A little intro: This is for a whirlwind intro to Group Theory as part of another class (QFT) in which we are not proving anything, simply introducing definitions and theorems. We are not using a text book, simply some notes the professor has written up for this intro and the notes are very incomplete. They also do not give any proof or explaination. So here is the problem:(adsbygoogle = window.adsbygoogle || []).push({});

Show that the number of group parameters for [tex] O(N) [/tex] and [tex] SO(N) [/tex] are [tex] \frac{N(N-1)}{2} [/tex]

Does this make sense as the solution:

We have [tex] n^2 [/tex] parameters. For [tex] O(N) [/tex] we have the condition that [tex] O^T O = 1[/tex]. This condition places [tex] \frac{n(n+1)}{2} [/tex] constraints on the parameters. This leaves [tex] n^2 - \frac{n(n+1)}{2} = \frac{n(n-1)}{2} [/tex]

Since [tex] O(N) [/tex] has [tex] detO(N) = \pm 1[/tex] and [tex] SO(N) [/tex] has [tex] detSO(N)=+1[/tex], simply picking the sign of the determinent doesn't force any additional constraint on the parameters.

I can only show this statement: "This condition places [tex] \frac{n(n+1)}{2} [/tex] constraints on the parameters." for specific examples, like N=2 and 3. I can see how it is extended to higher orders, but I cannot prove it in the general case for N. Any ideas? Also, any good books on Group Theory and Lie Algebras for Physicists?

Thanks,

Ryan

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# Homework Help: Group theory question- very basic

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