Self-dual Grassmannian, Wronski map, and representations of ${\mathfrak{gl}}_N$, ${\mathfrak{sp}}_{2r}$, ${\mathfrak{so}}_{2r+1}$

Abstract

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 $\hat{I}{\text{c◯}}_{\hat{\mathbf{I}}›}$ labeled by unordered sets $\hat{\mathbf{I}}›=(\hat{I}{»}^{(1)},\dots ,\hat{I}{»}^{(n)})$ of nonzero partitions with at most $N$ parts, satisfying a condition depending on $d$, and such that $({\otimes }_{i=1}^n{V}_{\hat{I}{»}^{(i)}}{)}^{{\mathfrak{sl}}_N}\ne 0$. Here $V_{\hat{I}{»}^{(i)}}$ is the irreducible ${\mathfrak{gl}}_N$-module with highest weight $\hat{I}{»}^{(i)}$. We show that the closure of a stratum $\hat{I}{\text{c◯}}_{\hat{\mathbf{I}}›}$ is the union of the strata $\hat{I}{\text{c◯}}_{\hat{\mathbf{I}}ž}$, $\hat{\mathbf{I}}ž=(\hat{I}{¾}^{(1)},\dots ,\hat{I}{¾}^{(m)})$, such that there is a partition $\{I_1,\dots ,I_m\}$ of $\{1,2,\dots ,n\}$ with ${\mathrm{Hom}}_{{\mathfrak{gl}}_N}(V_{\hat{I}{¾}^{(i)}},{\otimes }_{j\in I_i}{V}_{\hat{I}{»}^{(j)}})\ne 0$ 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 $N=2r$.

LaTeX, 30 pages, 2 figures