The faithfulness of an extension of Lawrence-Krammer representation on the group of conjugating automorphisms C_n in the cases n=3 and n=4

2023/08/23 by Mohamad N. Nasser, Nasser, Mohamad N.
Computer Science · Mathematics · #20F36 #Advanced Algebra and Geometry #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2308.12024

openalex publication_date 2023/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let C_n be the group of conjugating automorphisms. Valerij G. Bardakov defined a representation ρ of C_n, which is an extension of Lawrence-Krammer representation of the braid group B_n. Bardakov proved that the representation ρ is unfaithful for n ≥ 5. The cases n=3,4 remain open. M. N. Nasser and M. N. Abdulrahim made attempts towards the faithfulness of ρ in the case n=3. In this work, we prove that ρ is unfaithful in the both cases n=3 and n=4.

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