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There is a natural homomorphism that I am pretty sure is not zero in general that can be described most easily using the dual language of the Lie algebra $\frak g$: Recall that there is a differen …

This is a standard result in representation theory: Let the quadratic form on $\mathbb{H}$ be $\langle x,x\rangle> = x\bar x$ (which is positive definite). (Note that I am considering $\mathbb{R}$ …

Re-Amended Answer: My guess for $\mathrm{SO}(5)$ appears to have been correct in one sense, but not in another. It's true that a nonzero Weyl tensor in this case has to have stabilizer of dimension a …

This is a trick question, right? It's not true when $m=2$ because, then $\mathrm{SO}(m{-}1)=\mathrm{SO}(1)$ is trivial, so that all polynomials on $2$-by-$2$ symmetric matrices are invariants, and th …

I think you want to look at Faà di Bruno's formula, and the description of composition of formal power series, particularly the historical remarks. The linear representation of this group that allows …

You appear to have made a mistake in your calculation of the branching rules. The answer given in the wiki is correct, but it seems that you are using the 'wrong' subgroup of $\mathrm{SU}(6)$. Perha …

Maybe I'm misunderstanding something, but it seems that the answer is probably 'no', at least if $d = \dim_\mathbb{C} V$ is large enough. What really matters is the $n$-dimensional real subspace $\ …

I'll give an answer in a few parts: First, for the $n$-sphere $S^n\subset\mathbb{R}^{n+1}$: The obvious thing to do is to consider the vector fields $$ X_{ij} = x_i\frac{\partial }{\partial x_j} - x …

As per the OP's comment, we are to assume that $\mathrm{G}_2$ and $\mathrm{F}_4$ mean the complex simple Lie groups. Let's start with $\mathrm{G}_2\subset\mathrm{SO}(7,\mathbb{C})$, in its standard re …

The answer is 'no', though I don't know an easy way to see this without doing an explicit calculation. Here is where to look though, if you want to do the calculation yourself: Things work out a bit …

First, let me fix a misunderstanding: $\mathrm{Sp}(n)$ does not sit in $\mathrm{GL}(n,\mathbb{C})$, but in $\mathrm{GL}(2n,\mathbb{C})$, so I'll assume that you mean, for the second part that $A$ lie …

In the split cases of interest to you ($D=4,6$ and of arbitrary signature), you can find a discussion of the local existence and generality in a 2000 paper of mine entitled Pseudo-Riemannian metrics w …

This is easily computed via LiE: $Sym^2(\mathbb{R}^{16})$ breaks into three irreducible components: The trivial representation, i.e., $\mathbb{R}$, The standard representation of $\mathrm{SO}(9)$, …

This is about your specific question. For any vector space $V$ of dimension $n$, one has canonical decompositions $$ S^2(S^2V^*) \simeq S^4(V^*)\oplus K(V^*) $$ and $$ S^2(\Lambda^2V^*) \simeq \Lamb …

As Qiaochu Yuan wrote, these irreducible decompositions are classical. You can read about them in Hermann Weyl's The Classical Groups (for example), and the questions you are asking are basically exe …