On Krammer's representation of the braid group
Copyright policy )On Krammer's representation of the braid group
A connection is made between the Krammer representation and the Birman-Murakami-Wenzl algebra. Inspired by a dimension argument, a basis is found for a certain irrep of the algebra, and relations which generate the matrices are found. Following a rescaling and change of parameters, the matrices are found to be identical to those of the Krammer representation. The two representations are thus the same, proving the irreducibility of one and the faithfulness of the other.
15 pages, 4 figures. Submitted to Math Research Letters
- Barnard College United States
- King’s University United States
- Columbia University
- Columbia University
- Columbia University
Geometric Topology (math.GT), Group Theory (math.GR), Braid groups; Artin groups, Krammer representation, 20F36 (Primary) 20C15, 16K20 (Secondary), Mathematics - Geometric Topology, FOS: Mathematics, Knots and links in the \(3\)-sphere, Algebraic Topology (math.AT), irreducible representations, Mathematics - Algebraic Topology, Representation Theory (math.RT), Birman-Murakami-Wenzl algebras, Mathematics - Group Theory, Mathematics - Representation Theory, faithful representations
Geometric Topology (math.GT), Group Theory (math.GR), Braid groups; Artin groups, Krammer representation, 20F36 (Primary) 20C15, 16K20 (Secondary), Mathematics - Geometric Topology, FOS: Mathematics, Knots and links in the \(3\)-sphere, Algebraic Topology (math.AT), irreducible representations, Mathematics - Algebraic Topology, Representation Theory (math.RT), Birman-Murakami-Wenzl algebras, Mathematics - Group Theory, Mathematics - Representation Theory, faithful representations
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