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This MO answer cites the Goodman-Wallach book to affirm that: $$\mathrm{Sym}^k\left(\mathbb{R}^n\right)=\mathcal{H}^k\oplus q\mathcal{H}^{k-2}\oplus q^2\mathcal{H}^{k-4}\oplus\cdots$$ with $\mathrm{...

Question: Let $P\in \mathbb{R}^{d\times n}$ be a $d$-rank real matrix and $PP^T = c I_d$ with a certain constant $c > 0$. Under what additional conditions of $P$ does there exist an orthogonal ...

Let $SO(2n)$ denote the special orthogonal group of $2n\times 2n$ matrices over the complex numbers. Consider the action of $SO(2n)\times SO(2n)$ on the set of $2n\times 2n$ matrices : $ADB^{T}$, ...