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 $\mathrm{Sym}\left(V\oplus \wedge^2 V\right)\cong \sum\limits_{\lambda\text{ is a partition}} \mathrm{Schur}_{\lambda}\left(V\right)$ using the "weak" identity for Schur polynomials. Is there any better argument known? Maybe even one yielding a canonical isomorphism?

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?

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 (ed.??), 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 $\mathrm{Sym}\left(V\oplus \wedge^2 V\right)\cong \sum\limits_{\lambda\text{ is a partition}} \mathrm{Schur}_{\lambda}\left(V\right)$ using the "weak" identity for Schur polynomials. Is there any better argument known? Maybe even one yielding a canonical isomorphism?

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 $\mathrm{Sym}\left(V\oplus \wedge^2 V\right)\cong \sum\limits_{\lambda\text{ is a partition}} \mathrm{Schur}_{\lambda}\left(V\right)$ using the "weak" identity for Schur polynomials. Is there any better argument known? Maybe even one yielding a canonical isomorphism?