Prof. Dr. Francisco Santos Priv.-Doz. Dr. Christian Stump
Combinatorics; Algebra; Discrete Geometry
516 Geometrie 512 Algebra
In 1986, Richard Stanley associated to a finite poset two polytopes, the order and chain polytope, which geometrically reflect the combinatorics of the underlying poset. In the first part of this thesis, we develop a similar theory for double posets, defined by Malvenuto and Reutenauer. We associate to every double poset P a double order polytopes TO(P) and a double chain polytope TC(P).
Chapter 1 treats double order polytopes. We show that for compatible double posets, the facets of TO(P) correspond to alternating chains in P. Moreover, we characterize the 2-level polytopes of the form TO(P) and we establish a connection to Geissinger’s valuation polytopes. In Chapter 2 we look at the toric ideals of TO(P). In the compatible case, we obtain a quadratic Gröbner basis and a corresponding unimodular regular trianguation of TO(P), as well as a description of the complete facial structure of TO(P). Chapter 3 studies TC(P). We work in the larger class of Cayley sums of anti-blocking polytopes, for which we describe all facets and a canonical subdivision. For the special case of TC(P), this yields a unimodular triangulation, a combinatorial interpretation of the volume and, more generally, of the Ehrhart polynomial. For compatible P, we define a transfer map which relates the triangulations and Ehrhart polynomials of TO(P) and TC(P).
The central objects in the second part are real varieties that are invariant under the action of a finite reflection group. For the special case of the symmetric group, Timofte’s degree principle states that every nonempty variety that can be defined in terms of the first k elementary symmetric polynomials always intersects a k-flat of the associated reflection arrangement. Our goal is to generalize this result to arbitrary reflection groups.
Chapter 4 treats the infinite families An, Bn and Dn. For these groups we prove that every nonempty variety defined by the first k basic invariants, ordered by their degree, intersects a k-flat of the reflection arrangement. We conjecture that this holds for all irreducible reflection groups. In Chapter 5 we prove the conjecture in the case k=n-1 for arbitrary reflection groups and moreover for all k for the groups H3 and F4. Furthermore, we prove a weaker version of the conjecture. We also establish a connection to Lie groups and their invariant varieties and we prove a first result for complex reflection groups.
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