###### TRACE DIAGRAMS, REPRESENTATIONS, AND LOW-DIMENSIONAL TOPOLOGY

###### Postgraduate

**ABSTRACT**

This thesis concerns a certain basis for the coordinate ring of the character variety of a surface. Let G be a connected reductive linear algebraic group, and let Σ be a surface whose fundamental group π is a free group. Then the coordinate ring C[Hom(π,G)] of the homomorphisms from π to G is isomorphic to C[G×r] ∼= C[G]⊗r for some r ∈ N. 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(π,G)]G, may be described in terms of the matrix coefficients of the maximal compact subgroup.

This correspondence provides a basis {χα} for C[Hom(π,G)]G, whose constituents will be called central functions. These functions may be expressed as labelled graphs called trace diagrams. This point-of-view permits diagram manipulation to be used to construct relations on the functions.

In the particular case G = SL(2,C), we give an explicit description of the central functions for surfaces. For rank one and two fundamental groups, the diagrammatic approach is used to describe the symmetries and structure of the central function basis, as well as a product formula in terms of this basis. For SL(3,C), we describe how to write down the central functions diagrammatically using the Littlewood-Richardson Rule, and give some examples. We also indicate progress for SL(n,C).

This correspondence provides a basis {χα} for C[Hom(π,G)]G, whose constituents will be called central functions. These functions may be expressed as labelled graphs called trace diagrams. This point-of-view permits diagram manipulation to be used to construct relations on the functions.

In the particular case G = SL(2,C), we give an explicit description of the central functions for surfaces. For rank one and two fundamental groups, the diagrammatic approach is used to describe the symmetries and structure of the central function basis, as well as a product formula in terms of this basis. For SL(3,C), we describe how to write down the central functions diagrammatically using the Littlewood-Richardson Rule, and give some examples. We also indicate progress for SL(n,C).

**INTRODUCTION**

The purpose of this work is to explore the use of diagrammatic techniques in studying the structure of certain character varieties. The space of representations is a useful tool for studying a particular group, even when restricting to the finite-dimensional irreducible representations. It should come as no surprise that the space of representations of the fundamental group of a surface encodes a lot of information about that surface. Indeed, this set of representations in some sense actually encodes the possible geometries on the surface. This thesis examines the algebraic structure of a particular basis of functions on the space of representations of the fundamental group.

Let G be a connected reductive linear algebraic group. If U < G is the maximal compact subgroup of G, then the coordinate ring C[G] may be identified with Calg(U), the algebra of matrix coefficients of finite-dimensional unitary representations of U. Moreover, for the action of G on C[G] by simultaneous conjugation, the ring of invariants C[G]G is generated by the characters of such representations [CSM].

Let Σ be a compact surface with boundary and consider

R = Hom(π1(Σ,x0),G),

the space of homomorphisms from the fundamental group of Σ into G. The Gcharacter variety of Σ is defined as the categorical quotient X = R//G. This space may be identified with conjugacy classes of completely reducible representations [Dol]. Since the fundamental group of Σ is a free group Fr of rank r ∈ N, the space R of homomorphisms is isomorphic to Gr. Hence C[R] ∼= C[Gr]. The coordinate ring of the character variety consists of the G-invariant functions on this space:

C[X] ∼= C[R]G ∼= C[Gr]G ∼= (C[G]⊗r)G ∼= (Calg(U)⊗r)G. (1.1)

An application of the Peter-Weyl Theorem gives a decomposition

M

Calg(U) = Vλ∗ ⊗ Vλ,

λ

where {Vλ} is the set of all irreducible finite-dimensional representations of U

[CSM]. An additive basis for C[X] is obtained by inserting this decomposition into (1.1) and decomposing the resulting tensors into irreducibles. This construction is described in detail in Chapter 5.

Let G be a connected reductive linear algebraic group. If U < G is the maximal compact subgroup of G, then the coordinate ring C[G] may be identified with Calg(U), the algebra of matrix coefficients of finite-dimensional unitary representations of U. Moreover, for the action of G on C[G] by simultaneous conjugation, the ring of invariants C[G]G is generated by the characters of such representations [CSM].

Let Σ be a compact surface with boundary and consider

R = Hom(π1(Σ,x0),G),

the space of homomorphisms from the fundamental group of Σ into G. The Gcharacter variety of Σ is defined as the categorical quotient X = R//G. This space may be identified with conjugacy classes of completely reducible representations [Dol]. Since the fundamental group of Σ is a free group Fr of rank r ∈ N, the space R of homomorphisms is isomorphic to Gr. Hence C[R] ∼= C[Gr]. The coordinate ring of the character variety consists of the G-invariant functions on this space:

C[X] ∼= C[R]G ∼= C[Gr]G ∼= (C[G]⊗r)G ∼= (Calg(U)⊗r)G. (1.1)

An application of the Peter-Weyl Theorem gives a decomposition

M

Calg(U) = Vλ∗ ⊗ Vλ,

λ

where {Vλ} is the set of all irreducible finite-dimensional representations of U

[CSM]. An additive basis for C[X] is obtained by inserting this decomposition into (1.1) and decomposing the resulting tensors into irreducibles. This construction is described in detail in Chapter 5.

The constituents of this basis are called central functions, and are the central object studied in this thesis. They may be described explicitly as spin networks, which are special types of labelled graphs. Spin networks may be identified canonically with functions in C[X], and provide enough algebraic horsepower to give explicit descriptions of central functions and some of their properties.

This point-of-view was originated by mathematical physicist John Baez, who interprets these spin networks as quantum mechanical “state vectors.” In [Ba], he shows that the space of square integrable functions on a certain space of smooth connections modulo gauge transformations is spanned by graphs similar to the ones given here. More recently, the work of Florentino [FMN] uses a similar basis to produce distributions related to geometric quantization of moduli spaces of flat connections on a surface. The application of spin network bases to the Fricke-Klein-Vogt problem, and in particular to character varieties, was considered by Adam Sikora [Sik]. The core problem, as described in Chapter 5, was first introduced to me by my advisor Bill Goldman. Notes based on his correspondences with Nicolai Reshetikhin [Res], Charles Frohman, and Joanna Kania-Bartoszyn´ska provided the foundation for the explicit description of central functions for SL(2,C) given in Chapter 6.

Outline

This thesis describes in detail the case G = SL(2,C) and rank r = 2. To a lesser extent, higher rank SL(2,C) cases and the G = SL(3,C) case are considered. There is also some discussion of the most general case.

Chapter 2 gives necessary background from representation theory, including the classification of SU(n)-representations.

In Chapters 3 and 4, spin networks and trace diagrams are formally introduced, with special emphasis on SL(2,C) and SL(3,C).

Chapter 5 describes in detail the construction of the central functions of a surface, and explicitly demonstrates how spin networks may be used to construct a basis for C[X]. The role of the topology of the surface in this construction is strongly emphasized.

This point-of-view was originated by mathematical physicist John Baez, who interprets these spin networks as quantum mechanical “state vectors.” In [Ba], he shows that the space of square integrable functions on a certain space of smooth connections modulo gauge transformations is spanned by graphs similar to the ones given here. More recently, the work of Florentino [FMN] uses a similar basis to produce distributions related to geometric quantization of moduli spaces of flat connections on a surface. The application of spin network bases to the Fricke-Klein-Vogt problem, and in particular to character varieties, was considered by Adam Sikora [Sik]. The core problem, as described in Chapter 5, was first introduced to me by my advisor Bill Goldman. Notes based on his correspondences with Nicolai Reshetikhin [Res], Charles Frohman, and Joanna Kania-Bartoszyn´ska provided the foundation for the explicit description of central functions for SL(2,C) given in Chapter 6.

Outline

This thesis describes in detail the case G = SL(2,C) and rank r = 2. To a lesser extent, higher rank SL(2,C) cases and the G = SL(3,C) case are considered. There is also some discussion of the most general case.

Chapter 2 gives necessary background from representation theory, including the classification of SU(n)-representations.

In Chapters 3 and 4, spin networks and trace diagrams are formally introduced, with special emphasis on SL(2,C) and SL(3,C).

Chapter 5 describes in detail the construction of the central functions of a surface, and explicitly demonstrates how spin networks may be used to construct a basis for C[X]. The role of the topology of the surface in this construction is strongly emphasized.

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