Aufgaben:Problem 7
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Exercise
Compute the character table of \(S_4\)
Solution
Let \(G\) be a finite group
Let \(g \in G\)
Let \(\psi \text{ and } \phi \in \mathbb{C}^G\)
There is an inner product defined as \( (\psi , \phi)_G = \frac{1}{\left|G\right|} \sum_{y \in G} \psi (y) \phi (y)^{*} \)
We know that \( \psi_r , \psi_s \) are orthonormal if \( (\psi_r , \psi_s)_G = \delta_{rs} \)
Charactertable looks like:
\(S_4\) | \(e\) | \(6C_2\) | \(8C_3\) | \(6C_4\) | \(3C_{2,2}\) |
---|---|---|---|---|---|
\(U\) | 1 | 1 | 1 | 1 | 1 |
\(U'\) | 1 | -1 | 1 | -1 | 1 |
\(V\) | 3 | 1 | 0 | -1 | -1 |
\(V'\) | 3 | -1 | 0 | 1 | -1 |
\(W\) | 2 | 0 | -1 | 0 | 2 |
References
https://unapologetic.wordpress.com/2010/11/08/the-character-table-of-s4/
https://www.itp.uni-hannover.de/~flohr/lectures/symm/handout2.pdf