14
$\begingroup$

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.

$\endgroup$
6

1 Answer 1

22
$\begingroup$

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 $\mathrm{S}^2 k^n \cong \wedge^2 k^{n+1}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.