Pure and Applied Mathematics Quarterly Volume 13, Number 2, 291–335, 2017
Self-dual Grassmannian, Wronski map, and representations of glN , sp2r, so2r+1 Kang Lu, E. Mukhin, and A. Varchenko
Dedicated to Yuri Ivanovich Manin on the occasion of his 80th birthday
Abstract: We define a glN -stratification of the Grassmannian of N planes Gr(N,d). The glN -stratification consists of strata ΩΛ labeled by unordered sets Λ =(λ(1),...,λ(n)) of nonzero parti- tions with at most N parts, satisfying a condition depending on ⊗n slN d,andsuchthat( i=1Vλ(i) ) = 0. Here Vλ(i) is the irreducible (i) glN -module with highest weight λ . We show that the closure of (1) (m) astratumΩΛ is the union of the strata ΩΞ, Ξ =(ξ ,...,ξ ), { } { } such that there is a partition I1,...,Im of 1, 2,...,n with (i) ⊗ ∈ (j) HomglN (Vξ , j Ii Vλ =0fori =1,...,m.TheglN -stratification of the Grassmannian agrees with the Wronski map. We introduce and study the new object: the self-dual Grass- mannian sGr(N,d) ⊂ Gr(N,d). Our main result is a similar gN - stratification of the self-dual Grassmannian governed by represen-
tation theory of the Lie algebra g2r+1 := sp2r if N =2r +1 andof the Lie algebra g2r := so2r+1 if N =2r.
1. Introduction
The Grassmannian Gr(N,d)ofN-dimensional subspaces of the complex d- dimensional vector space has the standard stratification by Schubert cells Ωλ labeled by partitions λ =(d − N λ1 ... λN 0). A Schubert cycle is the closure of a cell Ωλ.ItiswellknownthattheSchubertcycleΩλ is the union of the cells Ωξ such that the Young diagram of λ is inscribed into the Young diagram of ξ. This stratification depends on a choice of a full flag in the d-dimensional space. In this paper we introduce a new stratification of Gr(N,d)governedby representation theory of glN and called the glN -stratification, see Theorem (1) (n) 3.5.TheglN -strata ΩΛ are labeled by unordered sets Λ =(λ ,...,λ )of
Received 11 May 2017.
291 292 Kang Lu et al.
(i) − (i) (i) nonzero partitions λ =(d N λ1 ... λN 0) such that