# Character table of $S_7$

Is there any reference where I can find the character table of the symmetric group $S_7$? A simple search in google gave me a GAP program that computes the character table, but I don't understand the table that the program produces, since it does not tell me each column in the table is associated to which conjugacy class. I would like to have a table like enter link description here

I would be extremely thankful if you provide me reference.

As you already discovered, GAP can easily compute this. The character table looks like this:

gap> G:=SymmetricGroup(7);
Sym( [ 1 .. 7 ] )
gap> ct := CharacterTable(G);;
CharacterTable( Sym( [ 1 .. 7 ] ) )
gap> Display(ct);
CT2

2  4  4  4  4  3  2  3  1  3  3   2  1   1  1  .
3  2  1  1  1  2  1  1  2  1  .   1  .   .  1  .
5  1  1  .  .  .  .  .  .  .  .   .  1   1  .  .
7  1  .  .  .  .  .  .  .  .  .   .  .   .  .  1

1a 2a 2b 2c 3a 6a 6b 3b 4a 4b 12a 5a 10a 6c 7a
2P 1a 1a 1a 1a 3a 3a 3a 3b 2b 2b  6b 5a  5a 3b 7a
3P 1a 2a 2b 2c 1a 2a 2b 1a 4a 4b  4a 5a 10a 2c 7a
5P 1a 2a 2b 2c 3a 6a 6b 3b 4a 4b 12a 1a  2a 6c 7a
7P 1a 2a 2b 2c 3a 6a 6b 3b 4a 4b 12a 5a 10a 6c 1a

X.1      1 -1  1 -1  1 -1  1  1 -1  1  -1  1  -1 -1  1
X.2      6 -4  2  .  3 -1 -1  . -2  .   1  1   1  . -1
X.3     14 -6  2 -2  2  .  2 -1  .  .   . -1  -1  1  .
X.4     14 -4  2  . -1 -1 -1  2  2  .  -1 -1   1  .  .
X.5     15 -5 -1  3  3  1 -1  . -1 -1  -1  .   .  .  1
X.6     35 -5 -1 -1 -1  1 -1 -1  1  1   1  .   . -1  .
X.7     21 -1  1  3 -3 -1  1  .  1 -1   1  1  -1  .  .
X.8     21  1  1 -3 -3  1  1  . -1 -1  -1  1   1  .  .
X.9     20  . -4  .  2  .  2  2  .  .   .  .   .  . -1
X.10    35  5 -1  1 -1 -1 -1 -1 -1  1  -1  .   .  1  .
X.11    14  4  2  . -1  1 -1  2 -2  .   1 -1  -1  .  .
X.12    15  5 -1 -3  3 -1 -1  .  1 -1   1  .   .  .  1
X.13    14  6  2  2  2  .  2 -1  .  .   . -1   1 -1  .
X.14     6  4  2  .  3  1 -1  .  2  .  -1  1  -1  . -1
X.15     1  1  1  1  1  1  1  1  1  1   1  1   1  1  1


In order to understand how to read this, one needs to know (e.g. by a quick look at the documentation) that the rows labelled X.1 to X.15 correspond to the irreducible characters, while the columns correspond to the conjugacy classes of the group. So, let's ask GAP for the classes and a representative of each:

gap> cc:=ConjugacyClasses(ct);
[ ()^G, (1,2)^G, (1,2)(3,4)^G, (1,2)(3,4)(5,6)^G, (1,2,3)^G, (1,2,3)(4,5)^G, (1,2,3)(4,5)(6,7)^G,
(1,2,3)(4,5,6)^G, (1,2,3,4)^G, (1,2,3,4)(5,6)^G, (1,2,3,4)(5,6,7)^G, (1,2,3,4,5)^G, (1,2,3,4,5)(6,7)^G,
(1,2,3,4,5,6)^G, (1,2,3,4,5,6,7)^G ]
gap> for i in [1..15] do Print(i, ": ", Representative(cc[i]), "\n"); od;
1: ()
2: (1,2)
3: (1,2)(3,4)
4: (1,2)(3,4)(5,6)
5: (1,2,3)
6: (1,2,3)(4,5)
7: (1,2,3)(4,5)(6,7)
8: (1,2,3)(4,5,6)
9: (1,2,3,4)
10: (1,2,3,4)(5,6)
11: (1,2,3,4)(5,6,7)
12: (1,2,3,4,5)
13: (1,2,3,4,5)(6,7)
14: (1,2,3,4,5,6)
15: (1,2,3,4,5,6,7)


Of course you can get a lot more information from this, too. More on this can be found in the GAP manual.

• Let me see. Do you mean that for instance X.1 is the character we get from the partition $\lambda=(1,1,1,1,1,1,1)$? If this is true then do you mean for instance the sixth column of X.1 is associated to the conjugacy class (1,2,3)(4,5)? – M.B Apr 5 '14 at 0:03
• The rows correspond to characters. – Mariano Suárez-Álvarez Apr 5 '14 at 4:26
• Sorry, of course Mariano is right and the rows correspond to the characters, the columns to the conjugacy classes! I'll correct my answer. – Max Horn Apr 5 '14 at 11:11

The ATLAS of Conway et al. has this information for $S_n$ for each $n = 5, 6, 7, \ldots, 13$. For $S_7$, look up $A_7$ in the main table to find the characters of Aut$(A_7)=S_7$, then check the appendix for the bijection between representations and partitions.

Page 350 of James and Kerber's symmetric group book, tables up to $S_{10}$ are there.

I am adding this because I am surprised no one has mentioned the wonderful CHEVIE package! This does precisely what the OP wants. For example.

gap> W:=CoxeterGroup("A",6);
CoxeterGroup("A",6)
gap> Display(CharTable(W));
A6

2       4       4       4       4       3      2     3       1     3
3       2       1       1       1       2      1     1       2     1
5       1       1       .       .       .      .     .       .     .
7       1       .       .       .       .      .     .       .     .

1111111  211111   22111    2221   31111   3211   322     331  4111
2P 1111111 1111111 1111111 1111111   31111  31111 31111     331 22111
3P 1111111  211111   22111    2221 1111111 211111 22111 1111111  4111
5P 1111111  211111   22111    2221   31111   3211   322     331  4111
7P 1111111  211111   22111    2221   31111   3211   322     331  4111

1111111          1      -1       1      -1       1     -1     1       1    -1
211111           6      -4       2       .       3     -1    -1       .    -2
22111           14      -6       2      -2       2      .     2      -1     .
2221            14      -4       2       .      -1     -1    -1       2     2
31111           15      -5      -1       3       3      1    -1       .    -1
3211            35      -5      -1      -1      -1      1    -1      -1     1
322             21      -1       1       3      -3     -1     1       .     1
331             21       1       1      -3      -3      1     1       .    -1
4111            20       .      -4       .       2      .     2       2     .
421             35       5      -1       1      -1     -1    -1      -1    -1
43              14       4       2       .      -1      1    -1       2    -2
511             15       5      -1      -3       3     -1    -1       .     1
52              14       6       2       2       2      .     2      -1     .
61               6       4       2       .       3      1    -1       .     2
7                1       1       1       1       1      1     1       1     1

2     3    2       1      1    1       .
3     .    1       .      .    1       .
5     .    .       1      1    .       .
7     .    .       .      .    .       1

421   43     511     52   61       7
2P 22111  322     511    511  331       7
3P   421 4111     511     52 2221       7
5P   421   43 1111111 211111   61       7
7P   421   43     511     52   61 1111111

1111111        1   -1       1     -1   -1       1
211111         .    1       1      1    .      -1
22111          .    .      -1     -1    1       .
2221           .   -1      -1      1    .       .
31111         -1   -1       .      .    .       1
3211           1    1       .      .   -1       .
322           -1    1       1     -1    .       .
331           -1   -1       1      1    .       .
4111           .    .       .      .    .      -1
421            1   -1       .      .    1       .
43             .    1      -1     -1    .       .
511           -1    1       .      .    .       1
52             .    .      -1      1   -1       .
61             .   -1       1     -1    .      -1
7              1    1       1      1    1       1


One can also do something similar with the newly written Pycox package!

• Sweet! It always annoyed be that GAP displays the rows in order of dimension and it's not always easy to see which partition each character corresponds to! – daveh Apr 5 '14 at 15:45

see page 6 of these lecture notes.

• Well $S_7$ has 15 representation, but that table just 9 representation. Beside I don't see the definition of 1A, 2A, .... – M.B Apr 4 '14 at 23:46
• In response to OP comment, the character table is displayed up to a multiplication by the character of "sign representation". If you count correctly, six of the non-one-dimensional representations (characters) can be tensored (multiplied) to give new representations (characters), which gives you the 15 you want. The notation of ccl's follows what ATLAS gives me, which I agree is slightly problematic to read. – Aaron Chan Apr 5 '14 at 2:44