Fachbereich Mathematik und Informatik der Freien Universität Berlin Double posets and real invariant varieties Two interactions between combinatorics and geometry Dissertation eingereicht von Tobias Friedl Berlin 2017 Advisor and first reviewer: Prof. Dr. Raman Sanyal Second reviewer: Prof. Dr. Francisco Santos Third reviewer: Priv.-Doz. Dr. Christian Stump Date of the defense: May 19, 2017 Acknowledgements My deepest thanks go to my advisor Raman Sanyal. A PhD-student can only hope for an advisor who is as dedicated and enthusiastic about mathematics as you are. Thank you for getting your hands dirty and spending many hours in front of the blackboard teaching me how to do research. I spent the last years in the amazing and inspiring work group directed by Günter Ziegler at FU Berlin. Thank you for providing such a welcoming and challenging work environment. I really enjoyed the time with my friends and colleagues at the "villa", most of all Francesco Grande, Katharina Jochemko, Katy Beeler, Albert Haase, Lauri Loiskekoski, Philip Brinkmann, Nevena Palić and Jean-Philippe Labbé. Thanks also to Elke Pose, who provides valuable support and keeps bureaucracy at a low level. I want to thank my coauthor Cordian Riener for many fruitful mathematical discussions and for an interesting and enjoyable week of research in Helsinki at Aalto University. Moreover, I’d like to express my gratitude towards my coauthor and friend Tom Chappell. Thanks to the reviewers Paco Santos and Christian Stump for their helpful com- ments and to Christian additionally for many discussions regarding the combinatorics of reflection groups and posets throughout the last years. I also want to thank Chris- tian Haase, Florian Frick, Michał Lason and Achill Schürmann for their inspiring input. During the past years I have received funding from several sources. In chronolog- ical order, I was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013), ERC grant agreement no 247029, the Dahlem Research School at FU Berlin and the DFG-Collaborative Re- search Center, TRR 109 "Discretization in Geometry and Dynamics". I also want to thank the Berlin Mathematical School for lots of mathematical input and for funding my research trip to Korea. I am very grateful to my girlfriend Maren Ring. Thank you for everything, including listening to and thinking about my mathematical problems whenever I couldn’t get them out of my head. Finally, I’d like to thank my parents for their support and for always being by my side when I needed them. v vi Introduction Mathematics is often most intriguing when we leave the boarders of classical disci- plines and instead connect different areas. Such correspondences will help us to shed new light on mathematical questions, as they allow the translation of problems and techniques between different mathematical languages. The topic of this thesis is, simply said, located in combinatorics, algebra and discrete and real algebraic geometry. The thesis consists of two parts. In Part I, we study polytopes associated to posets and double posets. These polytopes are very combinatorial in nature and much of their geometry can be described in terms of the combinatorics of the underlying (double) poset. Our main objects of study in Part II are real varieties invariant under the action of a finite reflection group and, in particular, relations to the discrete geometry of the associated reflection arrange- ment. In the following, we will give a concise overview over both topics. Rigorous definitions will be given later on. A finite partially ordered set (or poset, for short) is a finite set P together with a reflexive, transitive and anti-symmetric relation . The notion of partial order pervades all of mathematics and the enumerative and algebraic combinatorics of posets is underlying in computations in virtually all areas. Stanley [73] studied two convex polytopes for every poset P which, in quite different ways, geometrically capture combinatorial properties of P . These "poset polytopes" are very natural objects that appear in a variety of contexts in combinatorics and beyond (see for instance [2, 26, 55, 71, 80]). The order polytope O(P ) is the set of all order-preserving functions into the interval [0; 1]. That is, all functions f : P ! R such that 0 ≤ f(a) ≤ f(b) ≤ 1 for all a; b 2 P with a b. Many properties of P are encoded in the geometry of O(P ). Facets of O(P ) correspond to cover relations and maximal and minimal elements in P and the vertices of O(P ) are in bijection with order ideals in P . More generally, faces of arbitrary dimension relate to quotients of P , that is, posets ob- tained by consecutively contracting cover relations in P . In particular, faces of order polytopes are again order polytopes. Stanley [73] moreover describes a canonical unimodular triangulation of O(P ) whose simplices arise from chains of order ideals. Maximal simplices correspond to linear extensions of P , which also yields a simple combinatorial formula for the volume of O(P ). This bridge between geometry and combinatorics can, for example, be used to show that computing volume is hard (cf. [11]) and, conversely, geometric inequalities can be used on partially ordered vii viii sets (see [56, 73]). Generalizing the volume formula, even the Ehrhart polynomial P EhrO(P )(n) = jnO(P )\Z j can be described in terms of P : Up to a shift, it coincides with the order polynomial ΩP (n) of P (see [74, Sec. 3.15.2]). The chain polytope C(P ) is the collection of functions g : P ! R such that 0 ≤ g(a1) + g(a2) + ··· + g(ak) ≤ 1 for all chains a1 ≺ a2 ≺ · · · ≺ ak in P . In contrast to the order polytope, C(P ) does not determine P . In fact, C(P ) only depends on the comparability graph of P and bears strong relations to stable set polytopes of perfect graphs. Order and chain polytopes are closely related. Stanley defined a piecewise linear homeomorphism ΦP between O(P ) and C(P ) whose domains of linearity are exactly the simplices of the canonical triangulation. This transfer map takes the canonical triangulation of O(P ) to a triangulation of C(P ) and, since ΦP is lattice-preserving, it follows that EhrO(P )(n) = EhrC(P )(n). In particular, vol(O(P )) = vol(C(P )), which, on the combinatorial side, shows that the number of linear extensions of P only depends on its comparability graph. A double poset P is a triple consisting of a finite ground set P and two partial order relations + and − on P . We will write P+ = (P; +) and P− = (P; −) to refer to the two underlying posets. Double posets were introduced by Malvenuto and Reutenauer [60], generalizing Stanley’s labelled posets [72]. The combinatorial study of general double posets gained momentum in recent years with a focus on algebraic aspects (see for example [24, 25]). Our goal is to build a bridge to geometry by introducing "two double poset polytopes" that, like the chain- and the order polytope, geometrically reflect the combinatorial properties of double posets and, in particular, the interaction between the two partial orders. To any double poset P, we associate the double order polytope TO(P) := conv (2O(P+) × f1g) [ (−2O(P−) × {−1g) : P In other words, we embed 2O(P+) and −2O(P−) at heights 1 and −1 in R × R, respectively, and take their convex hull. Analogously, the double chain polytope is TC(P) := conv (2C(P+) × f1g) [ (−2C(P−) × {−1g) : Both polytopes are full-dimensional and the vertices of TO(P) (resp. TC(P)) are in bijection with filters (resp. antichains) in P+ and P−. Analogous to the case of ordi- nary poset polytopes, we aim for a combinatorial description of faces, triangulations, volume and Ehrhart polynomials of TO(P) and TC(P). Also, it is natural to ask whether Stanley’s transfer map extends to the case of double posets (see Figure 1). Our main results are the following. After treating basics regarding posets and double posets in Section 1.1 and order polytopes in Section 1.2, Section 1.3 is devoted to studying the facet structure of double order polytopes. We show that facets of TO(P) relate to alternating chains and alternating cycles in the underlying double poset P. For the important case of compatible double posets, that is, double posets where P+ and P− have a common linear extension, this yields a complete description of the facets of TO(P) in terms of the combinatorics of P. Furthermore, we determine which of these polytopes are 2-level, a geometric property which arises, for instance, in the context of centrally symmetric polytopes, optimization and statistics [36, 37, INTRODUCTION ix ΦP+ 2O(P+) 2C(P+) ? ΦP − −2O(P−) −2C(P−) Figure 1: Stanley’s transfer map relates the grey top and bottom facets. Does this extend to the whole double poset polytopes? 41, 68, 69, 78]. Moreover, using polars we prove a surprising connection between certain double order polytopes and valuation polytopes, another class of polytopes associated to distributive lattices, that was introduced by Geissinger [32]. Chapter 2 is again devoted to the study of double order polytopes, but we take a more algebraic perspective. Every integral polytope P has an associated toric ideal IP , an algebraic object that captures much of the geometry of P. For example, Sturmfels’ correspondence [77] relates initial ideals of IP to regular triangulations of P. We recall some of these well-known basics in Section 2.1. In Section 2.2, we apply this machinery to double order polytopes. The toric ideals corresponding to ordinary order polytopes are closely related to Hibi rings, which have been introduced in the context of algebras with straightening laws.