Of matroid polytopes, chow rings and character polynomials

Matroids are combinatorial structures that capture various notions of independence. Recently there has been great interest in studying various matroid invariants. In this thesis, we study two such invariants: Volume of matroid base polytopes and the Tutte polynomial. We gave an approach to computing volume of matroid base polytopes using cyclic flats and apply it to the case of sparse paving matroids. For the Tutte polynomial, we recover (some of) its coefficients as degrees of certain forms in the Chow ring of underlying matroid. Lastly, we study the stability of characters of the symmetric group via character polynomials. We show a combinatorial identity in the ring of class functions that implies stability results for certain class of Kronecker coefficients

On Polytopes Arising in Cluster Algebras & Finite Frames

Polytopes appear in many contexts, two being cluster algebras and finite frames. At first we study graph theoretic properties of polytopes arising in the context of cluster algebras of finite type. We introduce the basic terms and constructions for cluster algebras of finite type, then we consider their exchange graphs and give a conjecture about the Hamiltonicity of the exchange graphs. Then we study polytopes, which arise in the construction of finite frames with given lengths of frame vectors and given spectrum of the frame operator. After an introduction to finite frames, we give a non-redundant description of those polytopes for equal norm tight frames in terms of equations and inequalities. From this, we derive the dimension and number of facets of the polytopes. In this process we combinatorially obtain two isomorphisms between polytopes associated to frames. Afterwards we discuss how these isomorphisms are described by reversing the order of frame vectors and taking Naimark complements