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Totally positive spaces: topology and applications by Pavel Galashin Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2019 Massachusetts Institute of Technology 2019. All rights reserved. Signature redacted A uthor ...... .................... Department of Mathematics May 3, 2019 Signature redacted Certified by........................ Alexander Postnikov Professor of Applied Mathematics Signature redacted Thesis Supervisor Accepted by .... Davesh Maulik Chairman, Department Committee on Graduate Theses C1UN5E2019 LLBRARIES ARCHIVES 77 Massachusetts Avenue Cambridge, MA 02139 MITLibraries http://Iibraries.mit.edu/ask DISCLAIMER NOTICE Due to the condition of the original material, there are unavoidable flaws in this reproduction. We have made every effort possible to provide you with the best copy available. Thank you. The images contained in this document are of the best quality available. 2 Totally positive spaces: topology and applications by Pavel Galashin Submitted to the Department of Mathematics on May 3, 2019, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract This thesis studies topological spaces arising in total positivity. Examples include the to- tally nonnegative Grassmannian Gr>o(k, n), Lusztig's totally nonnegative part (G/P)>o of a partial flag variety, Lam's compactification of the space of electrical networks, and the space of (boundary correlation matrices of) planar Ising networks. We show that all these spaces are homeomorphic to closed balls. In addition, we confirm conjectures of Postnikov and Williams that the CW complexes Gr>o(k, n) and (G/P)>o are regular. This implies that the closure of each positroid cell inside Gr>o(k, n) is homeomorphic to a closed ball. We discuss the close relationship between the above spaces and the physics of scattering amplitudes, which has served as a motivation for most of our results. In the second part of the thesis, we investigate the space of planar Ising networks. We give a simple stratification-preserving homeomorphism between this space and the totally nonnegative orthogonal Grassmannian, describing boundary correlation matrices of the pla- nar Ising model by inequalities. Under our correspondence, Kramers-Wannier's high/low temperature duality transforms into the cyclic symmetry of Gr>o(k, n). Thesis Supervisor: Alexander Postnikov Title: Professor of Applied Mathematics 3 4 Acknowledgments I would like to thank my advisor Alex Postnikov for explaining lots of beautiful mathematics to me, providing me with support and advice for the past 5 years, and for teaching me how to be more "chill". He is mostly responsible for my decision to enter the field of Algebraic Combinatorics. I am also indebted to my coauthors Thomas Lam and Pasha Pylyavskyy. I learned a great deal from each of them, and had a lot of fun collaborating with them. Next, I am grateful to Sergey Fomin for sharing his infinite wisdom with me, and for the fact that the majority of my mathematical work so far has been centered around objects and questions introduced by him and his coauthors. I thank Igor Pak, Lauren Williams, Nicolai Reshetikhin, Alexei Borodin, and Pavel Etingof for their advice and help with my job search. In addition to the above people, the works of Nima Arkani-Hamed, Allen Knutson, George Lusztig, Robert Marsh, Konni Rietsch, Jeanne Scott, David Speyer, Richard Stanley, Andrei Zelevinsky, and many others have served as a constant source of inspiration for my research. This thesis is based on joint papers with Steven Karp, Thomas Lam, and Pasha Pylyavskyy. I thank them for letting me reproduce our work here. Throughout these years, I have benefited from conversations and collaborations with many of my colleagues, including Pavle Blagojevi6, Jake Bourjaily, Cesar "Misha" Cuenca, Philippe Di Francesco, Miriam Farber, Darij Grinberg, Patricia Hersh, Sam Hopkins, Rinat Kedem, Alisa Knizel, Gaku Liu, Thomas McConville, Gleb Nenashev, Nevena Pali, Vic Reiner, Tom Roby, Hugh Thomas, Nathan Williams, and Gfinter Ziegler. Before coming to MIT, I had a wonderful time learning from my mentors Gaiane Panina, Mikhail Lifshits, Sergei Ivanov, and Anatoly Vershik, as well as my peers Vladimir Zolotov and Andrei Alpeev - the regular participants of the all night long geometry seminar held in my car. Finally, I would like to acknowledge my family and friends, most importantly, my wife and parents. Special thanks go to my wife Valentina for keeping me in check throughout my PhD journey, listening patiently to my theorems, and being supportive of my slightly irregular working schedule. I am also particularly grateful to my mother Irina for the enormous effort that she has put into my education. 5 6 Contents 1 Introduction 11 1.1 The totally nonnegative Grassmannian ..... ......... ....... 11 1.2 The totally nonnegative part of a partial flag variety ............ 12 1.3 The totally nonnegative part of the unipotent radical ............ 13 1.4 The cyclically symmetric amplituhedron .. .............. .... 14 1.5 The space of planar electrical networks ........ ......... .... 14 1.6 The space of planar Ising networks .................. ..... 15 1.7 C onnections ..... ............. .............. .... 17 1.8 O utline ............ ........................... 18 2 The cyclic shift vector field 19 2.1 Contractive flows ............................ ..... 19 2.2 The totally nonnegative Grassmannian .......... ........... 23 2.3 The cyclically symmetric amplituhedron .................... 31 2.4 The space of planar electrical networks ..................... 34 3 Regularity theorem for totally nonnegative flag varieties 39 3.1 Stars, links, and the Fomin-Shapiro atlas ..... .............. 39 3.2 Overview of the proof .... ......... ......... ......... 42 3.3 Topological results .... ....... ....... ....... ...... 45 3.4 G/P: preliminaries ....... ....... ...... ....... ..... 53 3.5 Subtraction-free parametrizations ...... ....... ...... ..... 68 3.6 Bruhat projections and total positivity ..... ..... ...... ..... 79 7 3.7 Affine Bruhat atlas for the projected Richardson stratification . ....... 90 3.8 From Bruhat atlas to Fomin-Shapiro atlas .. ..... .... ..... ... 98 3.9 The case G = SL. ...... ......... ......... ......... 101 3.10 Further directions . ....... ....... ....... ...... ..... 120 3.A Appendix: Kac-Moody flag varieties ... ..... ..... ...... ... 122 4 Ising model 127 4.1 M ain results ... ....... ...... ....... ....... ...... 128 4.2 Consequences of the main construction ...... ..... ...... .... 132 4.3 Background on the combinatorics of the totally nonnegative Grassmannian . 147 4.4 The totally nonnegative orthogonal Grassmannian ..... ....... ... 153 4.5 From the Ising model to the orthogonal Grassmannian .... ...... .. 165 4.6 Cyclic symmetry and a homeomorphism with a ball ..... ...... ... 178 4.7 Generalized Griffiths' inequalities ... ....... ....... ...... 182 4.8 Open problems and future directions ....... ...... ....... .. 190 8 List of Figures 1-1 Connections between various totally positive spaces. ............ .. 17 3-1 The map Cig for the case G = SL3 and P = B from Example 3.1.1. ...... 40 3-2 The map from Le diagrams to bounded affine permutations. ......... 107 4-1 Doubling map: an example. .............. 130 4-2 Transforming a graph G into a bipartite graph G' and a medial graph G'. 136 4-3 Some almost perfect matchings of Go. .. ........... 138 4-4 A planar graph G, its dual G*, and the corresponding plabic graph G". 139 4-5 A planar graph G, the medial graph G' and its medial strands. .. ..... 141 4-6 Uncrossing the pairs {i, j} and {i', J'}. ... ....... ...... ..... 158 4-7 The Hasse diagram of the poset P3 . ......... ......... 160 4-8 The map 0 between almost perfect matchings and minimal spinned flows. .. 176 9 10 1 Introduction Hocmenenno qeaooec ympanueaem cerno giopmy u cmanooumca wapom. 1 cmao uapom, neso6ex ympaUueaem ece c6ou .tcedanux. - iJaHHH1J Xapxic We investigate various topological spaces arising in the theory of total positivity. Each space comes equipped with a natural decomposition into cells, and is conjectured to be a regular CW complex homeomorphic to a closed ball. A regular CW complex is a topological space subdivided into cells, such that the closure of each cell is homeomorphic to a ball, and the boundary of each cell is homeomorphic to a sphere. A prototypical example of a regular CW complex is a convex polytope. We show that each of the totally nonnegative spaces introduced in Sections 1.1-1.6 is homeomorphic to a closed ball. Additionally, we show that the spaces from Sections 1.1-1.3 are regular CW complexes. As we discuss in Section 1.7, the spaces that we consider are surprisingly closely related to each other and to the physics of scattering amplitudes. This thesis is based on papers [GKL17, GKL19, GP18. 1.1 The totally nonnegative Grassmannian Let Gr(k, n) denote the Grassmannian of k-planes in R'. Postnikov fPos07] defined its totally nonnegative part Gr>o(k, n) as the set of X E Gr(k, n) whose Pl6cker coordinates arc all nonnegative. Postnikov conjectured that Gr>o(k, n) is a regular CW complex homeomorphic 11 to a closed ball. He gave a decomposition of Gr>o(k, n) into positroid cells, where each cell is specified by requiring some subset of the Plucker coordinates to be strictly positive, and requiring the rest to equal zero. Over the past decade, much work has been done towards
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