On Polytopes Arising in Cluster Algebras&Finite Frames

Abstract

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.