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created: 2023-03-26T11:58
updated: 2024-04-14T15:14
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# Definitions

$$\begin{aligned}
\mathbf{S L}(n, \mathbf{R}) & =\{A \in \mathbf{G} \mathbf{L}(n, \mathbf{R}) \mid \operatorname{det}(A)=1\} \quad \text { (real special linear group) } \\
\mathbf{S L}(n, \mathbf{C}) & =\{A \in \mathbf{G L}(n, \mathbf{C}) \mid \operatorname{det}(A)=1\} \quad \text { (complex special linear group) } \\
\mathbf{U}(n) & =\left\{A \in \mathbf{G L}(n, \mathbf{C}) \mid A A^*=I\right\} \quad \text { (unitary group) } \\
\mathbf{S O}(n) & =\{A \in \mathbf{O}(n) \mid \operatorname{det}(A)=1\} \quad \text { (special orthogonal group) } \\
\mathbf{S U}(n) & =\{A \in \mathbf{U}(n) \mid \operatorname{det}(A)=1\} \quad \text { (special unitary group) }
\end{aligned}$$

# Dimensions

- $\dim \GL_n(\RR) = n^2$
- $\dim \Orth_n(\RR) = {1\over 2}n(n-1)$ since $\Orth_n(\RR) = \ker\qty{M\mapsto M^t M}$.
	- Regard as $\dim \GL_n(\RR) - \dim S_n(\RR) = n^2 - {1\over 2}n(n+1)$ where $S_n$ is the space of symmetric matrices, the image of the above map.
- $\dim \SO_n(\RR)  = {1\over 2}n(n-1)$.
- $\dim \SL_n(\RR) = \dim \GL_n(\RR) - 1 = n^2 - 1$ since $\SL_n(\RR) = \ker \det$.
- $\dim_\CC \U_n(\CC) = n^2$
- $\dim_\CC \SU_n(\CC) = n^2-1$.
- $\dim \Sp_{2n}(\RR) = n(2n+1)$.
- $\dim \PGL_n(\RR) = n^2-1$.
- $\dim \PSL_n(\RR) = n^2-1$ since $\PSL_n = \SL_n/\gens{\pm I}$, a quotient by a finite group.