# Trace Diagrams, Representations, and Low-Dimensional Topology

 dc.contributor.advisor Goldman, William en_US dc.contributor.author Peterson, Elisha en_US dc.date.accessioned 2006-06-14T05:53:11Z dc.date.available 2006-06-14T05:53:11Z dc.date.issued 2006-04-26 en_US dc.identifier.uri http://hdl.handle.net/1903/3500 dc.description.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 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. This correspondence provides a basis {X_a} for C[Hom(pi,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). en_US dc.format.extent 845344 bytes dc.format.mimetype application/pdf dc.language.iso en_US dc.title Trace Diagrams, Representations, and Low-Dimensional Topology en_US dc.type Dissertation en_US dc.contributor.publisher Digital Repository at the University of Maryland en_US dc.contributor.publisher University of Maryland (College Park, Md.) en_US dc.contributor.department Mathematics en_US dc.subject.pqcontrolled Mathematics en_US dc.subject.pquncontrolled trace diagrams en_US dc.subject.pquncontrolled spin networks en_US dc.subject.pquncontrolled central functions en_US dc.subject.pquncontrolled character variety en_US
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