Pure and Applied Mathematics Quarterly

Volume 13 (2017)

Number 2

Special Issue in Honor of Yuri Manin: Part 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: http://dx.doi.org/10.4310/PAMQ.2017.v13.n2.a4

Authors

Kang Lu (Department of Mathematical Sciences, Indiana University–Purdue University Indianapolis, Indiana, U.S.A.)

E. Mukhin (Department of Mathematical Sciences, Indiana University–Purdue University Indianapolis, Indiana, U.S.A.)

A. Varchenko (Department of Mathematics, University of North Carolina, Chapel Hill, N.C., U.S.A.)

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 $N = 2r$.

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Received 11 May 2017

Published 14 September 2018