The Burau representations of loop braid groups
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We give a simple topological construction of the Burau representations of the loop braid groups. There are four versions: defined either on the non-extended or extended loop braid groups, and in each case there is an unreduced and a reduced version. Three are not surprising, and one could easily guess the correct matrices to assign to generators. The fourth is more subtle, and does not seem combinatorially obvious, although it is topologically very natural.
Ordinary representations and characters, Homological methods in group theory, Topological methods in group theory, 2020 MSC: 20C12 20F36 20J05 57M07 57M10, Homological representations, Geometric Topology (math.GT), 20C12, 20F36, 20J05, 57M07, 57M10, Braid groups; Artin groups, 540, Covering spaces, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], 004, Mathematics - Geometric Topology, [MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT], [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], QA1-939, FOS: Mathematics, Algebraic Topology (math.AT), Loop braid groups, Mathematics - Algebraic Topology, Covering spaces and low-dimensional topology, Mathematics
Ordinary representations and characters, Homological methods in group theory, Topological methods in group theory, 2020 MSC: 20C12 20F36 20J05 57M07 57M10, Homological representations, Geometric Topology (math.GT), 20C12, 20F36, 20J05, 57M07, 57M10, Braid groups; Artin groups, 540, Covering spaces, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], 004, Mathematics - Geometric Topology, [MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT], [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], QA1-939, FOS: Mathematics, Algebraic Topology (math.AT), Loop braid groups, Mathematics - Algebraic Topology, Covering spaces and low-dimensional topology, Mathematics
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