Immanants are a generalization of the well-studied determinant and permanent. Although the combinatorial interpretations for the determinant and permanent have been studied in excess, there remain few combinatorial interpretations for the immanant. The main objective of this thesis is to consider the immanant, and its possible combinatorial interpretations, in terms of recursive structures on the character. This thesis presents a comprehensive view of previous interpretations of immanants. Furthermore, it discusses algebraic techniques that may be used to investigate further into the combinatorial aspects of the immanant. We consider the Temperley-Lieb algebra and the class of immanants over the elements of this algebra. Combinatorial tools including the Temperley-Lieb algebra and Kauffman diagrams will be used in a number of interpretations. In particular, we extend some results for the permanent and determinant based on the $R$-weighted planar network construction, where $R$ is a convenient ring, by Clearman, Shelton, and Skandera. This thesis also presents some cases in which this construction cannot be extended. Finally, we present some extensions to combinatorial interpretations on certain classes of tableaux, as well as certain classes of matrices.

Immanants are a generalization of the well-studied determinant and permanent. Although the combinatorial interpretations for the determinant and permanent have been studied in excess, there remain few combinatorial interpretations for the immanant. The main objective of this thesis is to consider the immanant, and its possible combinatorial interpretations, in terms of recursive structures on the character. This thesis presents a comprehensive view of previous interpretations of immanants. Furthermore, it discusses algebraic techniques that may be used to investigate further into the combinatorial aspects of the immanant. We consider the Temperley-Lieb algebra and the class of immanants over the elements of this algebra. Combinatorial tools including the Temperley-Lieb algebra and Kauffman diagrams will be used in a number of interpretations. In particular, we extend some results for the permanent and determinant based on the $R$-weighted planar network construction, where $R$ is a convenient ring, by Clearman, Shelton, and Skandera. This thesis also presents some cases in which this construction cannot be extended. Finally, we present some extensions to combinatorial interpretations on certain classes of tableaux, as well as certain classes of matrices.

We study the geometry of immanants, which are polynomials on n^2 variables that are defined by irreducible representations of the symmetric group Sn. We compute stabilizers of immanants in most cases by computing Lie algebras of stabilizers of immanants. We also study tensor network states, which are special tensors defined by contractions. We answer a question about tensor network states asked by Grasedyck. Both immanants and tensor network states are related to the Geometric Complexity Theory program, in which one attempts to use representation theory and algebraic geometry to solve an algebraic analogue of the P versus N P problem. We introduce the Geometric Complexity Theory (GCT) program in Section one and we introduce the background for the study of immanants and tensor network states. We also explain the relation between the study of immanants and tensor network states and the GCT program. Mathematical preliminaries for this dissertation are in Section two, including multilinear algebra, representation theory, and complex algebraic geometry. In Section three, we first give a description of immanants as trivial (SL(E) x SL(F )) ><| delta(Sn)-modules contained in the space S^n(E X F ) of polynomials of degree n on the vector space E X F , where E and F are n dimensional complex vectorspaces equipped with fixed bases and the action of Sn on E (resp. F ) is induced by permuting elements in the basis of E (resp. F ). Then we prove that the stabilizer of an immanant for any non-symmetric partition is T (GL(E) x GL(F )) ><| delta(Sn) ><| Z2, where T (GL(E) x GL(F )) is the group of pairs of n x n diagonal matrices with the product of determinants equal to 1, delta(Sn) is the diagonal subgroup of Sn x Sn. We also prove that the identity component of the stabilizer of any immanant is T (GL(E) x GL(F )). In Section four, we prove that the set of tensor network states associated to a triangle is not Zariski closed and we give two reductions of tensor network states from complicated cases to simple cases. In Section five, we calculate the dimension of the tangent space and weight zero subspace of the second osculating space of GL_(n^2) .[perm_n] at the point [perm_n] and determine the Sn x Sn-module structure of this space. We also determine some lines on the hyper-surface determined by the permanent polynomial. In Section six, we give a summary of this dissertation.