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 [reference-request](/questions/tagged/reference-request "show questions tagged 'reference-request'"), 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](https://mathoverflow.net/q/56162/41291) but it is not what I need... 


**Slightly later:**


As [user44191](https://mathoverflow.net/users/44191/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](https://en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29) combinatorics), although what algebra structures might be involved in this case is not clear to me.



Are there nice isomorphisms $\operatorname{S}^2(k^n)\cong\Lambda^2(k^{n+1})$?
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](https://www.sciencedirect.com/science/article/pii/S0022404906001812) 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 $\mathrm{S}^2 k^n \cong \wedge^2 k^{n+1}$. 


