ASKSAGE: Sage Q&A Forum - Individual question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Tue, 27 Aug 2019 06:14:34 -0500Automorphism group of Coxeter groupshttps://ask.sagemath.org/question/47507/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.Mon, 19 Aug 2019 05:05:49 -0500https://ask.sagemath.org/question/47507/automorphism-group-of-coxeter-groups/Answer by Christian Stump for <p>Coxeter groups can be invoked using the comment </p>
<pre><code>W = CoxeterGroup(['A',3],implementation='reflection')
</code></pre>
<p>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?</p>
<p>Thank you.</p>
https://ask.sagemath.org/question/47507/automorphism-group-of-coxeter-groups/?answer=47612#post-id-47612This 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]}Tue, 27 Aug 2019 04:24:24 -0500https://ask.sagemath.org/question/47507/automorphism-group-of-coxeter-groups/?answer=47612#post-id-47612Comment by Christian Stump for <p>This automorphism is realized by conjugation with the longest element in $W$:</p>
<pre><code>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]}
</code></pre>
https://ask.sagemath.org/question/47507/automorphism-group-of-coxeter-groups/?comment=47615#post-id-47615@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...Tue, 27 Aug 2019 06:14:34 -0500https://ask.sagemath.org/question/47507/automorphism-group-of-coxeter-groups/?comment=47615#post-id-47615Comment by jipilab for <p>This automorphism is realized by conjugation with the longest element in $W$:</p>
<pre><code>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]}
</code></pre>
https://ask.sagemath.org/question/47507/automorphism-group-of-coxeter-groups/?comment=47614#post-id-47614Although there is a subtlety with `['F',4]`, since in this case, this automorphism is the identity...Tue, 27 Aug 2019 06:10:34 -0500https://ask.sagemath.org/question/47507/automorphism-group-of-coxeter-groups/?comment=47614#post-id-47614