This dissertation concerns a particular basis for the coordinate ring of the character variety of a surface. Let G be a connected reductive linear algebraic group, and let S be a surface whose fundamental group pi is a free group. Then the coordinate ring C[Hom(pi,G)] of the homomorphisms from pi to G is isomorphic to C[G^r]=C[G]^{tensor r} for some r>=0. The coordinate ring C[G] may be identified with the ring of matrix coefficients of the maximal compact subgroup of G. Therefore, the coordinate ring on the character variety, which is also the ring of invariants C[Hom(pi,G)]^G, may be described in terms of the matrix coefficients of the maximal compact subgroup….

*Contents: Trace Diagrams, Representations, and Low-Dimensional Topology*

1 Introduction

2 Background from Representation Theory

2.1 Functions on Compact Lie Groups

2.2** Lie Algebra Representations**

2.3 Classiﬁcation of SU(n)-Representations

2.4 Representations of SL(2,C)

3 *Spin Networks*

3.1 Basic Deﬁnitions

3.2 Spin Network Component Maps

3.3 Symmetry Relations

3.4 The Spin Network Calculus

3.5 Trace Diagram Interpretation

3.6 Symmetrizers and Representations

3.7 Trivalent Spin Networks

3.8 6j-Symbols

4 Trace Diagrams

4.1 General Spin Networks

4.2 Trace Diagrams for Matrix Groups

4.3 3-Spin Networks

4.4 3-Trace Diagrams

4.5 Properties for General Groups

5 Central Functions of Hom(π,G)

5.1 The Character Variety

5.2 The Central Function Decomposition

5.3 Surface Cuts and Representations

5.4 Cut Triangulations

6 Central Functions for G = SL(2,C)

6.1 Rank One SL(2,C) Central Functions

6.2 Rank Two SL(2,C) Central Functions

6.3 Symmetries for Rank Two

6.4 A Recurrence Relation for Rank Two

6.5 Graded Structure for Rank Two

6.6 Multiplicative Structure for Rank Two

6.7 Direct Formula for Rank Two

7 Central Functions for Other Groups

7.1 Diagrams for SU(n) Representations…

Source: University of Maryland