ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Fri, 04 Nov 2016 07:54:50 -0500Order of symmetric group irrepshttp://ask.sagemath.org/question/35299/order-of-symmetric-group-irreps/ This is a key detail for me, but I can't find any information about it in the Sage or GAP documentation: if I use
SymmetricGroup(5).character_table()
I get a 7x7 matrix, with rows indexed by conjugacy classes and columns indexed by irreps. Both of these things can be indexed, for instance, by Young diagrams of size 5.
How are the rows and columns ordered? Is it safe to assume that they will be ordered in reverse-lexicographic order on the Young diagrams, i.e.
5 ; 4,1 ; 3,2 ; 3,1,1 ; 2,2,1 ; 2,1,1,1 ; 1,1,1,1,1
This seems to be the case in the examples I've checked but I want to be sure that this is guaranteed by the code and not a coincidence in small examples.Fri, 28 Oct 2016 18:52:33 -0500http://ask.sagemath.org/question/35299/order-of-symmetric-group-irreps/Comment by kcrisman for <p>This is a key detail for me, but I can't find any information about it in the Sage or GAP documentation: if I use</p>
<pre><code>SymmetricGroup(5).character_table()
</code></pre>
<p>I get a 7x7 matrix, with rows indexed by conjugacy classes and columns indexed by irreps. Both of these things can be indexed, for instance, by Young diagrams of size 5.</p>
<p>How are the rows and columns ordered? Is it safe to assume that they will be ordered in reverse-lexicographic order on the Young diagrams, i.e.</p>
<p>5 ; 4,1 ; 3,2 ; 3,1,1 ; 2,2,1 ; 2,1,1,1 ; 1,1,1,1,1</p>
<p>This seems to be the case in the examples I've checked but I want to be sure that this is guaranteed by the code and not a coincidence in small examples.</p>
http://ask.sagemath.org/question/35299/order-of-symmetric-group-irreps/?comment=35300#post-id-35300You may find https://ask.sagemath.org/question/9300/order-of-irreducible-representation-characters-wrong-in-sage/ helpful here.Sat, 29 Oct 2016 07:09:18 -0500http://ask.sagemath.org/question/35299/order-of-symmetric-group-irreps/?comment=35300#post-id-35300Answer by kcrisman for <p>This is a key detail for me, but I can't find any information about it in the Sage or GAP documentation: if I use</p>
<pre><code>SymmetricGroup(5).character_table()
</code></pre>
<p>I get a 7x7 matrix, with rows indexed by conjugacy classes and columns indexed by irreps. Both of these things can be indexed, for instance, by Young diagrams of size 5.</p>
<p>How are the rows and columns ordered? Is it safe to assume that they will be ordered in reverse-lexicographic order on the Young diagrams, i.e.</p>
<p>5 ; 4,1 ; 3,2 ; 3,1,1 ; 2,2,1 ; 2,1,1,1 ; 1,1,1,1,1</p>
<p>This seems to be the case in the examples I've checked but I want to be sure that this is guaranteed by the code and not a coincidence in small examples.</p>
http://ask.sagemath.org/question/35299/order-of-symmetric-group-irreps/?answer=35301#post-id-35301According to the [documentation on class functions](http://doc.sagemath.org/html/en/reference/groups/sage/groups/class_function.html#class-functions-of-groups),
> NOTE: The ordering of the columns of the character table of a group corresponds to the ordering of the list. However, in general there is no way to canonically list (or index) the conjugacy classes of a group. Therefore the ordering of the columns of the character table of a group is somewhat random.
I've used this functionality a fair amount but to my surprise there doesn't seem to be good direct documentation of `character_table` easily discoverable online. You have to know that it lives [here](http://doc.sagemath.org/html/en/reference/groups/sage/groups/perm_gps/permgroup.html#sage.groups.perm_gps.permgroup.PermutationGroup_generic.character_table), and it doesn't say much more about it than to ask how GAP orders it.Sat, 29 Oct 2016 07:16:59 -0500http://ask.sagemath.org/question/35299/order-of-symmetric-group-irreps/?answer=35301#post-id-35301Comment by kcrisman for <p>According to the <a href="http://doc.sagemath.org/html/en/reference/groups/sage/groups/class_function.html#class-functions-of-groups">documentation on class functions</a>, </p>
<blockquote>
<p>NOTE: The ordering of the columns of the character table of a group corresponds to the ordering of the list. However, in general there is no way to canonically list (or index) the conjugacy classes of a group. Therefore the ordering of the columns of the character table of a group is somewhat random.</p>
</blockquote>
<p>I've used this functionality a fair amount but to my surprise there doesn't seem to be good direct documentation of <code>character_table</code> easily discoverable online. You have to know that it lives <a href="http://doc.sagemath.org/html/en/reference/groups/sage/groups/perm_gps/permgroup.html#sage.groups.perm_gps.permgroup.PermutationGroup_generic.character_table">here</a>, and it doesn't say much more about it than to ask how GAP orders it.</p>
http://ask.sagemath.org/question/35299/order-of-symmetric-group-irreps/?comment=35420#post-id-35420I've done the same type of computation as you, I think, and usually can figure it out - note the documentation says "corresponds to the ordering of the list". So I guess you can just have it list the conjugacy classes `G.conjugacy_classes()`? I'd try that.Fri, 04 Nov 2016 07:54:50 -0500http://ask.sagemath.org/question/35299/order-of-symmetric-group-irreps/?comment=35420#post-id-35420Comment by xanderflood for <p>According to the <a href="http://doc.sagemath.org/html/en/reference/groups/sage/groups/class_function.html#class-functions-of-groups">documentation on class functions</a>, </p>
<blockquote>
<p>NOTE: The ordering of the columns of the character table of a group corresponds to the ordering of the list. However, in general there is no way to canonically list (or index) the conjugacy classes of a group. Therefore the ordering of the columns of the character table of a group is somewhat random.</p>
</blockquote>
<p>I've used this functionality a fair amount but to my surprise there doesn't seem to be good direct documentation of <code>character_table</code> easily discoverable online. You have to know that it lives <a href="http://doc.sagemath.org/html/en/reference/groups/sage/groups/perm_gps/permgroup.html#sage.groups.perm_gps.permgroup.PermutationGroup_generic.character_table">here</a>, and it doesn't say much more about it than to ask how GAP orders it.</p>
http://ask.sagemath.org/question/35299/order-of-symmetric-group-irreps/?comment=35409#post-id-35409I figured as much, but the fact that it is not (and cannot be done) for general groups says nothing about whether it gives a consistent result for symmetric groups in particular! If there is no consistent order of the columns, then there must at least be some way of determining the order. If you can't, then it would be impossible to use the character table for almost anything!
(In particular, I am computing the character of a certain representation as a vector and want to use the character table to decompose it.)Thu, 03 Nov 2016 20:27:41 -0500http://ask.sagemath.org/question/35299/order-of-symmetric-group-irreps/?comment=35409#post-id-35409