# Order of Irreducible Representation Characters wrong in Sage?

I am using Sage to calculate Irreducible Representations (IRR) of symmetry groups of graphs. Most of the time the order of the IRR characters match the order of the conjugacy class representatives. For example the trivial representation characters are the first row of the character table (1,1,1,1,...,1) and the first conjugacy class is ( ), the identity representation. But in the example below for a star graph (one central vertex, the 4 other vertices on spokes from the center) the trivial representation characters are the last row in the character table and the conjugacy class representatives have the identity as the first element.

Example (star graph).

ct=G.character_table()
print ct

[ 1 -1  1  1 -1]
[ 3 -1 -1  0  1]
[ 2  0  2 -1  0]
[ 3  1 -1  0 -1]
[ 1  1  1  1  1]  <--- trivial rep. is last

cc=G.conjugacy_classes_representatives()
print cc

[(), (1,2), (1,2)(3,4), (1,2,3), (1,2,3,4)]
^ trivial class is first


Is this a bug or known issue for Sage? How can I get around it so I know the orders of the characters and representatives match up properly?

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You are confusing rows and columns of the character table.

• The rows correspond to the irreducible representations
• The columns correspond to the conjugacy classes

For example:

sage: G = DihedralGroup(3)
sage: G.character_table()
[ 1 -1  1]
[ 2  0 -1]
[ 1  1  1]


So the last row in the character table is the trivial representation, which is the same ordering as

 sage: G.irreducible_characters()[-1] == G.trivial_character()
True


We also see that the trivial conjugacy class is the first column in the character table. For this, recall that the character evaluated at the trivial conjugacy class is just the dimension of the representation. As you already remarked, this is the same ordering that Sage (actually, GAP) returns the conjugacy classes in:

sage: G.conjugacy_classes_representatives()
[(), (1,2), (1,2,3)]

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Or is it possible that the OP is assuming (incorrectly, I believe) that there is a canonical way to associate characters to conjugacy classes? That was my first thought...

( 2012-09-07 17:11:54 -0500 )edit

I think I am mixing things up. Let me sit back and get it straight. Thank you for helping.

( 2012-09-10 07:42:41 -0500 )edit
Definition:     H.conjugacy_classes_representatives(self)
Docstring:
Returns a complete list of representatives of conjugacy classes in
a permutation group G. The ordering is that given by GAP.

Definition:     H.character_table(self)
Docstring:
Returns the matrix of values of the irreducible characters of a
permutation group G at the conjugacy classes of G. The columns
represent the conjugacy classes of G and the rows represent the
different irreducible characters in the ordering given by GAP.


So it seems that the rows correspond to characters only, and perhaps they aren't guaranteed to be in the same order as the conjugacy classes. After all, in your example above, which of the two 3-dimensional reps should correspond to which order 9 conjugacy class? And I do get the same order for the irreducible characters as in the table with H.irreducible_characters().

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I used irreducible_characters() and displayed them with [x.values() for x in irr]. The order appears correct now. Thank you.

( 2012-09-06 05:19:00 -0500 )edit

Running in Sage both character_table() and irreducible_characters() agree with conjugacy_classes_representatives() order. *But when I run from Python the order of characters from both calls are scrambled in the same way and no longer agree with the order as given in Sage or Python for the conjugacy_classes_representatives. So Python give same conjugacy_classes_representatives order as Sage, but not so for character tables done either way.

( 2012-09-06 05:26:54 -0500 )edit

Glad things helped. I'm a little confused by your second comment, though. What do you mean by "run from Python"? Do you mean you used a .py or .sage script to do this? We'd have to see the code - update your question, if there is still a problem.

( 2012-09-06 06:36:27 -0500 )edit

Yes, I made the mistake of confusing IRR characters and conjugacy classes. I'm sorry for the confusion and taking so long to respond. Thanks for the many replies. I have that cleared up now.

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