Automorphisms of three canonical extensions of symmetric groups

Abstract: The symmetric group $S_n$, $n \geq 2$, has a Coxeter presentation with generating set $X = \{\tau_1, \dots, \tau_{n-1}\}$ and defining relations

(1) Involutions: $\tau^2_i = 1$ for $1 \leq i \leq n − 1$;

(2) Braid relations: $\tau_i \tau_{i+1} \tau_i = \tau_{i+1} \tau_i \tau_{i+1}$ for $1 \leq i \leq n − 2$;

(3) Far commutativity: $\tau_i \tau_j = \tau_j \tau_i$ for $|i − j|\geq 2$.

By omitting all relations of type (1) (respectively type (2) ) from the preceding presentation of $S_n$, we get presentations of the Artin braid group $B_n$ (respectively the twin group $T_n$). Artin braid groups are well-studied objects with nice geometrical presentations in 3-space. Apart from mathematics, they have far-reaching applications in physics and biology. Twin groups can be thought of as planar analogues of braid groups. Recently these groups have attracted attention from (quantum) physicists. Thus, it is natural to ask about the remaining case. What kind of group do we get, if we omit all relations of the third type from the above presentation of $S_n$?

It follows that if we remove all relations of type (3), we get an odd Coxeter group whose associated Coxeter graph is a straight line on $n − 1$ vertices. In this talk, we will consider a general family of odd Coxeter groups whose associated Coxeter graphs are trees, discuss their automorphism groups and compare with the automorphism groups of Artin braid groups and twin groups.

group theory

Audience: researchers in the topic

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