Relations in a particular subgroup of the braid group. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T05:12:45Z http://mathoverflow.net/feeds/question/108648 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108648/relations-in-a-particular-subgroup-of-the-braid-group Relations in a particular subgroup of the braid group. Ed Segal 2012-10-02T15:54:50Z 2012-10-02T16:41:33Z <p>I think this should be a 10 minute exercise in a decent computer algebra package - unfortunately I'm hopelessly ignorant of such things, so I'm putting it up here in the hope that someone will be kind enough to do it for me...</p> <p>Here's the question: partition $n$ into two pieces, $n= p+q$, and let $S_p\times S_q \subset S_n$ be the associated Young subgroup. Now consider the braid group $B_n$. I'm interested in the subgroup of $B_n$ consisting of braids that preserve this partition of their endpoints, i.e. $$B_{p,q} := B_n \times_{S_n} (S_p\times S_q )$$ I can write down generators for $B_{p,q}$, namely $s_1,.., s_{p-1}, s_p^2, s_{p+1}, ..., s_{n-1}$ where the $s_i$ are the standard generators of $B_n$. My question is what are the relations? </p> <p>Obviously I need the usual braid relations on each piece of the partition, but are there any others? I'd be happy to extrapolate heuristically from low values of $p$ and $q$. </p> http://mathoverflow.net/questions/108648/relations-in-a-particular-subgroup-of-the-braid-group/108650#108650 Answer by Adrien for Relations in a particular subgroup of the braid group. Adrien 2012-10-02T16:41:33Z 2012-10-02T16:41:33Z <p>You probably already noticed that, but $B_{p,q}$ is the fundamental group of $$X_n/(S_p \times S_q)$$ where $X_n$ is the configuration space of $n$ points in the complex plane. Ths may help to guess some facts about these groups.</p> <p>So far I know these group are usually called "mixed braid groups" in the litterature, though this name is sometimes used for the group of braids with the $p$ first strands fixed. Anyway, a presentation of it is obtained in S. Manfredini <strong>Some subgroups of Artin's braid group</strong> (available <a href="http://www.sciencedirect.com/science/article/pii/S0166864196001526" rel="nofollow">here</a>). Results on their relation with the representation theory of $B_n$, as well as references you may find interesting, can be found in Bellingeri, Godelle, Guaschi, <strong>Exact sequences, lower central series and representations of surface braid groups</strong> (<a href="http://arxiv.org/abs/1106.4982" rel="nofollow">arXiv</a>).</p>