Braids and Permutations

2004/04/29 by Vladimir Lin, Lin, Vladimir · 1 citation
Computer Science · Mathematics · #55P #Advanced Graph Theory Research #Advanced Topology and Set Theory #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #math.GR #msc:55P

paper · pdf · doi:10.48550/arxiv.math/0404528

110pp

arxiv created 2004/04/29 · openalex publication_date 2004/04/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

E. Artin described all irreducible representations of the braid group B_k to the symmetric group S(k). We strengthen some of his results and, moreover, exhibit a complete picture of homomorphisms of B_k to S(n) for n<2k+1. We show that the image of such ahomomorphism f is cyclic whenever either (*) n6 there exist, up to conjugation, exactly 3 irreducible representations of B_k into S(2k) with non-cyclic images but they all are imprimitive. We use these results to prove that for n4 the intersection PB_k∩ B'_k of PB_k with the commutator subgroup B'_k=[B_k,B_k] is a completely characteristic subgroup of B'_k.

Cited by

• Counting homomorphisms onto finite solvable groups
• Braid groups are almost co-Hopfian

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