# Automorphism group of Coxeter groups

Coxeter groups can be invoked using the comment

W = CoxeterGroup(['A',3],implementation='reflection')


Now, W has three simple reflections S_1, S_2 and S_3. and I am interested in studying the automorphism S_1 -----> S_3, S_2 -----> S_2 and S_3 -----> S_1. Can any one please tell me, how to implement this automorphism of W in sage?

Thank you.

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This automorphism is realized by conjugation with the longest element in $W$:

sage: W = CoxeterGroup(['A',3],implementation="reflection")
sage: S = W.simple_reflections()
sage: Sauto = { i: W.w0*S[i]*W.w0 for i in S.keys() }
sage: S
Finite family {1: [-1  1  0]
[ 0  1  0]
[ 0  0  1], 2: [ 1  0  0]
[ 1 -1  1]
[ 0  0  1], 3: [ 1  0  0]
[ 0  1  0]
[ 0  1 -1]}
sage: Sauto
{1: [ 1  0  0]
[ 0  1  0]
[ 0  1 -1], 2: [ 1  0  0]
[ 1 -1  1]
[ 0  0  1], 3: [-1  1  0]
[ 0  1  0]
[ 0  0  1]}

more

Although there is a subtlety with ['F',4], since in this case, this automorphism is the identity...

( 2019-08-27 13:10:34 +0200 )edit

@jipilab: Yep, same in D4, where this is still an involution while the automorphism group is an $S_3$... But the question was about type A where it works...

( 2019-08-27 13:14:34 +0200 )edit