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$\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}$ Here's the new question:

$ \SU_2 $ is a subgroup of $ \GL_2(\mathbb{C}) $. $ \PSU_2 $ is not. However $ \PSU_2 \cong \SO_3(\mathbb{R}) $ is a subgroup of $ \GL_3(\mathbb{C}) $.

$ \SU_3 $ is a subgroup of $ \GL_3(\mathbb{C}) $. The smallest $ k $ such that $ \PSU_3 $ is a subgroup of $ \GL_k(\mathbb{C}) $ is $ k=8 $ see the answer by Ycor

https://math.stackexchange.com/questions/4394590/minimal-dimension-for-psu-3-as-a-matrix-group/4418592#4418592

In general, I would like to know what is the smallest dimension of a faithful complex representation of $ \PSU_n $?

For any $ n $, $ \SU_n $ has dimension $ n^2-1 $ so the adjoint representation realizes $ \PSU_n $ as a subgroup of $ \SO_{n^2-1}(\mathbb{R}) $ which in turn is contained inside of $ \GL_{n^2-1}(\mathbb{C}) $.

My intuition is that the adjoint representation $ n^2-1 $ should always be the smallest dimension containing $ \PSU_n $ as a matrix subgroup. However I know this is not true for all simple groups. For example the exceptional group $ G_2 $ has dimension $ 14 $ but already has a faithful representation in dimension $ 7 $.

If $ \SU_n $ is inside of $ \GL_k $ then it must be in the maximal compact subgroup $ \U_k $ and in fact in the maximal semi-simple compact $ \SU_k $. So the question really becomes what is the least $ k $ such that $ \PSU_n $ is a subgroup of $ \SU_k $?

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    $\begingroup$ Here's the link to the original MathSE question: math.stackexchange.com/questions/4394590. I've posted an answer there (yes, the smallest faithful representation has dimension 8) $\endgroup$
    – YCor
    Apr 2, 2022 at 12:52
  • $\begingroup$ That's very helpful! I'll just keep the MSE one the way it is and then change this one to be about the general case $ PSU_n $. $\endgroup$
    – Ian Gershon Teixeira
    Apr 2, 2022 at 13:04
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    $\begingroup$ By the same argument as in the linked answer, the question is the same as computing the smallest dimension of a faithful (or equivalently nontrivial) complex (holomorphic) representation of the complex Lie group $\mathrm{PGL}_n(\mathbf{C})$. $\endgroup$
    – YCor
    Apr 2, 2022 at 13:42
  • $\begingroup$ A side remark: there exist non-continuous representations of $\mathrm{SU}(n)\to\mathrm{GL}_m(\mathbf{C})$ (e.g., making use of non-continuous automorphisms of the field $\mathbf{C}$). I believe the question concerns continuous representations (also, I'd guess the conclusions to be the same for non-continuous representation although requiring additional arguments). $\endgroup$
    – YCor
    Apr 2, 2022 at 15:11

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