111 research outputs found
Generators, Relations and Symmetries in Pairs of 3x3 Unimodular Matrices
Denote the free group on two letters by F2 and the SL(3,C)-representation
variety of F2 by R = Hom(F2, SL(3,C)). There is a SL(3,C)-action on the
coordinate ring of R, and the geometric points of the subring of invariants is
an affine variety X. We determine explicit minimal generators and defining
relations for the subring of invariants and show X is a degree 6 hyper-surface
in C9 mapping onto C8. Our choice of generators exhibit Out(F2) symmetries
which allow for a succinct expression of the defining relations.Comment: 21 pages. This fourth version is the author's final version which
includes new comments and some correction
E-polynomial of the SL(3,C)-character variety of free groups
We compute the E-polynomial of the character variety of representations of a
rank r free group in SL(3, C). Expanding upon existing techniques, we stratify
the space of representations and compute the E-polynomial of each geometrically
described stratum using fibrations. Consequently, we also determine the
E-polynomial of its smooth, singular, and abelian loci and the corresponding
Euler characteristic in each case. Along the way, we give new proofs of the
SL(2,C), PGL(2,C), and PGL(3,C) cases.Comment: 21 pages; accepted for publication at the Pacific Journal of
Mathematic
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
Fundamental Groups of Character Varieties: Surfaces and Tori
We compute the fundamental group of moduli spaces of Lie group valued
representations of surface and torus groups.Comment: v2: 12 pages, minor edits, accepted for publication at Mathematische
Zeitschrif
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