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Representation Theory and Polytopes Representation Theory and Polytopes Habilitationsschrift zur Erlangung des akademischen Grades doctor rerum naturalium habilitatus (Dr. rer. nat. habil.) der Mathematisch-Naturwissenschaftlichen Fakultät der Universität Rostock. vorgelegt von Frieder Ladisch, geboren am 25. März 1977 in Karlsruhe, aus Rostock. Eingereicht: 1. Juni 2017 Verteidigt: 18. Januar 2018 Frieder Ladisch Universität Rostock Institut für Mathematik 18051 Rostock (Germany) [email protected] Research supported by the Deutsche Forschungsgemeinschaft (DFG), Project: SCHU 1503/6-1. Gutachter: Prof. Dr. rer. nat. habil. Achill Schürmann Institut für Mathematik, Universität Rostock. Peter McMullen, Emeritus Professor, Ph.D., D.Sc. University College London. Prof. Dr. rer. nat. habil. Rudolf Scharlau Fakultät für Mathematik, Technische Universität Dortmund. Eingereicht: 01. Juni 2017 Probevorlesung: 21. Dezember 2017 Kolloquium: 18. Januar 2018 2010 Mathematics Subject Classifcation: 52B15 Symmetry properties of polytopes 05E18 Group actions on combinatorial structures 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures 20C10 Integral representations of fnite groups 20C15 Ordinary representations and characters 52B05 Combinatorial properties of polytopes and polyhedra 52B20 Lattice polytopes 90C10 Integer programming Keywords and Phrases: Orbit polytope, group representation, afne symmetry, generic symmetry, repre- sentation polytope, permutation polytope, character, core point, lattice polytope, integer linear programming, Birkhof polytope, abstract regular polytope Preface The present habilitation thesis concerns applications of the representation theory of fnite groups to polytopes and their symmetries, and in particular, orbit polytopes. This is a cumulative thesis, and so the main part of this thesis consists of papers (listed on Page 23) which are already published, or submitted for publication. All papers are also available from the preprint server arXiv. The papers contained in this thesis have been written at the Institute for Mathematics of the University of Rostock between 2014 and 2017. I wish to thank the members of the Geometry group in Rostock for their collegiality and for stimulating discussions. In particular, I wish to thank Achill Schürmann for his steady support and encouragement during the writing of this thesis, and Erik Friese for the many discussions that led to our two joint papers. Since 2015, the author is supported by the Deutsche Forschungsgemeinschaft (DFG), Project SCHU 1503/6-1. Frieder Ladisch Rostock May 2017. For the publication of this thesis, the references were updated, and the comments of the referees of the corresponding papers have been incorporated (in ChaptersII, IV andVI). F. L. January 2018 iii Contents Preface iii I. Introduction1 1. Polytopes and their diferent symmetries..............2 2. Orbit polytopes............................4 3. Afne symmetries of orbit polytopes................8 4. Groups with a nontrivial ideal kernel................ 13 5. Core points.............................. 16 6. Two properties of the Birkhof polytope............... 19 7. Realizations of abstract regular polytopes.............. 20 8. Contributions to this thesis..................... 22 Papers contained in this thesis...................... 23 References................................. 24 II. Linear Symmetries of Orbits and Orbit Polytopes 31 (Erik Friese and Frieder Ladisch) 1. Introduction.............................. 31 2. Afne and linear symmetries..................... 39 3. Computing linear symmetries.................... 41 4. Generic points............................. 43 5. The generic symmetry group..................... 51 6. Representation polytopes....................... 55 7. Generic symmetries and left ideals.................. 58 8. Orbit polytopes as subsets of the group algebra.......... 59 9. Representation polytopes as subsets of the group algebra..... 64 10. Character criteria........................... 68 References................................. 71 III. Groups with a Nontrivial Nonideal Kernel 75 (Frieder Ladisch) 1. Introduction.............................. 75 2. Skew-linear characters........................ 77 3. The nonideal kernel.......................... 80 4. Dedekind groups........................... 83 5. Classifcation over the reals..................... 85 6. Classifcation over the rational numbers............... 88 iv Acknowledgment............................. 95 References................................. 95 IV. Classifcation of Orbit Symmetry Groups for some Fields 97 (Erik Friese and Frieder Ladisch) 1. Orbit polytopes of elementary abelian 2-groups.......... 97 2. Classifcation of afne symmetry groups of orbit polytopes.... 101 3. Classifcation of afne symmetry groups of rational orbit polytopes 106 4. Open questions and conjectures................... 109 Acknowledgments............................. 112 References................................. 112 V. Equivalence of Lattice Orbit Polytopes 115 (Frieder Ladisch and Achill Schürmann) 1. Introduction.............................. 115 2. Equivalence for core points...................... 117 3. Preliminaries on orders........................ 119 4. Finiteness of equivalence classes................... 121 5. Rationally irreducible......................... 124 6. Application to integer linear optimization.............. 132 References................................. 135 VI. Uniqueness of the Birkhof Polytope 139 (Barbara Baumeister and Frieder Ladisch) 1. Introduction.............................. 139 2. Preliminaries on permutation actions on a group.......... 141 3. The combinatorial symmetry group of the Birkhof polytope... 143 4. Characterization of the Birkhof polytope.............. 145 Acknowledgments............................. 147 References................................. 147 VII. Realizations of Abstract Regular Polytopes from a Representation Theoretic View 149 (Frieder Ladisch) 1. Introduction.............................. 149 2. Realizations as G-homomorphisms.................. 151 3. The structure of the realization cone................ 155 4. Counterexamples to a result of Herman and Monson........ 160 5. Orthogonality............................. 164 6. Cosine vectors and spherical functions............... 166 7. On the realizations of the 600-cell.................. 169 References................................. 173 v Chapter I. Introduction The character theory of fnite groups was invented by Ferdinand Georg Frobenius on April 12, 1896 in a letter to Richard Dedekind1, motivated by fnding the factorization of the group determinant. Soon thereafter, Frobenius introduced the general notion of a representation of an abstract group. The theory was further developed and simplifed by William Burnside, Issai Schur, Emmy Noether, and Richard Brauer, to name only a few. On the one hand, the representation theory of fnite groups is a beautiful subject in its own right. On the other hand, it was soon realized that this theory provides a powerful tool to prove theorems about fnite groups. For example, Burnside used character theory to prove that groups of order paqb (p and q primes) are solvable, or that a transitive permutation group of prime degree p is either doubly transitive, or permutation isomorphic to a proper subgroup of AGL(1, Fp), the group of afne transformations Fp → Fp. But the range of applications of representation theory is not limited to pure mathematics, and includes subjects such as physics, chemistry, engineering and statistics. Indeed, whenever some object is given, which is or can be embedded into a linear space, and when this object has symmetries, then representation theory can usually be applied in proftable ways. In this thesis, we apply this philosophy to the study of symmetry properties of polytopes. Symmetric polytopes in dimension 3 have fascinated mathematicians since antiquity, the most famous examples being the fve Platonic solids. The Platonic solids are polytopes that are “as symmetric as possible”. Making precise this statement leads to the defnition of regular polytopes in arbitrary dimension [26]. In the papers contained in this thesis, we are mostly concerned with a larger class of symmetric polytopes, namely orbit polytopes. These can be characterized as polytopes whose (linear) symmetry group acts transitively on their vertices. Equivalently, an orbit polytope is the convex hull of an orbit of a point v ∈ Rd under a fnite group G ⊂ AGL(d, R). Of course, all of the well known regular polytopes are orbit polytopes. By (one) defnition, the Archimedean solids are orbit polytopes in dimension 3 such that all faces are regular polygons (By another defnition, the pseudo-rhombicuboctahedron 1This is one of the few instances where one can assign a specifc date to the birthday of a whole mathematical theory. 1 2 I. Introduction is also an Archimedean solid, which is not vertex transitive. Indeed, many authors do not clearly distinguish these defnitions [35].) Another classical example is the permutahedron [17], which is an orbit polytope of the symmetric group Sn in its standard representation (permuting coordinates of Rn). Other permutation groups yield orbit polytopes which are important in combinatorial optimization, for example the (symmetric or asymmetric) traveling salesman polytope [6, 68]. A particularly interesting case are orbit polytopes of fnite refection groups, which are often called generalized permutahedra, or simply permutahedra [17, 43,
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