Spin networks and SL(2,C)-Character varieties

Denote the free group on 2 letters by F_2 and the SL(2,C)-representation variety of F_2 by R=Hom(F_2,SL(2,C)). The group SL(2,C) acts on R by conjugation. We construct an isomorphism between the coordinate ring C[SL(2,C)] and the ring of matrix coefficients, providing an additive basis of C[R]^SL(2,C) in terms of spin networks. Using a graphical calculus, we determine the symmetries and multiplicative structure of this basis. This gives a canonical description of the regular functions on the SL(2,C)-character variety of F_2 and a new proof of a classical result of Fricke, Klein, and Vogt.Comment: Updated historical treatment of the subject. Figures drawn with PGF/TikZ; Handbook of Teichmuller Theory II, A. Papadopoulos (ed), EMS Publishing House, Zurich, 200

Combinatorial Proofs of Generalizations of Sperner\u27s Lemma

In this thesis, we provide constructive proofs of serveral generalizations of Sperner\u27s Lemma, a combinatorial result which is equivalent to the Brouwer Fixed Point Theorem. This lemma makes a statement about the number of a certain type of simplices in the triangulation of a simplex with a special labeling. We prove generalizations for polytopes with simplicial facets, for arbitrary 3-polytopes, and for polygons. We introduce a labeled graph which we call a nerve graph to prove these results. We also suggest a possible non-constructive proof for a polytopal generalization

Trace Diagrams, Representations, and Low-Dimensional Topology

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)

Four-Person Envy-Free Chore Division

No abstract provided in this articl