Let $V$ be a finite-dimensional $K$-vector space. Then, the symmetric power $\mathrm{Sym}\left(V\oplus \wedge^2 V\right)$ is isomorphic to the direct sum of all Schur functors applied to $V$ (each one time only) as a Schur functor.

This follows by comparison of characters using the so-called Schur identity

$\sum\limits_{\lambda\text{ is a partition}} s_{\lambda} = \prod\limits_i \left(1-\xi_i\right)^{-1} \cdot \prod\limits_{i < j} \left(1-\xi_i\xi_j\right)^{-1}$,

where $s_{\lambda}$ denote the Schur "polynomials" and $\xi_i$ are countably many indeterminates (see, for instance, chapter 5.4 in "M. Lothaire", *Algebraic Combinatorics on Words*). While it is easy to like the Schur identity, it is hard not to dislike the proof of the "strong" isomorphism $\mathrm{Sym}\left(V\oplus \wedge^2 V\right)\cong \bigoplus\limits_{\lambda\text{ is a partition}} \mathrm{Schur}_{\lambda}\left(V\right)$ using the "weak" identity for Schur polynomials. Is there any better argument known? Maybe even one yielding a *canonical* isomorphism?