Algebraic and combinatorial aspects of two symmetric polytopes

Equivariant Ehrhart theory is an extension of Ehrhart theory that considers lattice polytopes under group actions. Ehrhart theory tells us that the lattice points in a lattice (or rational) polytope are counted by polynomials (or quasi-polynomials). In the equivariant analog, we consider the Ehrhart theory of the subsets of the polytope fixed by the action. This first part of this thesis, a joint project with Andres Vindas, focuses on the equivariant Ehrhart theory of IIn under the action of Sn. We prove that the fixed sub-polytopes of IIn are zonotopes and are combinatorially equivalent to permutahedra. We provide vertex and hyperplane descriptions. We also compute the equivariant Ehrhart theory for 111, 112, 113, and II4. This thesis also includes work in spectral graph theory. Motivated by a tropical approach to the Hodge conjecture, we compute the spectrum of the tropical Laplacian matrix of the root polytope An, confirming a special case of a result of Babaee and Huh.