New characterizations of partition functions using connection matrices

In this thesis we expand upon a line of research pioneered by Freedman, Lovász and Schrijver, and Szegedy, that uses algebraic methods to characterize families of partition functions. We introduce two new types of partition functions: skew partition functions and mixed partition functions. We give two algebraic characterizations of skew partition functions and we show that a mixed partition functions satisfy certain algebraic relationships that are related to the invariant of the symmetric group and to the invariant theory of the Orthosymplectic Supergroup. We furthermore give a characterization of real-valued partition functions for 3-graphs and for virtual link diagrams in terms of positive semidefiniteness of the associated connection matrices.