  We define a $\mathfrak{gl}_N$-stratification of the Grassmannian of $N$
planes $\mathrm{Gr}(N,d)$. The $\mathfrak{gl}_N$-stratification consists of
strata $\Omega_{\mathbf{\Lambda}}$ labeled by unordered sets
$\mathbf{\Lambda}=(\lambda^{(1)},\dots,\lambda^{(n)})$ of nonzero partitions
with at most $N$ parts, satisfying a condition depending on $d$, and such that
$(\otimes_{i=1}^n V_{\lambda^{(i)}})^{\mathfrak{sl}_N}\ne 0$. Here
$V_{\lambda^{(i)}}$ is the irreducible $\mathfrak{gl}_N$-module with highest
weight $\lambda^{(i)}$. We show that the closure of a stratum
$\Omega_{\mathbf{\Lambda}}$ is the union of the strata $\Omega_{\mathbf\Xi}$,
$\mathbf{\Xi}=(\xi^{(1)},\dots,\xi^{(m)})$, such that there is a partition
$\{I_1,\dots,I_m\}$ of $\{1,2,\dots,n\}$ with $ {\rm {Hom}}_{\mathfrak{gl}_N}
(V_{\xi^{(i)}}, \otimes_{j\in I_i}V_{\lambda^{(j)}}\big)\neq 0$ for
$i=1,\dots,m$. The $\mathfrak{gl}_N$-stratification of the Grassmannian agrees
with the Wronski map.
  We introduce and study the new object: the self-dual Grassmannian
$\mathrm{sGr}(N,d)\subset \mathrm{Gr}(N,d)$. Our main result is a similar
$\mathfrak{g}_N$-stratification of the self-dual Grassmannian governed by
representation theory of the Lie algebra $\mathfrak
{g}_{2r+1}:=\mathfrak{sp}_{2r}$ if $N=2r+1$ and of the Lie algebra $\mathfrak
g_{2r}:=\mathfrak{so}_{2r+1}$ if $N=2r$.
