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It is well known that the group $Spin(9)$ acts linearly on the vector space $\mathbb{R}^{16}$ (see for example "Spinors and calibrations" by R. Harvey).

Consider the induced representation of $Spin(9)$ in the space of symmetric quadratic forms on $\mathbb{R}^{16}$, i.e. in $Sym^2(\mathbb{R}^{16})$.

I am interested in a decomposition of (the complexification of) this space into irreducible components. In particular, is it multiplicity free? How many irreducible components? Description in terms of highest weights might also be useful.

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This is easily computed via LiE: $Sym^2(\mathbb{R}^{16})$ breaks into three irreducible components:

  1. The trivial representation, i.e., $\mathbb{R}$,
  2. The standard representation of $\mathrm{SO}(9)$, i.e., $\mathbb{R}^9$, and
  3. The irreducible representation of highest weight $(0,0,0,2)$, of dimension 126, which happens to be isomorphic to $\Lambda^4(\mathbb{R}^9)$.
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