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It is a well known fact that the Riemann Curvature Tensor ${R^{\alpha}}_{\beta\gamma\lambda}$ in $n$ dimensions have

$\frac{n^2(n^2-1)}{12}$

independent components. I have seen the combinatorial proof of this fact (for instance in this MS post or in Caroll's GR book section 3.7). I was wondering if there are easier ways of seeing this fact.

My question: Is there a way of seeing this formula in terms of group actions? (e.g. as some group acting on the space of tensors?)

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