Decomposition into irreducible components of a representation of $Spin(9)$

Question

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.

Answer 1

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 $\Lambda^4(\mathbb{R}^9)$.