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.**



Decomposition into irreducible components of a representation of $Spin(9)$
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 $\Lambda^4(\mathbb{R}^9)$.


