On Some Aspects of Cluster Algebras and Combinatorial Hopf Algebra

This dissertation deals with problems in cluster algebras and combinatorial Hopf algebras. Total positivity has been closely related to cluster algebras since their inception. Postnikov's totally nonnegative Grassmannian is a concrete example of total positivity with rich combinatorics. Our first problem is the computation of Plucker coordinates inside a generalization of the totally nonnegative Grassmannian. We provide a combinatorial formula in terms of edge weighted directed graphs embedded on a surface. The next problem we consider is the equality of a cluster algebra and its upper cluster algebra. Particular attention is paid to the coefficient ring of the cluster algebra. We give a sufficient condition for the cluster algebra and upper cluster algebra to coincide while allowing greater generality of coefficient ring than was previous known. The final problem we consider in cluster algebras is showing that log-canonical coordinates are as simple as possible (in a certain precise sense). Log-canonical coordinates are a fundamental part of the Poisson geometry approach to cluster algebras put forth by Gekhtman, Shapiro, and Vainshtein. In the theory of combinatorial Hopf algebras we compute a formula for the antipode in a Hopf algebra on simplicial complexes. This antipode formula generalizes Humpert and Martin's formula for graphs. We then use the character theory of Aguiar, Bergeron, and Sottile to realize a version of Stanley's chromatic symmetric function for simplicial complexes. We prove that the degree sequence of a uniform hypertree can be recovered from its chromatic symmetric function. We also show the chromatic symmetric function is not a complete invariant for uniform hypertrees.