Relations in a particular subgroup of the braid group.

Question

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...

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?

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 $q$.

Answer 1

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.

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 Some subgroups of Artin's braid group (available here). 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, Exact sequences, lower central series and representations of surface braid groups (arXiv).