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Hopf Monoids and the Theory of Valuations


Many families of combinatorial objects have natural ''merging" and ''breaking" operators which endow the family with a Hopf algebra structure. The theory of Hopf monoids is a refinement of the Hopf algebras that appear in combinatorics that are better equipped to deal with labeled objects and produce stronger results. Since their inception, these objects have unified various disparate results in combinatorics. In this dissertation, we present three results in this direction.

In the second chapter, we initiate the study of an extension of Hopf monoids to the category of posets. These objects model combinatorial families with a poset structure compatible with the Hopf structure. This compatibility is best understood as the multiplication maps and comultiplication maps forming an adjunction - a fact generalizing the restriction-induction adjunction of the representations of the symmetric group. With this, many aspects of Hopf monoids, such as duality, primitives, and the antipode, simplify to purely poset-theoretic aspects such as Galois connections, Mobius inversion, and characteristic polynomials. With this method, we give new calculations of the primitives of hypergraphs, set partitions, and simplicial complexes as well as giving formulas for their antipodes.

The third chapter presents joint work with Ardila that gives a new Hopf-theoretic framework for the theory of valuations on generalized permutahedra. Valuations are functions on polytopes which generalize measures. In particular, they satisfy certain inclusion-exclusion relations coming from polytope subdivisions. Using this new framework, we give a powerful method for constructing valuations from simpler ones. With this, we recover many of the previously studied valuations on matroid polytopes as well as construct new valuations for matroid polytopes, poset cones, and nestohedra. Of particular note, we show that many algebro-geometric invariants arising from the theory of wonderful compactifications of hyperplane arrangements are valuations. These include the Kazhdan-Lusztig polynomial of a matroid, the motivic zeta functions of a hyperplane arrangement, and the Eur's volume polynomial of the combinatorial Chow ring of a matroid.

In the final chapter, we discuss the relationship between Hopf monoids and the type A_n root system as well as efforts to extend Hopf monoids to arbitrary root systems. While properly defining Coxeter Hopf monoid is currently out-of-reach, we present joint work with Eur and Supina that describes a universal valuation for Coxeter matroids. In the type A_n case, this valuation is a Hopf monoid morphism - a fact which is responsible for the relationship between Hopf monoids and valuations. Extending this result to arbitrary root systems would lead to new applications in Coxeter combinatorics and in the geometry of the flag varieties of arbitrary Lie type.

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