ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Fri, 29 Sep 2023 19:56:01 +0200character table of normalizerhttps://ask.sagemath.org/question/73641/character-table-of-normalizer/Consider the character table of the cyclic permutation group $\mathbb{Z}_4$
Z4 = CyclicPermutationGroup(4)
Z4.character_table()
gives
[ 1 1 1 1]
[ 1 -1 1 -1]
[ 1 zeta4 -1 -zeta4]
[ 1 -zeta4 -1 zeta4]
if $e$ is the identity and $r$ is a rotation then the columns correspond to transformations $e$, $r$, $r^2$ and $r^3$, which of course is intuitive.
$\mathbb{Z}_4$ can for example be obtained from the centralizer of G((1,2,3,4)) with respect to the dihedral group $D_4$
G = DihedralGroup(4)
n = G.centralizer(G((1,2,3,4)))
ctable = n.character_table()
yields the same table as above with the second and fourth columns swapped.
[ 1 1 1 1]
[ 1 -1 1 -1]
[ 1 -zeta4 -1 zeta4]
[ 1 zeta4 -1 -zeta4]
`n.list()` yields `[(), (1,3)(2,4), (1,4,3,2), (1,2,3,4)]` or $[e,r^2,r^3,r]$ while the columns above are ordered as $[e,r^3,r^2,r]$, indicating that the columns are not given standard ordering nor are they given ordering with respect to the normalizer list. Generally, what is the convention for ordering?Thu, 28 Sep 2023 08:37:46 +0200https://ask.sagemath.org/question/73641/character-table-of-normalizer/Comment by John Palmieri for <p>Consider the character table of the cyclic permutation group $\mathbb{Z}_4$</p>
<pre><code>Z4 = CyclicPermutationGroup(4)
Z4.character_table()
</code></pre>
<p>gives</p>
<pre><code>[ 1 1 1 1]
[ 1 -1 1 -1]
[ 1 zeta4 -1 -zeta4]
[ 1 -zeta4 -1 zeta4]
</code></pre>
<p>if $e$ is the identity and $r$ is a rotation then the columns correspond to transformations $e$, $r$, $r^2$ and $r^3$, which of course is intuitive. </p>
<p>$\mathbb{Z}_4$ can for example be obtained from the centralizer of G((1,2,3,4)) with respect to the dihedral group $D_4$</p>
<pre><code>G = DihedralGroup(4)
n = G.centralizer(G((1,2,3,4)))
ctable = n.character_table()
</code></pre>
<p>yields the same table as above with the second and fourth columns swapped. </p>
<pre><code>[ 1 1 1 1]
[ 1 -1 1 -1]
[ 1 -zeta4 -1 zeta4]
[ 1 zeta4 -1 -zeta4]
</code></pre>
<p><code>n.list()</code> yields <code>[(), (1,3)(2,4), (1,4,3,2), (1,2,3,4)]</code> or $[e,r^2,r^3,r]$ while the columns above are ordered as $[e,r^3,r^2,r]$, indicating that the columns are not given standard ordering nor are they given ordering with respect to the normalizer list. Generally, what is the convention for ordering?</p>
https://ask.sagemath.org/question/73641/character-table-of-normalizer/?comment=73658#post-id-73658How can you tell whether columns 2 and 4 have been switched, or whether rows 3 and 4 have been switched? Couldn't it be the latter?Fri, 29 Sep 2023 19:56:01 +0200https://ask.sagemath.org/question/73641/character-table-of-normalizer/?comment=73658#post-id-73658Comment by ayodan for <p>Consider the character table of the cyclic permutation group $\mathbb{Z}_4$</p>
<pre><code>Z4 = CyclicPermutationGroup(4)
Z4.character_table()
</code></pre>
<p>gives</p>
<pre><code>[ 1 1 1 1]
[ 1 -1 1 -1]
[ 1 zeta4 -1 -zeta4]
[ 1 -zeta4 -1 zeta4]
</code></pre>
<p>if $e$ is the identity and $r$ is a rotation then the columns correspond to transformations $e$, $r$, $r^2$ and $r^3$, which of course is intuitive. </p>
<p>$\mathbb{Z}_4$ can for example be obtained from the centralizer of G((1,2,3,4)) with respect to the dihedral group $D_4$</p>
<pre><code>G = DihedralGroup(4)
n = G.centralizer(G((1,2,3,4)))
ctable = n.character_table()
</code></pre>
<p>yields the same table as above with the second and fourth columns swapped. </p>
<pre><code>[ 1 1 1 1]
[ 1 -1 1 -1]
[ 1 -zeta4 -1 zeta4]
[ 1 zeta4 -1 -zeta4]
</code></pre>
<p><code>n.list()</code> yields <code>[(), (1,3)(2,4), (1,4,3,2), (1,2,3,4)]</code> or $[e,r^2,r^3,r]$ while the columns above are ordered as $[e,r^3,r^2,r]$, indicating that the columns are not given standard ordering nor are they given ordering with respect to the normalizer list. Generally, what is the convention for ordering?</p>
https://ask.sagemath.org/question/73641/character-table-of-normalizer/?comment=73649#post-id-73649(I made a new post just to point out the inconsistency)Fri, 29 Sep 2023 06:09:58 +0200https://ask.sagemath.org/question/73641/character-table-of-normalizer/?comment=73649#post-id-73649Comment by ayodan for <p>Consider the character table of the cyclic permutation group $\mathbb{Z}_4$</p>
<pre><code>Z4 = CyclicPermutationGroup(4)
Z4.character_table()
</code></pre>
<p>gives</p>
<pre><code>[ 1 1 1 1]
[ 1 -1 1 -1]
[ 1 zeta4 -1 -zeta4]
[ 1 -zeta4 -1 zeta4]
</code></pre>
<p>if $e$ is the identity and $r$ is a rotation then the columns correspond to transformations $e$, $r$, $r^2$ and $r^3$, which of course is intuitive. </p>
<p>$\mathbb{Z}_4$ can for example be obtained from the centralizer of G((1,2,3,4)) with respect to the dihedral group $D_4$</p>
<pre><code>G = DihedralGroup(4)
n = G.centralizer(G((1,2,3,4)))
ctable = n.character_table()
</code></pre>
<p>yields the same table as above with the second and fourth columns swapped. </p>
<pre><code>[ 1 1 1 1]
[ 1 -1 1 -1]
[ 1 -zeta4 -1 zeta4]
[ 1 zeta4 -1 -zeta4]
</code></pre>
<p><code>n.list()</code> yields <code>[(), (1,3)(2,4), (1,4,3,2), (1,2,3,4)]</code> or $[e,r^2,r^3,r]$ while the columns above are ordered as $[e,r^3,r^2,r]$, indicating that the columns are not given standard ordering nor are they given ordering with respect to the normalizer list. Generally, what is the convention for ordering?</p>
https://ask.sagemath.org/question/73641/character-table-of-normalizer/?comment=73648#post-id-73648It just seems inconsistent. Like `G.conjugacy_classes_representatives()` for `G = CyclicPermutationGroup(4)` and `n.conjugacy_classes_representatives()` for above both yield `[(), (1,2,3,4), (1,3)(2,4), (1,4,3,2)]`. But the 2nd and 4th columns for `n.character_table()` are swapped relative to `G.character_table()`Fri, 29 Sep 2023 06:08:51 +0200https://ask.sagemath.org/question/73641/character-table-of-normalizer/?comment=73648#post-id-73648Comment by John Palmieri for <p>Consider the character table of the cyclic permutation group $\mathbb{Z}_4$</p>
<pre><code>Z4 = CyclicPermutationGroup(4)
Z4.character_table()
</code></pre>
<p>gives</p>
<pre><code>[ 1 1 1 1]
[ 1 -1 1 -1]
[ 1 zeta4 -1 -zeta4]
[ 1 -zeta4 -1 zeta4]
</code></pre>
<p>if $e$ is the identity and $r$ is a rotation then the columns correspond to transformations $e$, $r$, $r^2$ and $r^3$, which of course is intuitive. </p>
<p>$\mathbb{Z}_4$ can for example be obtained from the centralizer of G((1,2,3,4)) with respect to the dihedral group $D_4$</p>
<pre><code>G = DihedralGroup(4)
n = G.centralizer(G((1,2,3,4)))
ctable = n.character_table()
</code></pre>
<p>yields the same table as above with the second and fourth columns swapped. </p>
<pre><code>[ 1 1 1 1]
[ 1 -1 1 -1]
[ 1 -zeta4 -1 zeta4]
[ 1 zeta4 -1 -zeta4]
</code></pre>
<p><code>n.list()</code> yields <code>[(), (1,3)(2,4), (1,4,3,2), (1,2,3,4)]</code> or $[e,r^2,r^3,r]$ while the columns above are ordered as $[e,r^3,r^2,r]$, indicating that the columns are not given standard ordering nor are they given ordering with respect to the normalizer list. Generally, what is the convention for ordering?</p>
https://ask.sagemath.org/question/73641/character-table-of-normalizer/?comment=73647#post-id-73647The documentation for GAP at https://docs.gap-system.org/doc/ref/chap39.html#X7D474F8F87E4E5D9 does not provide further information on the ordering.Thu, 28 Sep 2023 21:55:00 +0200https://ask.sagemath.org/question/73641/character-table-of-normalizer/?comment=73647#post-id-73647Comment by John Palmieri for <p>Consider the character table of the cyclic permutation group $\mathbb{Z}_4$</p>
<pre><code>Z4 = CyclicPermutationGroup(4)
Z4.character_table()
</code></pre>
<p>gives</p>
<pre><code>[ 1 1 1 1]
[ 1 -1 1 -1]
[ 1 zeta4 -1 -zeta4]
[ 1 -zeta4 -1 zeta4]
</code></pre>
<p>if $e$ is the identity and $r$ is a rotation then the columns correspond to transformations $e$, $r$, $r^2$ and $r^3$, which of course is intuitive. </p>
<p>$\mathbb{Z}_4$ can for example be obtained from the centralizer of G((1,2,3,4)) with respect to the dihedral group $D_4$</p>
<pre><code>G = DihedralGroup(4)
n = G.centralizer(G((1,2,3,4)))
ctable = n.character_table()
</code></pre>
<p>yields the same table as above with the second and fourth columns swapped. </p>
<pre><code>[ 1 1 1 1]
[ 1 -1 1 -1]
[ 1 -zeta4 -1 zeta4]
[ 1 zeta4 -1 -zeta4]
</code></pre>
<p><code>n.list()</code> yields <code>[(), (1,3)(2,4), (1,4,3,2), (1,2,3,4)]</code> or $[e,r^2,r^3,r]$ while the columns above are ordered as $[e,r^3,r^2,r]$, indicating that the columns are not given standard ordering nor are they given ordering with respect to the normalizer list. Generally, what is the convention for ordering?</p>
https://ask.sagemath.org/question/73641/character-table-of-normalizer/?comment=73646#post-id-73646Isn't this the same question as in https://ask.sagemath.org/question/73582/read-character-table/? The columns are in the order given by the list of conjugacy classes: `n.conjugacy_classes()` or `n.conjugacy_classes_representatives()`. The documentation for the second of those methods says "The ordering is that given by GAP."Thu, 28 Sep 2023 21:49:54 +0200https://ask.sagemath.org/question/73641/character-table-of-normalizer/?comment=73646#post-id-73646