Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus
The self-dual spaces of polynomials are related to Bethe vectors in the Gaudin model associated to the Lie algebras of types B and C. In this paper, we give lower bounds for the number of real self-dual spaces in intersections of Schubert varieties related to osculating flags in the Grassmannian. Th...
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Lu, K. 2025-11-24T10:40:49Z 2018 Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus / K. Lu // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 20 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14N99; 17B80; 82B23 arXiv: 1710.06534 https://nasplib.isofts.kiev.ua/handle/123456789/209526 https://doi.org/10.3842/SIGMA.2018.046 The self-dual spaces of polynomials are related to Bethe vectors in the Gaudin model associated to the Lie algebras of types B and C. In this paper, we give lower bounds for the number of real self-dual spaces in intersections of Schubert varieties related to osculating flags in the Grassmannian. The higher Gaudin Hamiltonians are self-adjoint with respect to a nondegenerate indefinite Hermitian form. Our bound comes from the computation of the signature of this form. The author thanks E. Mukhin and V. Tarasov for useful discussions. The author also thanks the referees for their comments and suggestions that substantially improved the first version of this paper. This work was partially supported by the Zhejiang Province Science Foundation, grant No. LY14A010018. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus Article published earlier |
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Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus |
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Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus Lu, K. |
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Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus |
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Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus |
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Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus |
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Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus |
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lower bounds for numbers of real self-dual spaces in problems of schubert calculus |
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Lu, K. |
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Lu, K. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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The self-dual spaces of polynomials are related to Bethe vectors in the Gaudin model associated to the Lie algebras of types B and C. In this paper, we give lower bounds for the number of real self-dual spaces in intersections of Schubert varieties related to osculating flags in the Grassmannian. The higher Gaudin Hamiltonians are self-adjoint with respect to a nondegenerate indefinite Hermitian form. Our bound comes from the computation of the signature of this form.
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1815-0659 |
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https://nasplib.isofts.kiev.ua/handle/123456789/209526 |
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Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus / K. Lu // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 20 назв. — англ. |
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AT luk lowerboundsfornumbersofrealselfdualspacesinproblemsofschubertcalculus |
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2025-12-03T09:54:13Z |
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2025-12-03T09:54:13Z |
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1850480424150630400 |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 046, 15 pages
Lower Bounds for Numbers of Real Self-Dual Spaces
in Problems of Schubert Calculus
Kang LU
Department of Mathematical Sciences, Indiana University – Purdue University Indianapolis,
402 North Blackford St, Indianapolis, IN 46202-3216, USA
E-mail: lukang@iupui.edu
Received November 27, 2017, in final form May 07, 2018; Published online May 14, 2018
https://doi.org/10.3842/SIGMA.2018.046
Abstract. The self-dual spaces of polynomials are related to Bethe vectors in the Gaudin
model associated to the Lie algebras of types B and C. In this paper, we give lower bounds for
the numbers of real self-dual spaces in intersections of Schubert varieties related to osculating
flags in the Grassmannian. The higher Gaudin Hamiltonians are self-adjoint with respect
to a nondegenerate indefinite Hermitian form. Our bound comes from the computation of
the signature of this form.
Key words: real Schubert calculus; self-dual spaces; Bethe ansatz; Gaudin model
2010 Mathematics Subject Classification: 14N99; 17B80; 82B23
1 Introduction
It is well known that the problem of finding the number of real solutions to algebraic systems
is very difficult, and not many results are known. In particular, the counting of real points in
problems of Schubert calculus in the Grassmannian has received a lot of attention, see [2, 5, 6,
7, 13, 19, 20] for example. In this paper, we give lower bounds for the numbers of real self-dual
spaces in intersections of Schubert varieties related to osculating flags in the Grassmannian.
We define the Grassmannian Gr(N, d) to be the set of all N -dimensional subspaces of the
d-dimensional space Cd[x] of polynomials in x of degree less than d. In other words, we always
assume for X ∈ Gr(N, d), we have X ⊂ Cd[x]. Set P1 = C ∪ {∞}. Then, for any z ∈ P1,
we have the osculating flag F(z), see (4.1), (4.2). Denote the Schubert cells corresponding
to F(z) by Ωξ(F(z)), where ξ = (d − N > ξ1 > ξ2 > · · · > ξN > 0) are partitions. Then the
set Ωξ,z consists of spaces X ∈ Gr(N, d) such that X belongs to the intersection of Schubert
cells Ωξ(i)(F(zi)) for z = (z1, . . . , zn) and ξ =
(
ξ(1), . . . , ξ(n)
)
, where all zi ∈ P1 are distinct
and ξ(i) are partitions, see (4.3). A point X ∈ Gr(N, d) is called real if it has a basis consisting
of polynomials with all coefficients real. A lower bound for the number of real points in Ωξ,z is
given in [13].
Let X ∈ Gr(N, d) be an N -dimensional subspace of polynomials in x. Let X∨ be the N -
dimensional space of polynomials which are Wronskian determinants of N − 1 elements of X
X∨ =
{
det
(
di−1ϕj/dx
i−1
)N−1
i,j=1
, ϕj(x) ∈ X
}
.
The space X is called self-dual if X∨ = ψ ·X for some polynomial ψ(x), see [16]. We define sΩξ,z
the subset of Ωξ,z consisting of all self-dual spaces. Our main result of this paper is a lower
bound for the number of real self-dual spaces in Ωξ,z, see Corollary 7.4, i.e., a lower bound for
the number of real points in sΩξ,z, by following the idea of [13].
Let gN be the Lie algebra so2r+1 if N = 2r or the Lie algebra sp2r if N = 2r + 1. We also
set g3 = sl2. It is known from [10], see also [16, Section 6.1], that if sΩξ,z is nonempty, then
mailto:lukang@iupui.edu
https://doi.org/10.3842/SIGMA.2018.046
2 K. Lu
ξ
(s)
i − ξ
(s)
i+1 = ξ
(s)
N−i − ξ
(s)
N−i+1 for i = 1, . . . , N − 1. Hence the slN -weight corresponding to the
partition ξ(s) has certain symmetry and thus induces a gN -weight λ(s), cf. (4.4). Therefore, the
sequence of partitions ξ with nonempty sΩξ,z can be expressed in terms of a sequence of dominant
integral gN -weights λ =
(
λ1, . . . , λ(n)
)
and a sequence of nonnegative integers k = (k1, . . . , kn),
see Lemma 4.1. In particular, ki = ξ
(i)
N . We call ξ, z or λ,k, z the ramification data.
As a subset of Ωξ,z, sΩξ,z can be empty even if Ωξ,z is infinite. However, if sΩξ,z is nonempty,
then sΩξ,z is finite if and only if Ωξ,z is finite. More precisely, if
|ξ| :=
n∑
i=1
∣∣ξ(i)
∣∣ = N(d−N),
then the number of points in sΩξ,z counted with multiplicities equals the multiplicity of the
trivial gN -module in the tensor product Vλ(1) ⊗ · · · ⊗ Vλ(n) of irreducible gN -modules of highest
weights λ(1), . . . , λ(n). Since we are interested in the counting problem, from now on, we always
assume that |ξ| = N(d−N).
For brevity, we consider∞ to be real. If all z1, . . . , zn are real, it follows from [14, Theorem 1.1]
that all points in sΩξ,z are real. Hence the number of real points is maximal possible in this
case. Moreover, it follows from [15, Corollary 6.3] that all points in sΩξ,z are multiplicity-free.
Then we want to know how many real points we can guarantee in other cases. In general,
a necessary condition for the existence of real points is that the set {z1, . . . , zn} should be
invariant under the complex conjugation and the partitions at the complex conjugate points are
the same. In other words,
(
λ(i), ki
)
=
(
λ(j), kj
)
provided zi = z̄j . In this case we say that z, λ, k
are invariant under conjugation. Moreover, the greatest common divisor of X ∈ sΩξ,z in this case
is a real polynomial. Hence we reduce the problem to the case that ki = 0, for all i = 1, . . . , n.
The derivation of the lower bounds is based on the identification of the self-dual spaces of
polynomials with points of spectrum of higher Gaudin Hamiltonians of types B and C (gN ,
N > 4) built in [10] and [16], see Theorem 5.2. We show that higher Gaudin Hamiltonians of
types B and C have certain symmetry with respect to the Shapovalov form which is positive
definite Hermitian, see Proposition 6.1. In particular, these operators are self-adjoint with
respect to the Shapovalov form for real z1, . . . , zn and hence have real eigenvalues. Therefore, it
follows from Theorem 5.2 that self-dual spaces with real z1, . . . , zn are real.
If some of z1, . . . , zn are not real, but the data z, λ, k are invariant under the complex
conjugation, the higher Gaudin Hamiltonians are self-adjoint with respect to a nondegenerate
(indefinite) Hermitian form. One of the key observations for computing the lower bound for the
number of real points in sΩξ,z is the fact that the number of real eigenvalues of such operators
is at least the absolute value of the signature of the Hermitian form, see Lemma 6.4.
The computation of the signature of the form is reduced to the computation of the character
values of products of symmetric groups on products of commuting transpositions. The formula
for such character, similar to the Frobenius formula in [4] and [13, Proposition 2.1], is given in
Proposition 3.1. Consequently, we obtain our main result, a lower bound for the number of real
points in sΩξ,z for N > 4, see Corollary 7.4. The case N = 2 is the same as that of [13] since
every 2-dimensional space of polynomials is self-dual. By the proof of [10, Theorem 4.19], the
case N = 3 is reduced to the case of [13], see Section 7.2.
Based on the identification of the self-self-dual spaces of polynomials with points of spectrum
of higher Gaudin Hamiltonians of type G2 built in [1] and [8], we expect that lower bounds for
the numbers of real self-self-dual spaces in Ωξ,z with N = 7 can also be given in a similar way
as conducted in this paper.
It is also interesting to find an algorithm to compute all (real) self-dual spaces with prescribed
ramification data. The solutions to the Bethe ansatz equations described in [9] can be used to
find nontrivial examples of self-dual spaces.
Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus 3
The paper is organized as follows. We start with the standard notation of Lie theory in
Section 2 and computations of characters of a product of symmetric groups in Section 3. Then we
recall notation and definitions for osculating Schubert calculus and self-dual spaces in Section 4.
In Section 5 we recall the connections between Gaudin model of types B, C and self-dual spaces
of polynomials. The symmetry of higher Gaudin Hamiltonians with respect to Shapovalov form
and the key lemma from linear algebra are discussed in Section 6. In Section 7 we prove our
main results, see Theorem 7.2 and Corollary 7.4. Finally, we display some simple data computed
from Corollary 7.4 in Section 8.
2 Simple Lie algebras
Let g be a simple Lie algebra over C with Cartan matrix A = (ai,j)
r
i,j=1, where r is the rank
of g. Let D = diag(d1, . . . , dr) be the diagonal matrix with positive relatively prime integers di
such that DA is symmetric.
Let h ⊂ g be a Cartan subalgebra with the Cartan decomposition g = n−⊕h⊕n+. Fix simple
roots α1, . . . , αr in h∗. Let α̌1, . . . , α̌r ∈ h be the corresponding coroots. Fix a nondegenerate
invariant bilinear form ( , ) on g such that (α̌i, α̌j) = ai,j/dj . The corresponding bilinear form
on h∗ is given by (αi, αj) = diai,j . We have 〈λ, α̌i〉 = 2(λ, αi)/(αi, αi) for λ ∈ h∗. In particular,
〈αj , α̌i〉 = ai,j . Let ω1, . . . , ωr ∈ h∗ be the fundamental weights, 〈ωj , α̌i〉 = δi,j .
Let P = {λ ∈ h∗ | 〈λ, α̌i〉 ∈ Z, i = 1, . . . , r} and P+ = {λ ∈ h∗ | 〈λ, α̌i〉 ∈ Z>0, i = 1, . . . , r} be
the weight lattice and the cone of dominant integral weights.
Let e1, . . . , er ∈ n+, α̌1, . . . , α̌r ∈ h, f1, . . . , fr ∈ n− be the Chevalley generators of g.
Given a g-module M , denote by (M)g the subspace of g-invariants in M . The subspace (M)g
is the multiplicity space of the trivial g-module in M .
A sequence of nonnegative integers ξ = (ξ1, . . . , ξk) such that ξ1 > ξ2 > · · · > ξk > 0 is called
a partition with at most k parts. Set |ξ| =
k∑
i=1
ξi.
For λ ∈ h∗, let Vλ be the irreducible g-module with highest weight λ. For any g-weights λ
and µ, it is well known that dim(Vλ ⊗ Vµ)g = δλ,µ for g = so2r+1, sp2r.
For any Lie algebra g, denote by U(g) the universal enveloping algebra of g.
3 Characters of the symmetric groups
Let gN be the Lie algebra so2r+1 if N = 2r or the Lie algebra sp2r if N = 2r + 1, r > 2. We
also set g3 = sl2. Let GN be the respective classical group with Lie algebra gN .
Let Sk be the symmetric group permuting a set of k elements. In this section we deduce
a formula for characters of a product of the symmetric groups acting on a tensor product of
finite-dimensional irreducible gN -modules.
For each dominant integral gN -weight λ, denote by λ̄ = (λ̄1, . . . , λ̄r) the partition with at
most r parts such that
2〈λ, α̌i〉 = λ̄i − λ̄i+1, i = 1, . . . , r − 1, and λ̄r =
{
〈λ, α̌r〉, if N = 2r,
2〈λ, α̌r〉, if N = 2r + 1.
Define an anti-symmetric Laurent polynomial ∆N in x1, . . . , xr as follows
∆N = det
(
xN+1−2j
i − x−(N+1−2j)
i
)r
i,j=1
. (3.1)
We call ∆N the Vandermonde determinant of gN .
4 K. Lu
Let λ be a dominant integral gN -weight. It is well known that the character of the module Vλ
is given by
SNλ (x1, . . . , xr) = trVλ XN =
det
(
x
λ̄j+N+1−2j
i − x−(λ̄j+N+1−2j)
i
)r
i,j=1
∆N
, (3.2)
where XN ∈ GN is given by
XN =
{
diag
(
x2
1, . . . , x
2
r , 1, x
−2
r , . . . , x−2
1
)
, if N = 2r,
diag
(
x2
1, . . . , x
2
r , x
−2
r , . . . , x−2
1
)
, if N = 2r + 1.
We call SNλ the Schur function of gN associated with the weight λ.
Note that SNλ are symmetric Laurent polynomials in x1, . . . , xr, SNλ ∈
(
C
[
x±1
1 , . . . , x±1
r
])Sr .
Let λ(1), . . . , λ(s) be a sequence of dominant integral gN -weights and k1, . . . , ks a sequence of
positive integers. Consider the tensor product of gN -modules
Vλ = V ⊗k1
λ(1)
⊗ V ⊗k2
λ(2)
⊗ · · · ⊗ V ⊗ks
λ(s)
and its decomposition into irreducible gN -submodules
Vλ =
⊕
µ
Vµ ⊗Mλ,µ.
By permuting the corresponding tensor factors of Vλ, the product of symmetric groups Sk =
Sk1 × Sk2 × · · · × Sks acts naturally on Vλ. Note that the Sk-action commutes with the
gN -action, therefore the group Sk acts on the multiplicity space Mλ,µ for all µ.
For σ = σ1×σ2×· · ·×σs ∈ Sk, σi ∈ Ski . Suppose all σi are written as a product of disjoint
cycles. Denote by ci the number of cycles in the product representing σi and lij , j = 1, . . . , ci,
the lengths of cycles. Note that li,1 + · · ·+ li,ci = ki.
We then consider the value of the character of Sk corresponding to the representation Mλ,µ
on σ. Let χλ,µ = trMλ,µ .
Proposition 3.1. The character value χλ,µ(σ) equals the coefficient of the monomial
xµ̄1+N−1
1 xµ̄2+N−3
2 · · ·xµ̄r+N+1−2r
r
in the Laurent polynomial
∆N ·
s∏
i=1
ci∏
j=1
SN
λ(i)
(
x
lij
1 , . . . , x
lij
r
)
.
Proof. The proof of the statement is similar to that of [13, Proposition 2.1]. �
4 Osculating Schubert calculus and self-dual spaces
Let N , d ∈ Z>0 be such that N 6 d. Consider P1 := C ∪ {∞}. Set
P̊n :=
{
z = (z1, . . . , zn) ∈ (P1)n | zi 6= zj for 1 6 i < j 6 n
}
,
RP̊n :=
{
z = (z1, . . . , zn) ∈ P̊n | zi ∈ R or zi =∞, for 1 6 i 6 n
}
.
Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus 5
4.1 Osculating Schubert calculus
Let Cd[x] be the space of polynomials in x with complex coefficients of degree less than d. We
have dimCd[x] = d. Let Gr(N, d) be the Grassmannian of all N -dimensional subspaces in Cd[x].
The Grassmannian Gr(N, d) is a smooth projective complex variety of dimension N(d−N).
Let Rd[x] ⊂ Cd[x] be the set of polynomials in x with real coefficients of degree less than d. Let
GrR(N, d) ⊂ Gr(N, d) be the set of subspaces which have a basis consisting of polynomials with
all coefficients real. ForX ∈ Gr(N, d) we haveX ∈ GrR(N, d) if and only if dimR(X∩Rd[x]) = N .
We call such points X real.
For a complete flag F = {0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ Fd = Cd[x]} and a partition ξ = (ξ1, . . . , ξN )
such that ξ1 6 d−N , the Schubert cell Ωξ(F) ⊂ Gr(N, d) is given by
Ωξ(F) =
{
X ∈ Gr(N, d) | dim(X ∩Fd−j−ξN−j ) = N − j,
dim(X ∩Fd−j−ξN−j−1) = N − j − 1
}
.
Note that codim Ωξ(F) = |ξ|.
Let F(∞) be the complete flag given by
F(∞) =
{
0 ⊂ C1[x] ⊂ C2[x] ⊂ · · · ⊂ Cd[x]
}
. (4.1)
The subspace X is a point of Ωξ(F(∞)) if and only if for every i = 1, . . . , N , it contains
a polynomial of degree d− i− ξN+1−i.
For z ∈ C, consider the complete flag
F(z) =
{
0 ⊂ (x− z)d−1C1[x] ⊂ (x− z)d−2C2[x] ⊂ · · · ⊂ Cd[x]
}
. (4.2)
The subspace X is a point of Ωξ(F(z)) if and only if for every i = 1, . . . , N , it contains
a polynomial with a root at z of order exactly ξi +N − i.
A point z ∈ C is called a base point for a subspace X ⊂ Cd[x] if ϕ(z) = 0 for every ϕ ∈ X.
Let ξ=
(
ξ(1), . . ., ξ(n)
)
be a sequence of partitions with at most N parts and z=(z1, . . ., zn)∈P̊n.
Set |ξ| =
n∑
s=1
∣∣ξ(s)
∣∣.
Assuming |ξ| = N(d−N), denote by Ωξ,z the intersection of the Schubert cells
Ωξ,z =
n⋂
s=1
Ωξ(s)(F(zs)). (4.3)
Note that due to our assumption, Ωξ,z is a finite subset of Gr(N, d).
Define a sequence of polynomials T = (T1, . . . , TN ) by the formulas
Ti(x) =
n∏
s=1
(x− zs)ξ
(s)
i −ξ
(s)
i+1 , i = 1, . . . , N,
where ξ
(s)
N+1 = 0. Here and in what follows we use the convention that x − zs is considered as
the constant function 1 if zs =∞. We say that T is associated with ξ, z.
4.2 Self-dual spaces
Let X ∈ Gr(N, d) be an N -dimensional subspace of polynomials in x. Given a polynomial ψ
in x, denote by ψ ·X the space of polynomials of the form ψ · ϕ for all ϕ ∈ X.
6 K. Lu
Let X∨ be the N -dimensional space of polynomials which are Wronskian determinants of
N − 1 elements of X
X∨ =
{
det
(
di−1ϕj/dx
i−1
)N−1
i,j=1
, ϕj(x) ∈ X
}
.
The space X is called self-dual if X∨ = ψ ·X for some polynomial ψ(x), see [16].
Let sGr(N, d) be the set of all self-dual spaces in Gr(N, d). We call sGr(N, d) the self-dual
Grassmannian. The self-dual Grassmannian sGr(N, d) is an algebraic subset of Gr(N, d).
Denote by sΩξ,z the set of all self-dual spaces in Ωξ,z
sΩξ,z = Ωξ,z
⋂
sGr(N, d).
Let µ be a dominant integral gN -weight and k ∈ Z>0. Define a partition µA,k with at most N
parts by the rule: (µA,k)N = k and
(µA,k)i − (µA,k)i+1 =
{
〈µ, α̌i〉, if 1 6 i 6
[
N
2
]
,
〈µ, α̌N−i〉, if
[
N
2
]
< i 6 N − 1.
(4.4)
We call µA,k the partition associated with weight µ and integer k.
Let λ =
(
λ(1), . . ., λ(n)
)
be a sequence of dominant integral gN -weights and let k = (k1, . . ., kn)
be an n-tuple of nonnegative integers. Then denote λA,k =
(
λ
(1)
A,k1
, . . . , λ
(n)
A,kn
)
the sequence of
partitions associated with λ(s) and ks, s = 1, . . . , n.
We use the notation µA = µA,0 and λA = λA,(0,...,0).
Lemma 4.1 ([10]). If ξ is a sequence of partitions with at most N parts such that |ξ| = N(d−N)
and sΩξ,z is nonempty, then ξ has the form ξ = λA,k for a sequence of dominant integral gN -
weights λ =
(
λ(1), . . . , λ(n)
)
and a sequence of nonnegative integers k = (k1, . . . , kn). The pair
(λ,k) is uniquely determined by ξ.
In what follows we write Ωλ,z, Ωλ,k,z, sΩλ,z, sΩλ,k,z for ΩλA,z, ΩλA,k,z, sΩλA,z, sΩλA,k,z,
respectively.
Note that |λA,k| = |λA|+N |k|, where |k| = k1 + · · ·+ kn. Suppose |λA| = N(d−N), there
exists a bijection between Ωλ,z in Gr(N, d) and Ωλ,k,z in Gr(N, d+ |k|) given by
Ωλ,z → Ωλ,k,z, X 7→
n∏
s=1
(x− zs)ks ·X. (4.5)
Moreover, (4.5) restricts to a bijection between sΩλ,z in sGr(N, d) and sΩλ,k,z in sGr(N, d+ |k|).
5 Gaudin model
Let g[t] = g ⊗ C[t] be the Lie algebra of g-valued polynomials with the pointwise commutator.
We call it the current algebra of g. We identify the Lie algebra g with the subalgebra g ⊗ 1 of
constant polynomials in g[t].
It is convenient to collect elements of g[t] in generating series of a formal variable x. For
g ∈ g, set
g(x) =
∞∑
k=0
(
g ⊗ tk
)
x−k−1. (5.1)
For each a ∈ C, we have the evaluation homomorphism eva : g[t]→ g where eva sends g ⊗ ts
to asg for all g ∈ g and s ∈ Z>0. Its restriction to the subalgebra g ⊂ g[t] is the identity map.
Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus 7
For any g-module M , we denote by M(a) the g[t]-module, obtained by pulling M back through
the evaluation homomorphism eva. The g[t]-module M(a) is called an evaluation module. The
generating series g(x) acts on the evaluation module M(a) by g/(x− a).
The Bethe algebra B (the algebra of higher Gaudin Hamiltonians) for a simple Lie algebra g
was described in [3]. The Bethe algebra B is a commutative subalgebra of U(g[t]) which com-
mutes with the subalgebra U(g) ⊂ U(g[t]). An explicit set of generators of the Bethe algebra in
Lie algebras of types B, C, and D was given in [11].
Proposition 5.1 ([3, 11]). Let N > 3. There exist elements Fij ∈ gN , i, j = 1, . . . , N , and
polynomials Bs(x) in dkFij(x)/dxk, s = 1, . . . , N , k = 0, . . . , N , such that the Bethe algebra B
of gN is generated by the coefficients of Bs(x) considered as formal power series in x−1.
We denote M(∞) the gN -module M with the trivial action of the Bethe algebra B, see [10]
for more detail.
For a collection of gN -weights λ =
(
λ(1), . . . , λ(n)
)
and z = (z1, . . . , zn) ∈ P̊n, we set
Vλ,z =
n⊗
s=1
Vλ(s)(zs),
considered as a B-module. We also denote Vλ the module Vλ,z considered as a gN -module.
Let ∂x be the differentiation with respect to x. Define a formal differential operator
DB = ∂Nx +
N∑
i=1
Bi(x)∂N−ix ,
where
Bi(x) =
∞∑
j=i
Bijx
−j (5.2)
and Bij ∈ U(gN [t]), j ∈ Z>i, i = 1, . . . , N . The operator DB is called the universal operator.
Let z = (z1, . . . , zn) ∈ P̊n and let λ =
(
λ(1), . . . , λ(n)
)
be a sequence of dominant integral
gN -weights. For every g ∈ gN , the series g(x) acts on Vλ,z as a rational function of x.
Since the Bethe algebra B commutes with gN , B acts on the invariant space (Vλ,z)gN . For
b ∈ B, denote by b(λ, z) ∈ End((Vλ,z)gN ) the corresponding linear operator.
Given a common eigenvector v ∈ (Vλ,z)gN of the operators b(λ, z), denote by b(λ, z; v) the
corresponding eigenvalues, and define the scalar differential operator
Dv = ∂Nx +
N∑
i=1
∞∑
j=i
Bij(λ, z; v)x−j∂N−ix .
The following theorem connects self-dual spaces in the Grassmannian Gr(N, d) with the
Gaudin model associated to gN .
Theorem 5.2 ([10]). Let N > 3. There exists a choice of generators Bi(x) of the Bethe
algebra B, such that for any sequence of dominant integral gN -weights λ =
(
λ(1), . . . , λ(n)
)
, any
z ∈ P̊n, and any B-eigenvector v ∈ (Vλ,z)gN , we have
Ker
(
(T1 · · ·TN )1/2 · Dv · (T1 · · ·TN )−1/2
)
∈ sΩλ,z,
where T = (T1, . . . , TN ) is associated with λA, z.
Moreover, if |λA| = N(d−N), then this defines a bijection between the joint eigenvalues of B
on (Vλ,z)gN and sΩλ,z ⊂ Gr(N, d).
8 K. Lu
6 Shapovalov form and the key lemma
6.1 Shapovalov form
Define the anti-involution $ : gN → gN sending e1, . . . , er, α̌1, . . . , α̌r, f1, . . . , fr to f1, . . . , fr,
α̌1, . . . , α̌r, e1, . . . , er, respectively.
For any dominant integral gN -weight λ, the irreducible gN -module Vλ admits a positive
definite Hermitian form (·, ·)λ such that (gv, w)λ = (v,$(g)w)λ for any v, w ∈ Vλ and g ∈ gN .
Such a form is unique up to multiplication by a positive real number. We call this form the
Shapovalov form.
Let λ =
(
λ(1), . . . , λ(n)
)
be a sequence of dominant integral gN -weights. We define the positive
definite Hermitian form (·, ·)λ on the tensor product Vλ as the product of Shapovalov forms on
the tensor factors. The form (·, ·)λ induces a positive definite Hermitian form (·|·)λ on (Vλ,z)gN .
Proposition 6.1. For any i = 1, . . . , N , j ∈ Z>i, and any v, w ∈ (Vλ,z)gN , we have(
Bij(λ, z)v|w
)
λ
=
(
v|Bij(λ, z̄)w
)
λ
,
where Bij are given by (5.2), z̄ = (z̄1, . . . , z̄n) and the bar stands for the complex conjugation.
Proof. We prove the proposition in Section 6.3. �
If z ∈ RP̊n, then Bij(λ, z) are self-adjoint with respect to the Shapovalov form. Therefore
all Bij(λ, z) are simultaneously diagonalizable and all eigenvalues of Bij(λ, z) are real.
The following statement is also known.
Theorem 6.2 ([18]). For generic z ∈ P̊n, the action of the Bethe algebra B on (Vλ,z)gN is
diagonalizable and has simple spectrum. In particular, this statement holds for any sequence
z ∈ RP̊n.
If some of the partitions λ(1), . . . , λ(n) coincide, the operators b(λ, z) admit additional sym-
metry. Assume that λ(i) = λ(i+1) for some i. Let Pi ∈ End(Vλ) be the flip of the i-th and
(i+ 1)-st tensor factors and z̃(i) = (z1, . . . , zi−1, zi+1, zi, zi+2, . . . , zn).
Lemma 6.3. For any b ∈ B, we have Pib(λ, z)Pi = b
(
λ, z̃(i)
)
.
6.2 Self-adjoint operators with respect to indefinite Hermitian form
In this section we recall the key lemma from linear algebra, see [17].
Given a finite-dimensional vector space M , a linear operator T ∈ End(M), and a number
α ∈ C, let MT(α) = ker(T−α)dimM . When MT(α) is not trivial, it is the subspace of generalized
eigenvectors of T with eigenvalue α.
Lemma 6.4 ([17]). Let M be a complex finite-dimensional vector space with a nondegenerate
Hermitian form of signature κ, and let A ⊂ End(M) be a commutative subalgebra over R, whose
elements are self-adjoint operators. Let R =
⋂
T∈A
⊕
α∈RMT(α). Then the restriction of the
Hermitian form on R is nondegenerate and has signature κ. In particular, dimR > |κ|.
6.3 Proof of Proposition 6.1
In this section, we give the proof of Proposition 6.1. We follow the convention of [12]. We only
introduce the necessary notation and refer the reader to [11, Section 5] and [12, Section 3] for
more detail.
Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus 9
Proof of Proposition 6.1. We prove it for the case N = 2r first.
Let Eij with i, j = 1, . . . , 2r + 1 be the standard basis of gl2r+1. The Lie subalgebra
of gl2r+1 generated by the elements Fij = Eij − E2r+2−j,2r+2−i is isomorphic to the Lie al-
gebra so2r+1 = gN . With this isomorphism, the anti-involution $ : gN → gN is realized by
taking transposition, Fij 7→ Fji. To be consistent with the notation in [12], we write g for gN .
The number N in [12] is 2r + 1 in our notation.
We write Fij [s] for Fij ⊗ ts in the loop algebra g
[
t, t−1
]
. Consider the affine Lie algebra
ĝ = g
[
t, t−1
]
⊕ CK, which is the central extension of the loop algebra g
[
t, t−1
]
, where the
element K is central in ĝ and
[g1[k], g2[l]] = [g1, g2][k + l] + kδk,−l(g1, g2)K, g1, g2 ∈ g, k, l ∈ Z.
Consider the extended affine Lie algebra ĝ⊕ Cτ = g
[
t, t−1
]
⊕ CK ⊕ Cτ , where τ satisfies
[τ, Fij [s]] = −sFij [s− 1], [τ,K] = 0, s ∈ Z.
Set U = U(ĝ ⊕ Cτ) and fix m ∈ {1, . . . , N}. Introduce the element F [s]a of the algebra(
End
(
C2r+1
))⊗m ⊗ U , see [12, equation (3.5)], by
F [s]a =
2r+1∑
i,j=1
1⊗(a−1) ⊗ eij ⊗ 1⊗(m−a) ⊗ Fij [s],
where eij ∈ End
(
C2r+1
)
denote the standard matrix units. The map $ induces an anti-
involution
$ : U
(
t−1g
[
t−1
])
→ U
(
t−1g
[
t−1
])
, Fij [s] 7→ Fji[s], s ∈ Z6−1.
For 1 6 a < b 6 m, consider the operators Pab and Qab in
(
End
(
C2r+1
))⊗m
defined as
follows
Pab =
2r+1∑
i,j=1
1⊗(a−1) ⊗ eij ⊗ 1⊗(b−a−1) ⊗ eji ⊗ 1⊗(m−b),
Qab =
2r+1∑
i,j=1
1⊗(a−1) ⊗ eij ⊗ 1⊗(b−a−1) ⊗ e2r+2−i,2r+2−j ⊗ 1⊗(m−b).
Set
S(m) =
1
m!
∏
16a<b6m
(
1 +
Pab
b− a
− 2Qab
2r + 2b− 2a− 1
)
,
where the product is taken in the lexicographic order on the pairs (a, b). The element S(m) is the
symmetrizer of the Brauer algebra acting on
(
C2r+1
)⊗m
. In particular, for any 1 6 a < b 6 m,
the operator S(m) satisfies
S(m)Qab = QabS
(m) = 0, S(m)Pab = PabS
(m) = S(m).
Replacing τ with ∂x and Fij [−`− 1] with −∂`xFij(x)/`!, where Fij(x) is defined in (5.1), for
` ∈ Z>0, in the element
2r +m− 1
2r + 2m− 1
trS(m)(τ + F [−1]1) · · · (τ + F [−1]m),
10 K. Lu
see [12, formula (3.26)], where the trace is taken on all m copies of End
(
C2r+1
)
, we get a dif-
ferential operator
ϑm0(x)∂mx + ϑm1(x)∂m−1
x + · · ·+ ϑmm(x),
where ϑmi(x) is a formal power series in x−1 with coefficients in U(g[t]). The Bethe subalgebra B
of U(g[t]) is generated by the coefficients of ϑmi(x), m = 1, . . . , N , i = 0, . . . ,m, see [11,
Section 5].
Therefore, to prove the proposition, it suffices to show that the element
2r +m− 1
2r + 2m− 1
trS(m)(τ + F [−1]1) · · · (τ + F [−1]m), (6.1)
is stable under the anti-involution $. Here $ maps τ to τ .
Applying transposition on a-th and b-th components to the commutator relation
F [k]aF [l]b − F [l]bF [k]a = (Pab −Qab)F [k + l]b − F [k + l]b(Pab −Qab),
see the proof of [12, Lemma 3.6], we get
F>[k]aF
>[l]b − F>[l]bF
>[k]a = F>[k + l]b(Pab −Qab)− (Pab −Qab)F>[k + l]b,
for all 1 6 a < b 6 m. Here > stands for transpose, explicitly,
F>[s]a =
2r+1∑
i,j=1
1⊗(a−1) ⊗ eij ⊗ 1⊗(m−a) ⊗ Fji[s].
Thus one can use the same argument as in the proof of [12, Lemma 3.2] to show that the image
of (6.1) under the anti-involution $ equals
2r +m− 1
2r + 2m− 1
trS(m)
(
τ + F>[−1]1
)
· · ·
(
τ + F>[−1]m
)
. (6.2)
By applying the simultaneous transposition eij → eji to all m copies of End
(
C2r+1
)
we conclude
that (6.2) coincides with (6.1) because the transformation takes each factor τ + F>[−1]a to
τ + F [−1]a whereas the symmetrizer S(m) stays invariant. Hence we complete the proof of
Proposition 6.1 for the case N = 2r.
The case N = 2r + 1 is proved similarly, see for example [12, Lemma 3.9]. �
7 The lower bound
In this section we prove our main results – the lower bound for the number of real self-dual
spaces in Ωλ,z, see Theorem 7.2 and Corollary 7.4.
Recall the notation from Section 4. For positive integers N , d such that d > N we consider
the Grassmannian Gr(N, d) of N -dimensional planes in the space Cd[x] of polynomials of degree
less than d. A point X ∈ Gr(N, d) is called real if it has a basis consisting of polynomials with
all coefficients real.
7.1 The general case N >>> 4
Let us first consider the case N > 4.
Let λ =
(
λ(1), . . . , λ(n)
)
be a sequence of dominant integral gN -weights, k = (k1, . . . , kn) an
n-tuple of nonnegative integers, and z = (z1, . . . , zn) ∈ P̊n. Suppose that |λA,k| = N(d − N).
Denote by d(λ,k, z) the number of real points counted with multiplicities in sΩλ,k,z ⊂ Gr(N, d).
Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus 11
Clearly, d(λ,k, z) = 0 unless the set {z1, . . . , zn} is invariant under the complex conjugation
and
(
λ(i), ki
)
=
(
λ(j), kj
)
whenever zi = z̄j . In particular, the polynomial
n∏
s=1
(x− zs)ks has only
real coefficients. It follows from (4.5) that the number of real points in sΩλ,k,z ⊂ Gr(N, d) is
equal to that of sΩλ,z ⊂ Gr(N, d − |k|). From now on, we shall only consider the case that
k = (0, . . . , 0). We simply write d(λ, z) for d(λ,k, z) if k = (0, . . . , 0).
Let T = (T1, . . . , TN ) be associated with λA,k, z. Note that if z, λ, k is invariant under
conjugation, then the polynomial T1 · · ·TN also has only real coefficients.
In what follows we denote by c the number of complex conjugate pairs in the set {z1, . . . , zn}
and without loss of generality assume that z1 = z̄2, . . . , z2c−1 = z̄2c while z2c+1, . . . , zn are real
(one of them can be infinity). We will also always assume that λ(1) = λ(2), . . . , λ(2c−1) = λ(2c).
Recall that for any λ and generic z ∈ P̊n, all points of Ωλ,z are multiplicity-free. The same
also holds true with λ imposed above for any c.
Consider the decomposition of Vλ into irreducible gN -submodules
Vλ =
⊕
µ
Vµ ⊗Mλ,µ.
Then Mλ,0 = (Vλ)gN . Since λ(2i−1) = λ(2i) for i = 1, . . . , c, the flip P2i−1 of the (2i− 1)-st and
2i-th tensor factors of Vλ commutes with the gN -action and thus acts on (Vλ)gN . Denote by
Pλ,c ∈ End((Vλ)gN ) the action of the product P1P3 · · ·P2c−1 on (Vλ)gN .
The operator Pλ,c is self-adjoint relative to the Hermitian form (·|·)λ on (Vλ)gN given in
Section 6. Define a new Hermitian form (·, ·)λ,c on (Vλ)gN by the rule: for any v, w ∈ (Vλ)gN
(v, w)λ,c = (Pλ,cv|w)λ.
Denote by q(λ, c) the signature of the form (·, ·)λ,c.
Proposition 7.1. The signature q(λ, c) equals the coefficient of the monomial
xN−1
1 xN−3
2 · · ·xN+1−2r
r ,
in the Laurent polynomial
∆N ·
c∏
i=1
SN
λ(2i)
(
x2
1, . . . , x
2
r
) n∏
j=2c+1
SN
λ(j)
(x1, . . . , xr).
Here ∆N and SN
λ(s)
are given by (3.1) and (3.2), respectively.
Proof. Since P 2
λ,c = 1 and Mλ,0 = (Vλ)gN , we have q(λ, c) = trMλ,0 Pλ,c, and the claim follows
from Proposition 3.1. �
Theorem 7.2. The number d(λ, z) of real self-dual spaces in Ωλ,z is no less than |q(λ, c)|.
Proof. Our proof is parallel to that of [13, Theorem 7.2].
By Proposition 6.1 and Lemma 6.3, the operators Bij(λ, z) ∈ End((Vλ)gN ) are self-adjoint
relative to the form (·, ·)λ,c. By Lemma 6.4,
dim
(⋂
i,j
⊕
α∈R
(
(Vλ)gN
)
Bij(λ,z)
(α)
)
> |q(λ, c)|.
By Theorem 6.2, for any λ and generic z ∈ P̊n the operators Bij(λ, z) are diagonalizable and
the action of the Bethe algebra B on (Vλ)gN has simple spectrum. The same also holds true
with λ imposed above for any c. Thus for generic z, the operators Bij(λ, z) have at least |q(λ, c)|
common eigenvectors with distinct real eigenvalues, which provides |q(λ, c)| distinct real points
in sΩλ,z by Theorem 5.2. Hence, d(λ, z) > |q(λ, c)| for generic z, and therefore, for any z, due
to counting with multiplicities. �
12 K. Lu
Remark 7.3. If dim(Vλ)gN is odd, it follows from Theorem 7.2 by counting parity that
d(λ, z) > |q(λ, c)| > 1.
Therefore, there exists at least one real point in sΩλ,z. In particular, if dim(Vλ)gN = 1, then
the only point in sΩλ,z is always real.
The following corollary of Proposition 7.1 and Theorem 7.2 is our main result.
Corollary 7.4. The number d(λ, z) of real self-dual spaces in Ωλ,z (real points in sΩλ,z) is no
less than |a(λ, c)|, where a(λ, c) is the coefficient of the monomial xN−1
1 xN−3
2 · · ·xN+1−2r
r in the
Laurent polynomial
∆N ·
c∏
i=1
SN
λ(2i)
(
x2
1, . . . , x
2
r
) n∏
j=2c+1
SN
λ(j)
(x1, . . . , xr).
Here ∆N is the Vandermonde determinant of gN and SN
λ(s)
is the Schur function of gN associated
with λ(s), s = 1, . . . , n, see (3.1) and (3.2).
Remark 7.5. Recall that the total number of points (counted with multiplicities) in sΩλ,z
equals dim(Vλ)gN = q(λ, 0). Hence if z ∈ RP̊n, Theorem 7.2 claims that all points in sΩλ,z
are real. It is proved in [15, Corollary 6.3] that for z ∈ RP̊n all points in Ωλ,z are real and
multiplicity-free, so are the points in sΩλ,z.
7.2 The case N = 2,3
Now let us consider the case N = 2, 3. Note that sGr(2, d) = Gr(2, d), this case is the usual
Grassmannian, which has already been discussed in [13].
Let N = 3 and g3 = sl2. It suffices for us to consider the case that points in sΩλ,z have no
base points, see the beginning of Section 7.1 for more detail. We shall consider sGr(3, 2d − 1)
instead of sGr(3, d), see [10, Section 4.6]. We identify the dominant integral sl2-weights with
nonnegative integers. Let λ =
(
λ(1), . . . , λ(n)
)
be a sequence of nonnegative integers and z =
(z1, . . . , zn) ∈ P̊n. Then λA has coordinates
λ
(s)
A =
(
2λ(s), λ(s), 0
)
, s = 1, . . . , n.
We also assume |λA| = 6(d− 2).
Recall from [10, Theorem 4.19], if X ∈ sΩλ,z, then there exist monic polynomials ϕ and ψ
such that ϕ2, ϕψ, ψ2 form a basis of X. Denote by
√
X the space of polynomials spanned by ϕ
and ψ. Let ξ(i) be the partitions with at most two parts defined by
(
λ(i), 0
)
, i = 1, . . . , n. Set
ξ =
(
ξ(1), . . . , ξ(n)
)
, then |ξ| = 2(d − 2). It follows from the proof of [10, Theorem 4.19] that√
X ∈ Ωξ,z ⊂ Gr(2, d). The map Ωξ,z → sΩλ,z given by
√
X 7→ X is bijective.
Lemma 7.6. The self-dual space X is real if and only if
√
X is real.
Proof. It is obvious that X is real if
√
X is real.
Conversely, if X is real, then there exist complex numbers ai, bi, ci, i = 1, 2, 3, such that
aiϕ
2 + biϕψ + ciψ
2, i = 1, 2, 3,
are real polynomials and form a basis of X. Without loss of generality, we assume degϕ < degψ.
Since degϕ < degψ, we have ci ∈ R, i = 1, 2, 3. At least one of ci is nonzero. We assume c3 6= 0.
By subtracting a proper real multiple of a3ϕ
2 + b3ϕψ + c3ψ
2, we assume further c1 = c2 = 0.
Continuing with the previous step, we assume that b1 = 0, b2 6= 0, a1 6= 0 and hence obtain
that a1, b2, c3 ∈ R. Then a1ϕ
2 is a real polynomial, so is ϕ. Therefore, a2ϕ+ b2ψ is also a real
polynomial, which implies that the space of polynomials
√
X is also real. �
Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus 13
Because of Lemma 7.6, the case N = 3 is reduced to the lower bound for the number of
real solutions to osculating Schubert problems of Gr(2, d), see [13]. Moreover, Corollary 7.4 also
applies for this case by putting N = 3, r = 1, and gN = sl2.
8 Some data for small N
In this section, we give some data obtained from Corollary 7.4 when N is small. Since the cases
N = 2, 3 reduce to the cases of [13], we start with N = 4.
We always assume that λ, k, z are invariant under conjugation. By Remark 7.3, we shall only
consider the cases that dim(Vλ)gN > 2. We also exclude the cases that z ∈ RP̊n. In particular,
the cases that all pairs
(
λ(s), ks
)
, s = 1, . . . , n, are different.
We write the highest weights in terms of fundamental weights, for example (1, 0, 0, 1) =
ω1 + ω4. We also write
(
λ(1)
)
k1
, . . . ,
(
λ(n)
)
kn
for (λ,k) and simply write λ(s) for (λ(s))0. We
use
(
λ
(s)
1 , λ
(s)
2
)⊗m
ks
to indicate that the pair
((
λ
(s)
1 , λ
(s)
2
)
, ks
)
appears in (λ,k) exactly m times.
For instance, (0, 1)1, (0, 1)⊗3 represents the pair (λ,k) where λ = ((0, 1), (0, 1), (0, 1), (0, 1)) and
k = (1, 0, 0, 0).
8.1 The case N = 4,5
For each g4-weight λ = (λ1, λ2), denote by λC the g5-weight (λ2, λ1). Note that g4 = so5 is
isomorphic to g5 = sp4, the lower bound obtained from the ramification data λ =
(
λ(1), . . . , λ(n)
)
and k = (k1, . . . , kn) of g4 is the same as that obtained from the ramification data λC =(
λ
(1)
C , . . . , λ
(n)
C
)
and k = (k1, . . . , kn) of g5.
ramification data dimension c = 1 c = 2 c = 3
(0, 1)⊗6 14 2 2 6
(1, 0)⊗3, (0, 1)⊗2 4 0,2 2
(1, 0)⊗3, (1, 0)1 3 1
(1, 0)⊗4, (0, 0)1 3 1 3
(0, 2), (0, 1)⊗4 6 0 2
(0, 0)1, (0, 1)⊗4 3 1 3
(1, 0), (0, 1)⊗4 5 1 1
(1, 1), (0, 1)⊗3 2 0
(0, 1)1, (0, 1)⊗3 3 1
(0, 2)⊗2, (0, 1)⊗2 3 1 3
(1, 0)⊗2, (0, 1)⊗2 2 0 2
(0, 2), (1, 0), (0, 1)⊗2 2 0
(1, 0)1, (1, 0), (0, 1)⊗2 2 0
(1, 0)⊗2, (0, 1)1, (0, 1) 2 0
(1, 1), (1, 0)⊗2, (0, 1) 2 0
Table 1. The case N = 4, 5.
In Table 1, we give lower bounds for the cases from Gr(4, 7) and Gr(5, 10). By the observation
above, we transform the case from Gr(5, 10) to its counter part in Gr(4, d) for some d depending
on the ramification data. The number in the column of dimension is equal to dim(Vλ)g4 for the
corresponding ramification data λ in each row. The numbers in the column of c = i equal the
lower bounds computed from Corollary 7.4 with the corresponding c.
For a given c, there may exist several choices of complex conjugate pair corresponding to
different pairs of gN -weights. If the corresponding lower bounds are the same, we just write
14 K. Lu
one number. For example, in the case of (0, 2)⊗2, (0, 1)⊗2 and c = 1 of Table 1, the complex
conjugate pair may correspond to the weights (0, 2)⊗2 or (0, 1)⊗2. However, they give the same
lower bound 1. Hence we just write 1 for c = 1. If the bounds are different, we write the lower
bound with the conjugate pairs corresponding to the leftmost 2c weights first while the one
with the conjugate pairs corresponding to the rightmost 2c weights last, in terms of the order
of the ramification data displayed on each row. Since we have at most 3 cases, the possible
remaining case is clear. For instance, in the case (0, 1, 0)⊗4, (0, 0, 1)⊗4 and c = 2 of Table 2, the
two complex conjugate pairs corresponding to (0, 1, 0)⊗4 give the lower bound 12 while the two
complex conjugate pairs corresponding to (0, 0, 1)⊗4 give the lower bound 24. The remaining
case, where the two conjugate pairs corresponding to (0, 1, 0)⊗2 and (0, 0, 1)⊗2, gives the lower
bound 2.
8.2 The case N = 6
In what follows, we give lower bounds for ramification data consisting of fundamental weights
when N = 6. We follow the same convention as in Section 8.1.
ramification data dimension c = 1 c = 2 c = 3 c = 4
(0, 0, 1)⊗4 4 0 4
(0, 1, 0)⊗4 6 2 6
(1, 0, 0)⊗4 3 1 3
(0, 0, 1)⊗2, (0, 1, 0)⊗2 3 1 3
(0, 0, 1)⊗2, (1, 0, 0)⊗2 2 0 2
(0, 1, 0)⊗2, (1, 0, 0)⊗2 3 1 3
(0, 0, 1)⊗6 30 2 2 10
(0, 1, 0)⊗6 130 8 14 36
(1, 0, 0)⊗6 15 3 3 7
(0, 1, 0)⊗2, (0, 0, 1)⊗4 34 4,2 0,6 16
(0, 1, 0)⊗4, (0, 0, 1)⊗2 55 3,1 3,7 19
(1, 0, 0)⊗2, (0, 0, 1)⊗4 16 2 0,4 10
(1, 0, 0)⊗4, (0, 0, 1)⊗2 10 0,2 2,0 6
(1, 0, 0)⊗2, (0, 1, 0)⊗4 46 2 6 18
(1, 0, 0)⊗4, (0, 1, 0)⊗2 21 1,3 5,3 11
(1, 0, 0)⊗2, (0, 1, 0)⊗2, (0, 0, 1)⊗2 20 2 0,4,0 10
(0, 0, 1)⊗8 330 20 6 0 50
(0, 1, 0)⊗8 6111 69 59 113 311
(1, 0, 0)⊗8 105 15 9 7 25
(0, 1, 0)⊗4, (0, 0, 1)⊗4 984 22,28 12,2,24 0,38 108
(1, 0, 0)⊗4, (0, 0, 1)⊗4 116 6,12 8,2,12 0,10 32
(1, 0, 0)⊗4, (0, 1, 0)⊗4 510 6,12 22,4,18 28,18 74
Table 2. The case N = 6.
Acknowledgements
The author thanks E. Mukhin and V. Tarasov for useful discussions. The author also thanks
the referees for their comments and suggestions that substantially improved the first version of
this paper. This work was partially supported by Zhejiang Province Science Foundation, grant
No. LY14A010018.
Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus 15
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1 Introduction
2 Simple Lie algebras
3 Characters of the symmetric groups
4 Osculating Schubert calculus and self-dual spaces
4.1 Osculating Schubert calculus
4.2 Self-dual spaces
5 Gaudin model
6 Shapovalov form and the key lemma
6.1 Shapovalov form
6.2 Self-adjoint operators with respect to indefinite Hermitian form
6.3 Proof of Proposition 6.1
7 The lower bound
7.1 The general case N4
7.2 The case N=2,3
8 Some data for small N
8.1 The case N=4,5
8.2 The case N=6
References
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