Discrete geometry is a field of mathematics which encompasses the study of polyhedra, or intersections of closed half-spaces in some vector space. Algebraic combinatorics employs algebraic methods to learn about combinatorial objects. This thesis lies at the intersection of these two fields and traverses back and forth between them. We use algebra to prove results about polyhedra, and use polyhedral geometry to classify functions that are of interest in algebraic combinatorics. It is divided into four chapters: an introduction, and three sections of original results which each use different algebraic techniques.
In the first chapter, we provide some background on polyhedra and establish the main actors of this thesis, generalized permutahedra. Along the way we introduce orbit polytopes and matroids as examples, and we also show how the combinatorics of generalized permutahedra can be extended to arbitrary finite reflection groups to obtain generalized Coxeter permutahedra and Coxeter matroids. We further give some background on combinatorial Hopf monoids, Ehrhart theory, and valuations that hints at the various approaches explored in this thesis.
In the second chapter, we introduce the Hopf monoid of orbit polytopes, which is generated by the generalized permutahedra that are invariant under the action of the symmetric group. We show that modulo normal equivalence, these polytopes are in bijection with integer compositions. We interpret their Hopf structure through this lens, and we show that applying the first Fock functor to this Hopf monoid gives a Hopf algebra of compositions. We describe the character group of the Hopf monoid of orbit polytopes in terms of noncommutative symmetric functions, and we give a combinatorial interpretation of the basic character and its polynomial invariant.
In Chapter 3, we explore equivariant Ehrhart theory, a generalization of Ehrhart theory introduced by Stapledon that counts the lattice points in a polytope up to the action of some symmetry group. We describe the equivariant Ehrhart theory of the permutahedron, and we explore the relationship between the properties of its equivariant Ehrhart series and the permutahedral toric variety. Answering a question of Stapledon, we describe the equivariant Ehrhart theory of the permutahedron, and we prove his Effectiveness Conjecture in this special case.
In the last chapter, we give the universal valuative invariant of Coxeter matroids. In the process, we also describe the universal valuation of generalized Coxeter permutahedra. This chapter builds off of prior work of Derksen and Fink, but many of their techniques do not apply to the Coxeter setting, requiring new methods to extend their results. We borrow tools from the theory of 0-Hecke algebras to establish properties of Coxeter Schubert matroids.
