# Are there nice isomorphisms $\operatorname{S}^2(k^n)\cong\Lambda^2(k^{n+1})$?

This might be forced to migrate to math.SE but let me still risk it.

The spaces $$\operatorname{S}^2(k^n)$$ and $$\Lambda^2(k^{n+1})$$ from the title have equal dimensions. Is there a natural isomorphism between them?

To make the question more MOish - choosing a basis, elements of $$\operatorname{S}^2(V)$$ can be identified with symmetric matrices, so this does have a Jordan algebra structure. On the other hand, if $$V$$ has even dimension $$2k$$, the adjoint representation of the Lie algebra $$\mathfrak{sp}(2k)$$ can be identified with $$\operatorname{S}^2(V)$$, so that the latter has a Lie algebra structure. On the third hand, regardless of the dimension, the adjoint representation of $$\mathfrak{so}(m)$$ can be identified with $$\Lambda^2(k^m)$$, so the latter also has a Lie algebra structure. And on the fourth hand, some isomorphism as above would give it also some Jordan algebra structure.

So all in all we seem to get a Lie algebra structure and a Jordan algebra structure on each of these spaces. This more MOish question then is: understanding by "natural" an isomorphism interchanging these Lie and Jordan structures in some way - is there a nice description of such an isomorphism?

Slightly more generally, for some involutive automorphism of $$k$$ one might ask for similar isomorphisms with "symmetric" and "exterior" replaced by "Hermitian" and "skew-Hermitian".

I am tagging this with , surely this must be in the literature. I tried The Book of Involutions, but could not find it there. Maybe I did not look hard enough, don't know.

I also tried to look for it here on MO; the closest I could find is Symmetric matrices as a module over the skewsymmetric ones but it is not what I need...

Slightly later:

As user44191 notes in the comment below, there is a more general question about $$\operatorname{S}^i(k^n)\cong\Lambda^i(k^{n+i-1})$$ (pertaining to $$\left(\binom ni\right)=\binom{n+i-1}i$$ and the stars and bars combinatorics), although what algebra structures might be involved in this case is not clear to me.

Let $$E$$ be a $$2$$-dimensional $$k$$-vector space. The Wronksian isomorphism is an isomorphism of $$\mathrm{SL}(E)$$-modules $$\bigwedge^m \mathrm{S}^{m+r-1}(E)\cong \mathrm{S}^m \mathrm{S}^r(E)$$. It is easiest to deduce it from the corresponding identity in symmetric functions (specialized to $$1$$ and $$q$$), but it can also be defined explicitly: see for example Section 2.5 of this paper of Abdesselam and Chipalkatti.
In particular, identifying $$\mathrm{S}^n(E)$$ with the homogeneous polynomial functions on $$E$$ of degree $$n$$, their definition becomes the map $$\wedge^2 \mathrm{S}^n (E) \rightarrow \mathrm{S}^2 \mathrm{S}^{n-1}(E)$$ defined by
$$f \wedge g \mapsto \frac{\partial f}{\partial X} \frac{\partial g}{\partial Y} -\frac{\partial f}{\partial Y} \frac{\partial g}{\partial X}.$$
Now $$\mathrm{S}^{n}(E) \cong k^{n+1}$$ and $$\mathrm{S}^{n-1}(E) \cong k^n$$, so we have the required isomorphism $$\mathrm{S}^2 k^n \cong \wedge^2 k^{n+1}$$.