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 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.


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 C d [x] of polynomials in x of degree less than d. In other words, we always assume for X ∈ Gr(N, d), we have X ⊂ C d [x]. Set P 1 = C ∪ {∞}. Then, for any z ∈ P 1 , we have the osculating flag F(z), see (4.1), (4.2). Denote the Schubert cells corresponding to F(z) by Ω ξ (F(z)), where ξ = (d − N ξ 1 ξ 2 · · · ξ N 0) are partitions. Then the set Ω ξ,z consists of spaces X ∈ Gr(N, d) such that X belongs to the intersection of Schubert cells Ω ξ (i) (F(z i )) for z = (z 1 , . . . , z n ) and ξ = ξ (1) , . . . , ξ (n) , where all z i ∈ P 1 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 Ω ξ,z is given in [13].
Let X ∈ Gr(N, d) be an N -dimensional subspace of polynomials in x. Let X ∨ be the Ndimensional space of polynomials which are Wronskian determinants of N − 1 elements of X N −i+1 for i = 1, . . . , N − 1. Hence the sl N -weight corresponding to the partition ξ (s) has certain symmetry and thus induces a g N -weight λ (s) , cf. (4.4). Therefore, the sequence of partitions ξ with nonempty sΩ ξ,z can be expressed in terms of a sequence of dominant integral g N -weights λ = λ 1 , . . . , λ (n) and a sequence of nonnegative integers k = (k 1 , . . . , k n ), see Lemma 4.1. In particular, k i = ξ (i) N . We call ξ, z or λ, k, z the ramification data. As a subset of Ω ξ,z , sΩ ξ,z can be empty even if Ω ξ,z is infinite. However, if sΩ ξ,z is nonempty, then sΩ ξ,z is finite if and only if Ω ξ,z is finite. More precisely, if then the number of points in sΩ ξ,z counted with multiplicities equals the multiplicity of the trivial g N -module in the tensor product V λ (1) ⊗ · · · ⊗ V λ (n) of irreducible g N -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 z 1 , . . . , z n are real, it follows from [14, Theorem 1.1] that all points in sΩ ξ,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 sΩ ξ,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 {z 1 , . . . , z n } should be invariant under the complex conjugation and the partitions at the complex conjugate points are the same. In other words, λ (i) , k i = λ (j) , k j provided z i =z j . In this case we say that z, λ, k are invariant under conjugation. Moreover, the greatest common divisor of X ∈ sΩ ξ,z in this case is a real polynomial. Hence we reduce the problem to the case that k i = 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 (g N , 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 z 1 , . . . , z n and hence have real eigenvalues. Therefore, it follows from Theorem 5.2 that self-dual spaces with real z 1 , . . . , z n are real.
If some of z 1 , . . . , z n 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 sΩ ξ,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 sΩ ξ,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 G 2 built in [1] and [8], we expect that lower bounds for the numbers of real self-self-dual spaces in Ω ξ,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.
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.

Simple Lie algebras
Let g be a simple Lie algebra over C with Cartan matrix A = (a i,j ) r i,j=1 , where r is the rank of g. Let D = diag(d 1 , . . . , d r ) be the diagonal matrix with positive relatively prime integers d i such that DA is symmetric.
Let e 1 , . . . , e r ∈ n + ,α 1 , . . . ,α r ∈ h, f 1 , . . . , f r ∈ 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 .
For any Lie algebra g, denote by U(g) the universal enveloping algebra of g.

Characters of the symmetric groups
Let g N be the Lie algebra so 2r+1 if N = 2r or the Lie algebra sp 2r if N = 2r + 1, r 2. We also set g 3 = sl 2 . Let G N be the respective classical group with Lie algebra g N .
Let S k 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 g N -modules.
For each dominant integral g N -weight λ, denote byλ = (λ 1 , . . . ,λ r ) the partition with at most r parts such that Define an anti-symmetric Laurent polynomial ∆ N in x 1 , . . . , x r as follows We call ∆ N the Vandermonde determinant of g N .
Let λ be a dominant integral g N -weight. It is well known that the character of the module V λ is given by where X N ∈ G N is given by We call S N λ the Schur function of g N associated with the weight λ.
Let λ (1) , . . . , λ (s) be a sequence of dominant integral g N -weights and k 1 , . . . , k s a sequence of positive integers. Consider the tensor product of g N -modules and its decomposition into irreducible g N -submodules By permuting the corresponding tensor factors of V λ , the product of symmetric groups S k = S k 1 × S k 2 × · · · × S ks acts naturally on V λ . Note that the S k -action commutes with the g N -action, therefore the group S k acts on the multiplicity space M λ,µ for all µ.
For σ = σ 1 × σ 2 × · · · × σ s ∈ S k , σ i ∈ S k i . Suppose all σ i are written as a product of disjoint cycles. Denote by c i the number of cycles in the product representing σ i and l ij , j = 1, . . . , c i , the lengths of cycles. Note that l i,1 + · · · + l i,c i = k i .
We then consider the value of the character of S k corresponding to the representation M λ,µ on σ. Let χ λ,µ = tr M λ,µ .
Proposition 3.1. The character value χ λ,µ (σ) equals the coefficient of the monomial Proof . The proof of the statement is similar to that of [13, Proposition 2.1].

Osculating Schubert calculus
be the space of polynomials in x with complex coefficients of degree less than d. We be the set of polynomials in x with real coefficients of degree less than d. Let Gr R (N, d) ⊂ Gr(N, d) be the set of subspaces which have a basis consisting of polynomials with all coefficients real.
Let F(∞) be the complete flag given by The subspace X is a point of Ω ξ (F(∞)) if and only if for For z ∈ C, consider the complete flag The subspace X is a point of Ω ξ (F(z)) if and only if for every i = 1, . . . , N , it contains a polynomial with a root at z of order exactly Let ξ= ξ (1) , . . ., ξ (n) be a sequence of partitions with at most N parts and z=(z 1 , . . ., z n )∈P n .
Assuming |ξ| = N (d − N ), denote by Ω ξ,z the intersection of the Schubert cells Note that due to our assumption, Ω ξ,z is a finite subset of Gr(N, d).
Define a sequence of polynomials T = (T 1 , . . . , T N ) by the formulas Here and in what follows we use the convention that x − z s is considered as the constant function 1 if z s = ∞. We say that T is associated with ξ, z.

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.
Let X ∨ be the N -dimensional space of polynomials which are Wronskian determinants of N − 1 elements of 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).
Let µ be a dominant integral g N -weight and k ∈ Z 0 . Define a partition µ A,k with at most N parts by the rule: We call µ A,k the partition associated with weight µ and integer k. Let λ = λ (1) , . . ., λ (n) be a sequence of dominant integral g N -weights and let k = (k 1 , . . ., k n ) be an n-tuple of nonnegative integers. Then denote λ A,k = λ A,kn the sequence of partitions associated with λ (s) and k s , s = 1, . . . , n.

Gaudin model
Let 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 For each a ∈ C, we have the evaluation homomorphism ev a : g[t] → g where ev a sends g ⊗ t s to a s g for all g ∈ g and s ∈ Z 0 . Its restriction to the subalgebra g ⊂ g[t] is the identity map. 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 commutes 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].
We denote M (∞) the g N -module M with the trivial action of the Bethe algebra B, see [10] for more detail.
For a collection of g N -weights λ = λ (1) , . . . , λ (n) and z = (z 1 , . . . , z n ) ∈P n , we set considered as a B-module. We also denote V λ the module V λ,z considered as a g N -module.
Let ∂ x be the differentiation with respect to x. Define a formal differential operator and B ij ∈ U(g N [t]), j ∈ Z i , i = 1, . . . , N . The operator D B is called the universal operator. Let z = (z 1 , . . . , z n ) ∈P n and let λ = λ (1) , . . . , λ (n) be a sequence of dominant integral g N -weights. For every g ∈ g N , the series g(x) acts on V λ,z as a rational function of x.
Since the Bethe algebra B commutes with g N , B acts on the invariant space (V λ,z ) g N . For b ∈ B, denote by b(λ, z) ∈ End((V λ,z ) g N ) the corresponding linear operator.
Given a common eigenvector v ∈ (V λ,z ) g N of the operators b(λ, z), denote by b(λ, z; v) the corresponding eigenvalues, and define the scalar differential operator The following theorem connects self-dual spaces in the Grassmannian Gr(N, d) with the Gaudin model associated to g N .
There exists a choice of generators B i (x) of the Bethe algebra B, such that for any sequence of dominant integral g N -weights λ = λ (1) , . . . , λ (n) , any z ∈P n , and any B-eigenvector v ∈ (V λ,z ) g N , we have N ), then this defines a bijection between the joint eigenvalues of B on (V λ,z ) g N and sΩ λ,z ⊂ Gr(N, d).
6 Shapovalov form and the key lemma 6
For any dominant integral g N -weight λ, the irreducible g N -module V λ admits a positive definite Hermitian form (·, ·) λ such that (gv, w) λ = (v, (g)w) λ for any v, w ∈ V λ and g ∈ g N . Such a form is unique up to multiplication by a positive real number. We call this form the Shapovalov form.
Proof . We prove the proposition in Section 6.3.
If z ∈ RP n , then B ij (λ, z) are self-adjoint with respect to the Shapovalov form. Therefore all B ij (λ, z) are simultaneously diagonalizable and all eigenvalues of B ij (λ, z) are real.
The following statement is also known.

Theorem 6.2 ([18]
). For generic z ∈P n , the action of the Bethe algebra B on (V λ,z ) g N is diagonalizable and has simple spectrum. In particular, this statement holds for any sequence z ∈ RP n .

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 M T (α) = ker(T−α) dim M . When M T (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∈R M T (α). Then the restriction of the Hermitian form on R is nondegenerate and has signature κ. In particular, dim R |κ|.

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.
Proof of Proposition 6.1. We prove it for the case N = 2r first.
Let E ij with i, j = 1, . . . , 2r + 1 be the standard basis of gl 2r+1 . The Lie subalgebra of gl 2r+1 generated by the elements F ij = E ij − E 2r+2−j,2r+2−i is isomorphic to the Lie algebra so 2r+1 = g N . With this isomorphism, the anti-involution : g N → g N is realized by taking transposition, F ij → F ji . To be consistent with the notation in [12], we write g for g N . The number N in [12] is 2r + 1 in our notation.
We write F ij [s] for F ij ⊗ t s in the loop algebra g t, t −1 . Consider the affine Lie algebra g = 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 g and Consider the extended affine Lie algebra Set U = U( g ⊕ Cτ ) and fix m ∈ {1, . . . , N }. Introduce the element F [s] a of the algebra End C 2r+1 ⊗m ⊗ U, see [12, equation (3.5)], by where e ij ∈ End C 2r+1 denote the standard matrix units. The map induces an antiinvolution For 1 a < b m, consider the operators P ab and Q ab in End C 2r+1 ⊗m defined as follows 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 C 2r+1 ⊗m . In particular, for any 1 a < b m, the operator S (m) satisfies Replacing see [12, formula (3.26)], where the trace is taken on all m copies of End C 2r+1 , we get a differential operator where ϑ mi (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 ϑ mi (x), m = 1, . . . , N , i = 0, . . . , m, see [11,Section 5]. Therefore, to prove the proposition, it suffices to show that the element is stable under the anti-involution . Here maps τ to τ . Applying transposition on a-th and b-th components to the commutator relation see the proof of [12, Lemma 3.6], we get for all 1 a < b m. Here stands for transpose, explicitly, 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 By applying the simultaneous transposition e ij → e ji to all m copies of End C 2r+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].

The lower bound
In this section we prove our main results -the lower bound for the number of real self-dual spaces in Ω λ,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 C d [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.

The general case N 4
Let us first consider the case N 4.
Let T = (T 1 , . . . , T N ) be associated with λ A,k , z. Note that if z, λ, k is invariant under conjugation, then the polynomial T 1 · · · T N also has only real coefficients.
Recall that for any λ and generic z ∈P n , all points of Ω λ,z are multiplicity-free. The same also holds true with λ imposed above for any c.
Theorem 7.2. The number d(λ, z) of real self-dual spaces in Ω λ,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 B ij (λ, z) ∈ End((V λ ) g N ) are self-adjoint relative to the form (·, ·) λ,c . By Lemma 6.4, By Theorem 6.2, for any λ and generic z ∈P n the operators B ij (λ, z) are diagonalizable and the action of the Bethe algebra B on (V λ ) g N has simple spectrum. The same also holds true with λ imposed above for any c. Thus for generic z, the operators B ij (λ, z) have at least |q(λ, c)| common eigenvectors with distinct real eigenvalues, which provides |q(λ, c)| distinct real points in sΩ λ,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.
Therefore, there exists at least one real point in sΩ λ,z . In particular, if dim(V λ ) g N = 1, then the only point in sΩ λ,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 Ω λ,z (real points in sΩ λ,z ) is no less than |a(λ, c)|, where a(λ, c) is the coefficient of the monomial Here ∆ N is the Vandermonde determinant of g N and S N λ (s) is the Schur function of g N 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 sΩ λ,z equals dim(V λ ) g N = q(λ, 0). Hence if z ∈ RP n , Theorem 7.2 claims that all points in sΩ λ,z are real. It is proved in [15,Corollary 6.3] that for z ∈ RP n all points in Ω λ,z are real and multiplicity-free, so are the points in sΩ λ,z .

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].
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 a i , b i , c i , i = 1, 2, 3, such that are real polynomials and form a basis of X. Without loss of generality, we assume deg ϕ < deg ψ.
Since deg ϕ < deg ψ, we have c i ∈ R, i = 1, 2, 3. At least one of c i is nonzero. We assume c 3 = 0. By subtracting a proper real multiple of a 3 ϕ 2 + b 3 ϕψ + c 3 ψ 2 , we assume further c 1 = c 2 = 0. Continuing with the previous step, we assume that b 1 = 0, b 2 = 0, a 1 = 0 and hence obtain that a 1 , b 2 , c 3 ∈ R. Then a 1 ϕ 2 is a real polynomial, so is ϕ. Therefore, a 2 ϕ + b 2 ψ is also a real polynomial, which implies that the space of polynomials √ X is also real.
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 g N = sl 2 .
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(V λ ) g N 2. We also exclude the cases that z ∈ RP n . In particular, the cases that all pairs λ (s) , k s , s = 1, . . . , n, are different.

The case N = 4, 5
For each g 4 -weight λ = (λ 1 , λ 2 ), denote by λ C the g 5 -weight (λ 2 , λ 1 ). Note that g 4 = so 5 is isomorphic to g 5 = sp 4 , the lower bound obtained from the ramification data λ = λ (1) , . . . , λ (n) and k = (k 1 , . . . , k n ) of g 4 is the same as that obtained from the ramification data λ C = λ  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(V λ ) g 4 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 g N -weights. If the corresponding lower bounds are the same, we just write 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.

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.