven Neil | Advisor(s): Williams, Lauren K. | Abstract: Total positivity is the mathematical study of spaces and their positive parts, which can have interesting combinatorial properties as well as applications in areas such as analysis, representation theory, and theoretical physics. In this dissertation, I study total positivity in the Grassmannian Gr(k,n), which is the space of k-dimensional subspaces of R^n. The totally nonnegative Grassmannian is the subset of Gr(k,n) where all Plucker coordinates are nonnegative. In Chapter 2, I generalize a result of Gantmakher and Krein, who showed that an element V of Gr(k,n) is totally nonnegative if and only if every vector in V, when viewed as a sequence of n numbers and ignoring any zeros, changes sign at most k-1 times. I characterize when the vectors in V change sign at most k-1+m times for any m, in terms of the Plucker coordinates of V. I then apply this result to solve the problem of determining when Grassmann polytopes, generalizations of polytopes into the Grassmannian studied by Lam, are well defined. In Chapter 3, which is joint work with Lauren Williams, we study the (tree) amplituhedron A(n,k,m), the image in Gr(k,k+m) of the totally nonnegative part of Gr(k,n) under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in order to give a geometric basis for computing scattering amplitudes in N=4 supersymmetric Yang-Mills theory. We take an orthogonal point of view and define a related "B-amplituhedron" B(n,k,m), which we show is isomorphic to A(n,k,m), and use the results of Chapter 2 to describe the amplituhedron in terms of sign variation. Then we use this reformulation to give a cell decomposition of the amplituhedron in the case m=1, using the images of a collection of distinguished cells of the totally nonnegative Grassmannian. We also identify A(n,k,1) with the complex of bounded faces of a cyclic hyperplane arrangement, and deduce that A(n,k,1) is homeomorphic to a ball. In Chapter 4, I study the action of the cyclic group of order n on the totally nonnegative part of Gr(k,n). I show that the cyclic action has a unique fixed point, given by taking n equally spaced points on the trigonometric moment curve (if k is odd) or the symmetric moment curve (if k is even). More generally, I show that the cyclic action on the entire complex Grassmannian has exactly n choose k fixed points, corresponding to k-subsets of nth roots of (-1)^(k-1). I explain how these fixed points also appear in the study of the quantum cohomology ring of the Grassmannian.