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

TL;DR

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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 $Ω_{\mathbfΛ}$ labeled by unordered sets $\mathbfΛ=(λ^{(1)},\dots,λ^{(n)})$ of nonzero partitions with at most $N$ parts, satisfying a condition depending on $d$, and such that $(\otimes_{i=1}^n V_{λ^{(i)}})^{\mathfrak{sl}_N}\ne 0$. Here $V_{λ^{(i)}}$ is the irreducible $\mathfrak{gl}_N$-module with highest weight $λ^{(i)}$. We show that the closure of a stratum $Ω_{\mathbfΛ}$ is the union of the strata $Ω_{\mathbfΞ}$, $\mathbfΞ=(ξ^{(1)},\dots,ξ^{(m)})$, such that there is a partition $\{I_1,\dots,I_m\}$ of $\{1,2,\dots,n\}$ with $ {\rm {Hom}}_{\mathfrak{gl}_N} (V_{ξ^{(i)}}, \otimes_{j\in I_i}V_{λ^{(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 $N=2r$.

Theorems & Definitions (77)

• Lemma 2.1
• Lemma 3.1
• Theorem 3.2
• proof
• Lemma 3.3
• Proposition 3.4
• proof
• Theorem 3.5
• Theorem 3.6
• Corollary 3.7