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Pure and Applied Mathematics Quarterly
Volume 13 (2017)
Number 2
Special Issue in Honor of Yuri Manin: Part 2 of 2
Guest Editors: Lizhen Ji, Kefeng Liu, Yuri Tschinkel, and Shing-Tung Yau
Self-dual Grassmannian, Wronski map, and representations of $\mathfrak{gl}_N, \mathfrak{sp}_{2r}, \mathfrak{so}_{2r+1}$
Pages: 291 – 335
DOI: https://dx.doi.org/10.4310/PAMQ.2017.v13.n2.a4
Authors
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 $\Omega_{\Delta}$ labeled by unordered sets $\Delta = (\lambda^{(1)}, \dotsc , \lambda^{(n)})$ of nonzero partitions with at most $N$ parts, satisfying a condition depending on $d$, and such that $(\otimes^n_{i=1} V_{\lambda^{(i)}})^{\mathfrak{sl}_N} \neq 0$. Here $V_{\lambda^{(i)}}$ is the irreducible $\mathfrak{gl}_N$-module with highest weight $\lambda^{(i)}$. We show that the closure of a stratum $\Omega_{\Delta}$ is the union of the strata $\Omega_{\Xi} , \Xi = (\xi^{(1)}, \dotsc , \xi^{(m)})$, such that there is a partition $\lbrace I_1, \dotsc , I_m \rbrace$ of $\lbrace 1, 2, \dotsc , n \rbrace$ with $\mathrm{Hom}_{\mathfrak{gl}_N} (V_{\xi^{(i)}} , \otimes_{j \in I_i} V_{λ^{(j)}} \neq 0$ for $i = 1, \dotsc , m$. 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 $N = 2r$.
Received 11 May 2017
Published 14 September 2018