Difference between revisions of "Aufgaben:Problem 7"

Revision as of 11:39, 15 June 2015

Compute the character table of \(S_4\)

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

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 ==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: