Hi
I'm studying character tables in my representation theory module and don't understand how to create them for the cyclic groups S3 or S4. COuld anyone help?
I understand:
- that the set of each type of cycle forms a conjugacy class
eg. for S3: {1-cycles in S3}, {2-cycles in S3}, {3-cycles in S3} these sets are conjugacy classes
- the size of the centraliser for each conjugacy class is the size of the group divided by the size of the conjugacy class:
eg. |C(g)| = |G|/|x^G| for all g in G (x^G is the conjugacy class for x in G)
- the square of the entries in each column of the character table, sum to the value of the centraliser for the conjugacy class
- the cols and rows of the table are normal to each other
- the number of characters in the table is equal to the number of conjugacy classes
- for the linear character of degree 1, X1, the character of each conjugacy class is 1, so the top row is all ones in the table.
For S3, I get as far as:
conjugacy classes: | 1-cycles | 2-cycles | 3-cycles
|centraliser| | 6 | 2 | 3
X1 | 1 | 1 | 1
X2 | | |
X3 | | |
(This is meant to be a table...Sorry. Rows 2 and 3 are empty)
I can see that for the 1-cycle that a +/-2 and +/-1 is missing.
Also the other entries in the column of the 3-cycles should be +/-1 and +/-1.
The entries in the column of the 2-cycle should be 0 and +/-1.
How do I know which way round to place them? Is there a trick to it? Is there a way to verify the numbers in the rows?
For S4 I got as far as:
conjugacy classes: | 1-cycles | 2-cycles | 3-cycles | 4-cycles | 2-cycle,2-cycle
|centraliser| | 24 | 4 | 3 | 4 | 8
X1 | 1 | 1 | 1 | 1 | 1
X2 | | | | |
X3 | | | | |
X4 | | | | |
X5 | | | | |
Thanks for your help.