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Full Text (PDF Format) 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 n N (i) ⊗n slN − (1.1) ( i=1Vλ(i) ) =0, λj = N(d N), i=1 j=1 (i) where Vλ(i) is the irreducible glN -module with highest weight λ .Wehave dim ΩΛ = n. We call the closure of a stratum ΩΛ in Gr(N,d)aglN -cycle.The glN -cycle ΩΛ is an algebraic set in Gr(N,d). We show that ΩΛ is the union of (1) (m) { } the strata ΩΞ, Ξ =(ξ ,...,ξ ), such that there is a partition I1,...,Im { } (i) ⊗ ∈ (j) of 1, 2,...,n with HomglN (Vξ , j Ii Vλ =0fori =1,...,m, see Theo- rem 3.8. Thus we have a partial order on the set of sequences of partitions satis- { } { } fying (1.1). Namely Λ Ξ if there is a partition I1,...,Im of 1, 2,...,n (i) ⊗ ∈ (j) with HomglN (Vξ , j Ii Vλ =0fori =1,...,m. An example of the corre- sponding graph is given in Example 3.9.TheglN -stratification can be viewed as the geometrization of this partial order. Let us describe the construction of the strata in more detail. We identify the Grassmannian Gr(N,d) with the Grassmannian of N-dimensional sub- spaces of the d-dimensional space Cd[x] of polynomials in x of degree less than d. In other words, we always assume that for X ∈ Gr(N,d), we have 1 1 X ⊂ Cd[x]. Set P = C ∪{∞}. Then, for any z ∈ P ,wehavetheosculating flag F(z), see (3.3), (3.4). Denote the Schubert cells corresponding to F(z)by Ωλ(F(z)). Then the stratum ΩΛ consists of spaces X ∈ Gr(N,d) such that F X belongs to the intersection of Schubert cells Ωλ(i) ( (zi)) for some choice 1 of distinct zi ∈ P : n F ⊂ ΩΛ = Ωλ(i) ( (zi)) Gr(N,d). z1,...,zn i=1 zi=zj 1 n A stratum ΩΛ is a ramified covering over (P ) without diagonals quotient by the free action of an appropriate symmetric group, see Proposition 3.4.The ⊗n slN degree of the covering is dim( i=1Vλ(i) ) . For example, if N = 1, then Gr(1,d)isthe(d − 1)-dimensional projective space of the vector space Cd[x]. The strata Ωm are labeled by unordered sets m =(m1,...,mn) of positive integers such that m1 +···+mn = d−1. A stra- tum Ωm consists of all polynomials f(x) which have n distinct zeros of multi- plicities m1,...,mn. In this stratum we also include the polynomials of degree d−1−mi with n−1 distinct roots of multiplicities m1,...,mi−1,mi+1,...,mn. We interpret these polynomials as having a zero of multiplicity mi at infinity. Self-dual Grassmannian 293 The stratum Ω(1,...,1) is open in Gr(1,d). The union of other strata is classically called the swallowtail and the gl1-stratification is the standard stratification of the swallowtail; see, for example, Section 2.5 of Part 1 of [AGV]. The glN -stratification of Gr(N,d) agrees with the Wronski map Wr : Gr(N,d) → Gr(1,N(d − N)+1) which sends an N-dimensional subspace of polynomials to its Wronskian i−1 i−1 N det(d fj/dx )i,j=1,wheref1(x),...,fN (x) is a basis of the subspace. For − any gl1-stratum Ωm of Gr(1,N(d N) + 1), the preimage of Ωm under the Wronski map is the union of glN -strata of Gr(N,d) and the restriction of the Wronski map to each of those strata ΩΛ is a ramified covering over Ωm of ⊗n slN degree b(Λ)dim( i=1Vλ(i) ) , where b(Λ) is some combinatorial symmetry coefficient of Λ,see(3.9). The main goal of this paper is to develop a similar picture for the new ob- ject sGr(N,d) ⊂ Gr(N,d), called self-dual Grassmannian.LetX ∈ Gr(N,d) be an N-dimensional subspace of polynomials in x.LetX∨ be the N-dimen- sional space of polynomials which are Wronski determinants of N −1elements of X: N−1 ∨ i−1 i−1 X = {det d fj/dx ,fj(x) ∈ X}. i,j=1 The space X is called self-dual if X∨ = g · X for some polynomial g(x), see [MV1]. We define sGr(N,d) as the subset of Gr(N,d) of all self-dual spaces. It is an algebraic set. The main result of this paper is the stratification of sGr(N,d)governed by representation theory of the Lie algebras g2r+1 := sp2r if N =2r +1 and g2r := so2r+1 if N =2r. This stratification of sGr(N,d) is called the gN -stratification, see Theorem 4.11. The gN -stratification of sGr(N,d) consists of gN -strata sΩΛ,k labeled by (1) (n) unordered sets of dominant integral gN -weights Λ =(λ ,...,λ ), equipped ⊗n gN with nonnegative integer labels k =(k1,...,kn), such that ( i=1Vλ(i) ) =0 and satisfying a condition similar to the second equation in (1.1); see Section (i) 4.3.HereVλ(i) is the irreducible gN -module with highest weight λ . Different liftings of an slN -weight to a glN -weight differ by a vector (k,...,k) with integer k.Ourlabelki is an analog of this parameter in the case of gN . 1 n A gN -stratum sΩΛ,k is a ramified covering over (P ) without diagonals quotient by the free action of an appropriate symmetric group. The degree ⊗n gN of the covering is dim( i=1Vλ(i) ) and, in particular, dim sΩΛ,k = n;see 294 Kang Lu et al. Proposition 4.9. We call the closure of a stratum sΩΛ,k in sGr(N,d)agN - cycle.ThegN -cycle sΩΛ,k is an algebraic set. We show that sΩΛ,k is the (1) (m) union of the strata sΩΞ,l, Ξ =(ξ ,...,ξ ), such that there is a partition { } { } ⊗ I1,...,Im of 1, 2,...,n satisfying HomgN (Vξ(i) , j∈Ii Vλ(j) =0fori = 1,...,m, and the appropriate matching of labels; see Theorem 4.13. If N =2r, there is exactly one stratum of top dimension 2(d − N)= dim sGr(N,d). For example, the so5-stratification of sGr(4, 6) consists of 9 strata of dimensions 4, 3, 3, 3, 2, 2, 2, 2, 1, see the graph of adjacencies in Example 4.14.IfN =2r + 1, there are many strata of top dimension d − N (except in the trivial cases of d =2r +1andd =2r + 2). For example, the sp4-stratification of sGr(5, 8) has four strata of dimension 3; see Section 4.7. In all cases we have exactly one one-dimensional stratum corresponding to n =1,Λ =(0),andk =(d − N). Essentially, we obtain the gN -stratification of sGr(N,d) by restricting the glN -stratification of Gr(N,d)tosGr(N,d). For X ∈ sGr(N,d), the multiplicity of every zero of the Wronskian of X is divisible by r if N =2r and by N if N =2r + 1. We define the reduced Wronski map Wr : sGr(N,d) → Gr(1, 2(d − N)+1)ifN =2r and Wr : sGr(N,d) → Gr(1,d − N +1)if N =2r + 1 by sending X to the r-th root of its Wronskian if N =2r and to the N-th root if N =2r +1.
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