A braidlike presentation of Sp(n, p)
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Forn even andp an odd prime a symplectic group Sp(n, p) is a quotient of the Artin braid groupB n+1. Ifs 1, …,s n are standard generators ofB n+1 then the kernel of the corresponding epimorphism is the normal closure of just four elements:s 1 p ,(s 1 s 2)6,s 1 (p+1)/2 s 2 4 s 1 (p−1)/2 s 2 −2 s 1 −1 s 2 2 and (s 1 s 2 s 3)4 A −1 s 1 −2 A, whereA=s 2 s 3 −1 s 2 (p−1)/2 s 4 s 3 2 s 4, all of them lying in the subgroupB 5. Sp(n, p) acts on a vector space and the image of the subgroupB n ofB n+1 in Sp(n, p), denoted Sp(n−1,p), is a stabilizer of one vector. A sequence of inclusions …B k+1·B k … induces a sequence of inclusions …Sp(k,p)·Sp(k−1,p)…, which can be used to study some finite-valued invariants of knots and links in the 3-sphere via the Markov theorem.
KeywordsInduction Hypothesis Braid Group Symplectic Group Split Epimorphism Minimal Representative
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