An identity in the Bethe subalgebra of C[Sn]$\mathbb {C}[\mathfrak {S}_n]$

As part of the proof of the Bethe ansatz conjecture for the Gaudin model for gln$\mathfrak {gl}_n$ , Mukhin, Tarasov, and Varchenko described a correspondence between inverse Wronskians of polynomials and eigenspaces of the Gaudin Hamiltonians. Notably, this correspondence afforded the first proof of the Shapiro–Shapiro conjecture. In this paper, we give an identity in the group algebra of the symmetric group, which allows one to establish the correspondence directly, without using the Bethe ansatz.


Introduction
Let f 1 (u), . . ., f m (u) ∈ (u) be linearly independent rational functions.The Wronskian m , is also a rational function, which, up to a scalar multiple, depends only on the span of f 1 , . . ., f m .It is therefore reasonable to talk about the Wronskian of a finite dimensional subspace V ⊂ (u): if V = 〈 f 1 , . . ., f m 〉 is the subspace of (u) spanned by f 1 , . . ., f m , we define Wr V (u) ∈ (u) to the unique scalar multiple of Wr( f 1 , . . ., f m ) which is a monic rational function, i.e. a ratio of two monic polynomials.We will mainly be interested in the case where the basis elements f 1 , . . ., f m are polynomials, in which case Wr V is a monic polynomial.Given a polynomial w(u) = (u + z 1 ) the inverse Wronskian problem for w(u) is to find all subspaces of polynomials V ⊂ [u] such that Wr V = w.There are finitely many such V of any particular dimension m.Moreover, if one can find all the n-dimensional solutions, then it is straightforward to find all solutions of any other dimension (see Proposition 2.4); we will therefore focus on the case m = n.
The inverse Wronskian problem appears in many guises throughout mathematics.It can be reformulated as a Schubert intersection problem, or in terms of linear series on 1 , or in terms of rational curves in with with prescribed flexes.It is also a special case of the pole placement problem in control theory [1].The survey [14] discusses many of these alternate formulations along with a variety applications.
There is also a deep connection with representation theory and quantum integrable systems.Over a series of papers (see [6]), Mukhin, Tarasov and Varchenko showed that the problem of finding these solutions is equivalent to the problem of finding eigenvectors of the Bethe algebra for the Gaudin model.The Bethe algebra is defined as a commutative subalgebra of the universal enveloping algebra U gl m ( [t]) [7]; however by Schur-Weyl duality, it has a quotient n (z 1 , . . ., z n ) which can be identified with a commutative subalgebra of [S n ], the group algebra of the symmetric group [8].
Briefly, here's how the equivalence works.One concretely writes down certain operators (the Gaudin Hamiltonians), which in this paper are denoted β − k,l ∈ [S n ], k, l ≤ n.(The "−" in our notation requires some explanation; this will be provided shortly.)These operators commute pairwise, and they are generators of n (z 1 , . . ., z n ).We combine them to form a linear differential operator with coefficients in [S n ] ⊗ (u): One can then restrict this differential operator to any eigenspace E of n (z 1 , . . ., z n ), which gives a scalar valued differential operator − E of order n, with coefficients in (u).Theorem 1.1 (Mukhin-Tarasov-Varchenko).The kernel of − E is an n-dimensional vector space V E ⊂ [u], which is a solution of the inverse Wronskian problem for w(u).Furthermore all n-dimensional solutions to the inverse Wronskian problem are of this form.
Theorem 1.1 is far from obvious.Arguably the most mysterious part is the dimension of the space of polynomials in the kernel.In general, if one writes down a linear differential equation of order n with coefficients in (u), it is rare for it to have any rational solutions, let alone an n-dimensional space of polynomial solutions.Of course, one can write down equations for when this occurs, but these are difficult to work with explicitly, and checking directly that the operators β − k,l satisfy these equations seems to be impractical.Mukhin, Tarasov and Varchenko's proof of Theorem 1.1 is part of a larger body of work on the Bethe ansatz, a technique from mathematical physics for finding the eigenvectors to certain problems involving commuting operators.In a nutshell, they show that when one applies the Bethe ansatz method to the Gaudin model, the Bethe ansatz equations for finding the eigenvectors can be reinterpreted as equations for solving the inverse Wronskian problem.The formulation in terms of n (z 1 , . . ., z n ) ⊂ [S n ] is derived from theorems about the infinite dimensional Bethe algebra inside U gl m ( [t]) using Schur-Weyl duality.
The main goal of this paper is to give an account of Theorem 1.1, which is short, mostly self-contained, operates strictly inside [S n ], and does not involve finding the eigenvectors of the Bethe algebra.Our main result (Theorem 1.2) is an identity in n (z 1 , . . ., z n ), which accomplishes this.We introduce a second operator + n , which is related to − n by an anti-involution of the algebra of [S n ]-valued linear differential operators.All minus signs in the formula are changed to plusses, and the order of factors is reversed from left to right. .
The coefficients β + k,l are given by a similar formula to β − k,l , but again, without signs.We show that the elements β + k,l are also generators for n (z 1 , . . ., z n ).This means one can also restrict + n to any eigenspace E of n (z 1 , . . ., z n ) to get a scalar valued differential operator + E .
Theorem 1.2.In [S n ], the algebra of [S n ]-valued linear differential operators, we have the identity We can now argue as follows.If E is any eigenspace of n (z 1 , . . ., z n ), we obtain the scalar valued differential operator identity Since ker(∂ 2n u ) is a 2n-dimensional subspace of [u], and ker( + E ), ker( − E ) both have dimension at most n, we see that V E = ker( − E ) must be an n-dimensional subspace of ker(∂ 2n u ); in particular V E is an n-dimensional space of polynomials.It now follows readily (see Corollary 2.3) that V E is a solution to the inverse Wronskian problem for w(u).The fact that every solution arises in this way follows as well, because we know how many solutions there are to each of the two problems (see Remark 6.3).
An important consequence of Theorem 1.1 is the reality theorem, conjectured by B. and M. Shapiro in the mid-1990s and proved by Mukhin, Tarasov and Varchenko in [6] (see also [2,5,7,14]).If z 1 , . . ., z n are real, then the operators β − k,l are real and self-adjoint with respect to the standard inner product on [S n ] (for which the group elements form an orthonormal basis); hence n (z 1 , . . ., z n ) is diagonalizable over , and the entire argument above goes through with in place of .Theorem 1.3.If z 1 , . . ., z n ∈ , then all solutions to the inverse Wronskian problem for w(u) are real.
A natural question is whether there are analogous results for + n .For a partial answer, consider the inverse Wronskian problem for rational functions: given g(u) ∈ (u), find V ⊂ (u) such that Wr V = g.Theorem 1.2 implies that if E is an eigenspace of n (z 1 , . . ., z n ), then ker( + E ) is an n-dimensional subspace of (u), which is a solution to the inverse Wronskian problem for the rational function 1 w(u) .However, in this case we are not getting all rational solutions: unlike the polynomial inverse Wronskian problem, the rational inverse Wronskian has infinitely many solutions of any given dimension.We discuss this further in Section 8.
This paper is structured as follows.Sections 2 and 3 provide background on the fundamental differential operator of a subspace V ⊂ (u), and on the Bethe subalgebra of [S n ].The proof of Theorem 1.2 is given in Section 4. Sections 5, 6 and 7 establish additional properties of the algebra n (z 1 , . . ., z n ), beginning with a combinatorial proof of commutativity, and culminating in the fact that the operators β + k,l ∈ [S n ] are generators (Theorem 3.1).We conclude with a discussion of the mysterious operator + n , and other open questions in Section 8.In keeping with our stated objectives, our exposition includes proofs of known results whenever the original proof was based on the Bethe ansatz or derived from identities in algebras other than [S n ], e.g. using Schur-Weyl duality.
Acknowledgements.This work became possible thanks to discussions during the Fields Institute Thematic Program on Combinatorial Algebraic Geometry in 2016, with Joel Kamnitzer, Frank Sottile, and David Speyer.I thank Vitaly Tarasov and an anonymous referee, for pointing out some recent relevant references.Innumerable calculations for this project were carried out using Sage [12].

Fundamental differential operators
Let = (u)[∂ u ] denote the algebra of complex valued linear differential operators in variable u, with rational function coefficients.The algebra (u) of rational functions is a commutative subalgebra of , and has the commutation relations for g = g(u) ∈ (u).Every element Ψ ∈ can be expressed uniquely in the form where ψ 0 (u), . . ., ψ m (u) ∈ (u).If ψ m (u) = 0, then m = ord(Ψ) is called the order of Ψ, and we say Ψ is a monic operator if ψ m (u) = 1.Write 〈Ψ〉 j = ψ j (u), to mean the coefficient of ∂ j u in this canonical representation.
We view Ψ as a linear differential operator Ψ : (u) → (u), via Write ker(Ψ) ⊂ (u) for the kernel of this operator, and pker(Ψ) = ker(Ψ) ∩ [u] for the subspace of polynomials in ker(Ψ).Note that when we write Ψ g or Ψ g(u), this will always mean the product of Ψ and g in , and should not be confused with the rational function 〈Ψ g〉 0 obtained by applying the differential operator Ψ to g.
From the general theory of linear ordinary differential equations, we have the following basic inequalities (see e.g.[4, §3.32]).Let V ⊂ (u) be a finite dimensional -linear subspace of (u).Choose any basis ( f 1 , . . ., f m ) for V .The fundamental differential operator of V is the monic operator D V ∈ , defined by the determinantal formula This definition is independent of the choice of basis.Here, we use the convention that the determinant of a k × k matrix A with non-commuting entries is defined to be the "row-expansion" Equivalently, viewing as D V a differential operator (u) → (u), we have The numerator is zero if and only if f 1 , . . ., f m , g are linearly dependent, i.e. if and only if g ∈ V .Hence we see that ker(D V ) = V .Not every monic operator in is a fundamental differential operator.We have the following elementary characterization.Proposition 2.2.Suppose Ψ ∈ is a monic operator of order m.
(i) Ψ = D V for some finite dimensional V ⊂ (u) if and only if dim ker(Ψ) = m.
(ii) Ψ = D V for some finite dimensional V ⊂ [u] if and only if dim pker(Ψ) = m.
In either case, if Proof.If Ψ = D V then dim ker(Ψ) = ord(Ψ) = m.Conversely, if ker(Ψ) = V , and dim V = m, then Ψ and D V are both monic differential operators of order m, with kernel V .Therefore V ⊆ ker(D V − Ψ), so dim ker(D V − Ψ) > ord(D V − Ψ); by Proposition 2.1, this is only possible if D V − Ψ = 0.This proves (i) and a similar argument proves (ii).The final statement is a straightforward calculation, and follows directly from the definitions of D V and Wr V .
Corollary 2.3.Let g(u) ∈ (u) be a monic rational function, and let Ψ ∈ be a monic operator of order m.
, then ker(Ψ) is a solution to the (rational) inverse Wronskian problem for g(u).
, then pker(Ψ) is a solution to the (polynomial) inverse Wronskian problem for g(u).
We now describe the relationship between solutions of the (polynomial) inverse Wronskian problem of different dimensions.

solution to the inverse Wronskian problem for w(u) if and only if pker(D
Using Proposition 2.4, the inverse Wronskian problem for w(u) reduces to the case where dim V = deg(w) = n.If we can we find all n-dimensional solutions, then we obtain all m-dimensional solutions, m = 1, 2, 3, . . ., as follows: for m < n, take all subspaces In the remaining sections, our discussion of the inverse Wronskian problem will focus exclusively on the case where dim V = deg(w).

The Bethe subalgebra of [S n ]
Let S n denote the symmetric group of permutations of [n] = {1, . . ., n}, and let [S n ] denote the group algebra of S n .Write 1 S n for the identity element of S n .
As before, let w(u) where z 1 , . . ., z n are complex numbers.For a subset X ⊆ ∈ X } be the subgroup S n which permutes only the elements of X .Define elements α ± X ∈ [S n ], as follows: In particular, note that to be the subalgebra of [S n ] generated by the group algebra elements β − k,l (resp.β + k,l ), k, l ≤ n.Let n (z 1 , . . ., z n ) denote the algebra generated by both the β − k,l and β + k,l operators.
Theorem 3.1.The elements β ± k,l commute pairwise.Furthermore, The proof of Theorem 3.1 is given in Section 5.
The commutative algebra n (z 1 , . . ., z n ) is called is called the Bethe subalgebra of [S n ] of Gaudin type.Certain properties of this subalgebra depend on the numbers z 1 , . . ., z n .For example, the dimension of n (z 1 , . . ., z n ) depends on z 1 , . . ., z n ; in some cases n (z 1 , . . ., z n ) is semisimple, but not always.However, in all cases it contains From this, it is not hard to see that n (z 1 , . . ., z n ) is generated by the elements β ± k,n−k (t), k = 0, . . ., n, t ∈ .In particular the Bethe subalgebra is translation invariant, i.e.
Consider the algebra For ease of notation, we will implicitly identify Ψ ∈ , with Ψ ⊗1 The operators − n , + n ∈ [S n ] which appear in Theorem 1.2 can now be written more concisely as: Let λ ⊢ n be a partition, and let M λ denote the irreducible [S n ]-module associated to λ.An eigenspace of the Bethe algebra of type λ is a maximal linear subspace E ⊂ M λ , such that each operator γ ∈ n (z 1 , . . ., z n ) acts as a scalar γ E on E. In particular, for any eigenspace E of the Bethe algebra, we obtain scalars and scalar valued differential operators − E , + E ∈ : w(u) .Thus Theorem 1.2 and Corollary 2.3 imply that ker( − E ) is a solution to the inverse Wronskian problem for w(u), and ker( + E ) is a solution to the inverse Wronskian problem for1 w(u) .
Example 3.4.Take w(u , in which the elementary transpositions (1 2) and (2 3) are represented by the matrices acts as zero, and The eigenspaces 3 ( 3, − 3, 0) of type λ = 21 are therefore the eigenspaces of the matrix One can check that ker( − ) = 〈u 4 −4u 3 , u 2 +2u, 1〉, which are indeed solutions to the inverse Wronskian problem for u 3 −3u.There are two more solutions, which come from the 1-dimensional [S n ]-modules M 3 and M 111 .Remark 3.5.Our exposition differs from [8] in the following respect.In [8], the Bethe subalgebra of [S n ] is defined to be the algebra generated by the elements β − k,l , whereas here, we have defined it to be the algebra generated by all elements β ± k,l .Theorem 3.1 asserts that these definitions agree.The fact that − n (z 1 , . . ., z n ) is commutative is the content of [8, Proposition 2.4], and one can easily deduce that + n (z 1 , . . ., z n ) is also commutative.However, the fact that n (z 1 , . . ., z n ) is commutative does not seem to follow directly; we prove this in Section 5. From here we deduce Theorem 1.1, and use it to show that all three algebras are equal.This establishes that n (z 1 , . . ., z n ) is generated by the elements β − k,l , and it is also generated by the elements β + k,l , as asserted in the introduction.Theorem 3.2 is [8, Proposition 2.1], and we include a short proof in Section 6.

Proof of the main identity
In this section, we prove Theorem 1.2.
For each a ∈ [n] let q a (u) = 1 u+z a , and for a subset X ⊆ [n], let q X (u) = a∈X q a (u).The operators ± n can be rewritten as follows.
Given σ Z ∈ SP n and a subset A ⊆ Z, let σ Z ,A be the set of pairs of supported permutations (δ X , ǫ Y ), such that X ∪ Y = Z, X ∩ Y = A, and δǫ = σ.Thus σ Z ,A is the set of factorizations of σ into two supported permutations, with some conditions on the supports.
Consider the differential operators When we expand the product + n − n , and reorganize the terms, we get the following formula.
Proof.We have We now show that almost all of the terms on the right hand side of Lemma 4.1 are equal to zero.
Proof.As |A| ≥ 2, there exists a transposition τ ∈ S A .Then (δ X , ǫ Y ) ↔ ((δτ) X , (τǫ) Y ) defines a sign reversing involution on the set σ Z ,A , so To analyze the cases |A| ≤ 1, consider the -bilinear map Φ : [s, t] × (u) → , defined by Φ(s i t j , g) = ∂ i u g(u)∂ j u , for g(u) ∈ (u), i, j ≥ 0. Notice that the operator F σ Z ,A is equal to Φ(p σ Z ,A , q A q Z ), for the polynomial The following identity is a reformulation of the commutation relations (2.1).
Proposition 4.3.For any p(s, t) ∈ [s, t], and g(u) ∈ (u), Proof.Since Φ is bilinear, if suffices to prove this for p = s i t j , in which case we have To simplify some of the notation, we present the argument for the case Z = [n]; the other cases are proved by a conceptually identical argument, with S Z in place of S n .
Proof.Suppose σ ∈ S n has cycles γ 1 , . . ., γ m .Let ν i be the length of cycle γ i and for any subset K ⊆ [m], write ν K = i∈K ν i .We will show that the polynomials p σ [n] ,A , |A| ≤ 1, are related to the following polynomial: where K ⊆ [m − 1] and 1 ≤ i ≤ ν m .For this factorization, we have Finally, by Lemma 4.2 and Proposition 4.3, we have The result now follows, because q ′ [n] + n a=1 q a q [n] = 0. Proof of Theorem 1.2.By Lemma 4.4, the only σ Z ∈ SP n that produces a non-zero summand on the right hand side of Lemma 4.1 is the pair Z = , σ = 1 S n , which yields

Commutativity
In this section, we give a bijective proof of the fact that the operators β ± k,l all commute.The proof is is essentially identical for all sign combinations, so for ease of notation, we'll focus on We will treat z 1 , . . ., z n as formal indeterminates.Working formally, it suffices to prove the above identity in the case where l ′ = n − k ′ .This is enough because if we know that for all t ∈ , from which one can easily deduce the other commutation relations.
Let B k,l denote the set of pairs (σ X , Y ) where σ X ∈ SP n is a supported permutation and Y ⊆ (5.1) Define a preorder on S n as follows.For π, τ ∈ S n , we'll say π τ if every fixed point of τ is a fixed point of π.The following two lemmas are straightforward.
where ξ = ξσσ ′ ,Y ∪Y ′ .Then ρ is a bijection.Furthermore, ρ is weight preserving, in the sense that σσ We begin by verifying that ( σ′ The only part of this claim that's not clear is the assertion σ′ X ′ ∈ SP n .To see this, rewrite the formula for σ′ as σ′ = ξ(σ ′ π 2 ) −1 ξ−1 , and note that σ ′ , π ∈ S X ′ . (Remark: Here is where we need the assumption l We check that ρ is weight preserving.First of all, Next, since π ∈ S [n]\Z , and ξ ∈ S Z , Z = Y ∪ Y ′ is an invariant subset for both π and ξ and hence it is invariant for ξ.Therefore, The fact that sgn(σ) = sgn( σ) and sgn(σ ′ ) = sgn( σ′ ) is clear from (5.2).Finally, we check that ρ is a bijection.Since the domain and codomain have the same cardinality, it suffices to prove that ρ is injective.Suppose that ρ(σ We now give the proof of Theorem 3.1, with one small caveat: the final case in the proof uses Theorem 7.6, which is proved in Section 7.This does not lead to any circularity, since the final case is not used by any of the arguments in Section 6 or 7.The argument below establishes the commutativity of n (z 1 , . . ., z n ), and the equality of the different algebras in the case where (z 1 , . . ., z n ) is a general point of n .The final case, where (z 1 , . . ., z n ) ∈ n is arbitrary, is where we need Theorem 7.6.
We note that the commutativity of n (z 1 , . . ., z n ) is enough to infer Theorem 1.1 from Theorem 1.2: the equality of the three algebras is not needed for this argument.Therefore, in the remaining sections of the paper, we will freely use Theorem 1.1.
Proof of Theorem 3.1.The bijection ρ in Proposition 5.3 corresponds terms on the left hand side of (5.1) with terms on the right hand side, which proves commutativity.This shows that ± n (z 1 , . . ., z n ) and n (z 1 , . . ., This follows from the fact that for (z 1 , . . ., z n ) general, ± n (z 1 , . . ., z n ) is a deformation of the Gelfand-Tsetlin subalgebra of [S n ] [8, Proposition 2.5], which is a maximal commutative subalgebra (see [9]).We deduce that Proving this equality of algebras for arbitrary (z 1 , . . ., z n ) is a bit more involved.We need to show that for all k, l, there exists a polynomial function which expresses β + k,l in terms of the operators β − k ′ ,l ′ , and vice-versa.This is the content of Theorem 7.6.

Schubert cells
We may assume that our basis for V is chosen such that Then λ = (λ 1 , λ 2 , . . ., λ n ) is a partition.(Note that here, some of the "parts" λ i may be zero.)We say V has Schubert type λ, and the numbers d 1 , . . .d n are called the exponents of V at infinity.The space of all V of Schubert type λ is called a Schubert cell, and is denoted λ .Note that |λ| = deg(Wr V ).
The fundamental differential operator D V encodes the Schubert type V , as follows.If g(u) ∈ (u) is a non-zero rational function, we say that c ∈ × is the leading coefficient

and V ′ have the same exponents at infinity if and only if
Proof.The exponents of V at infinity are the roots of the indicial equation (see e.g.[4, §7.21]).Theorem 6.2.Let λ ⊢ n, and let E ⊂ M λ be an eigenspace of the Bethe algebra, of type λ.
for some sequence of rational numbers (c λ 0 , . . ., c λ n ).Up to a scalar multiple, (c λ 0 , . . ., Taking derivatives of both sides of (6.1), and using the fact we obtain the following recurrence for the numbers c λ k : In the last case, the sum is taken over all partitions µ ⊢ n − 1 such that µ i ≤ λ i for all i.
The elements β − n−k,0 ∈ − n (z 1 , . . ., z n ) do not depend on z 1 , . . ., z n , and are in the centre of [S n ].Hence β − n−k,0 acts as a scalar b λ k on M λ .Considering the trace, we find that dim Here χ λ : [S n ] → denotes the character of M λ , and s λ is the Schur symmetric function; the second equality above uses the Frobenius characteristic map.It is well-known that 〈s λ , φs 1 〉 = µ⋖λ 〈s µ , φ〉 for any symmetric function φ; hence the numbers dim M λ n! b λ k satisfy the same recurrence as the numbers c λ k , and we conclude that
Proof.Proposition 7.1 implies that there exists an eigenspace E * ⊂ M ⊗ , such that V * E = V E * .We must show that E * = E ⊗ , for (z 1 , . . ., z n ) general.By continuity, this implies the result for all (z 1 , . . ., z n ).Note that the relationship between E * and E is completely determined by what happens at any general point (z 1 , . . ., z n ), by parallel transport.So it is enough to prove this for (z 1 , . . ., z n ) belonging to some Zariski dense open subset of n .
Consider the degeneration of n (z 1 , . . ., z n ) to the Gelfand-Tsetlin algebra [8, Proposition 2.5], which can be obtained by substituting z i → t i z i , and letting t → ∞.For t large, this degeneration process allows us to assign standard Young tableaux [9,10,11,13,17]).In the case of V ⊂ [u], the tableau T V is defined in terms the asymptotics of the coordinates of V ; in the case of E ⊂ M λ , the limit is an eigenspace of the Gelfand-Tsetlin algebra, which naturally has an associated tableau.Each tableau uniquely identifies the subspace of [u] or eigenspace of n (z 1 , . . ., z n ) in question.Furthermore, these tableaux are related.Using the definition of T V from [11, §2.1], Theorem 6.2 implies that T V E = T E for any eigenspace E of the Bethe algebra (see also [17]).It follows from Proposition 7.1 that T V * E = T * V E .Finally, T E⊗ = T * E is a basic property of the Gelfand-Tsetlin algebra (see [9]).Here if T is a standard Young tableau, T * denotes the conjugate tableau, obtained by reflecting T along the main diagonal.Putting this all together, we have We now use duality to finish the proof of Theorem 3.1, showing that + n (z 1 , . . ., z n ) = − n (z 1 , . . ., z m ) = n (z 1 , . . ., z n ) for all (z 1 , . . ., z n ) ∈ n .Lemma 7.5.Let V ∈ λ , with canonical coordinates (v i j ) j≤λ i .For j ≤ λ i , let s i j be the coefficient of u e j in 〈Wr V D V u d i 〉 0 .Then s i j is given by a polynomial in the canonical coordinates with -coefficients, which is of the form s i j = c i j v i j + r i j , where c i j ∈ is a non-zero constant, and r i j is a polynomial involving only the coordinates Proof.Let ( f 1 , . . ., f n ) be the canonical basis for V .Up to an irrelevant non-zero scalar, s i j is equal to the coefficient of u e j in Wr( f 1 , . . ., f n , u d i ).This is a polynomial in the canonical coordinates with -coefficients.Rewriting the Wronskian as we see that s i j is a linear function of f i − u d i d i !, so each term in s i j must contain exactly one v i j ′ for some j ′ ≤ λ i .Now, think of v i j as an indeterminate of degree d i − e j ; hence f i is a homogeneous polynomial of degree d i .Then s i j is homogeneous of degree d i − e j , which means it can only involve indeterminates of degree d i − e j or less.This, together with the preceding remarks shows that s i j = c i j v i j + r i j , where r i j only involves indeterminates of degree less than d i − e j .Finally, which is non-zero, since the exponents d 1 , . . ., d n , e j are distinct.
Theorem 7.6.For all k, l ≤ n there exist polynomials with -coefficients which express the operators β + k,l as a function the operators β − k ′ ,l ′ , and vice-versa.
Proof.First note that if we have a formula for β + k,l in terms of the β − k ′ ,l ′ , then applying ⋆ to both sides gives a formula for β − k,l in terms of the β + k ′ ,l ′ , so the "vice-versa" statement will be automatic.
Let β ± k,l,λ ∈ End(M λ ) and ± λ ∈ ⊗ End(M λ ) denote the restrictions of operators β ± k,l and ± n to M λ .Let P λ ∈ Z( [S n ]) denote the central idempotent which acts as the identity on M λ , and as zero on M λ ′ , λ ′ = λ.By Theorem 3.2, P λ is given by some polynomial with -coefficients in β − 0,0 , . . ., β − n,0 .We will prove that there exist polynomials Q k,l,λ with -coefficients which express β + k,l,λ in terms of β − k ′ ,l ′ ,λ for each λ ⊢ n.This is sufficient, as λ⊢n Q k,l,λ P λ will then give a polynomial expression for Let n ⊂ be the vector space of differential operators Ψ of order at most n, such that 〈Ψ〉 i is a polynomial of degree at most i.Let Ω λ : λ → n be the map defined by Ω λ (V ) = Wr V D V .Both ∆ λ and Ω λ are defined by polynomials with -coefficients.Lemma 7.5 shows that Ω λ : λ → n has an left-inverse Υ λ : n → λ , defined by polynomials with -coefficients: given Ψ = Wr V D V , we can solve for the canonical coordinates (v i j ) j≤λ i of V , recursively, in increasing order of d i − e j .Now consider the composition which is a polynomial map from n to itself.By definition, for all V ∈ λ , Θ λ (Wr V D V ) = Wr V * D V * .Thus, if E ⊂ M λ is an eigenspace of n (z 1 , . . ., z n ), then by Theorem 7.4, Finally, since this is a polynomial identity which holds for (z 1 , . . ., z n ) general, it holds for all (z 1 , . . ., z n ) ∈ n .Therefore the required polynomials Q k,l,λ are just the coordinates of the map Θ λ .Remark 7.7.The maps Ω λ and Υ λ in the proof of Theorem 7.6 are essentially the isomorphism described in [8, Theorem 4.3(iv)] and its inverse.The proof of Lemma 7.5 is based on [7, Lemma 4.5].

Discussion
Theorem 1.1 and the results of Sections 6 and 7 mainly focus on the differential operators − n and − E .It not obvious what the corresponding story is for + n .Let E be an eigenspace of n (z 1 , . . ., z n ), and consider the subspace V + E = ker( + E ) ⊂ (u).We would like to know which subspaces of (u) are of this form.As already noted in the introduction, Theorem 1.2 tells us that V + E is n-dimensional, with Wr V + E = 1 w .We now state a slightly stronger necessary condition.), where f 1 , . . ., f n are polynomials of degree at most 2n.This shows that wV + E ⊂ [u], and its Schubert type λ satisfies λ 1 ≤ n.Finally we have the identity Wr(g f 1 , . . ., g f n ) = g n • Wr( f 1 , . . ., f n ) for any g(u) ∈ (u).Hence the fact that Wr V + E = 1 w implies that Wr wV + E = w n • 1 w .The converse is false.If V is an n-dimensional solution to the inverse Wronskian problem for w n−1 , with appropriate conditions on the Schubert type, it is not necessarily true that V = wV + E for some eigenspace of the Bethe algebra E. We have a pretty good guess what the right sufficient condition is.Conjecture 8.2.Suppose z 1 , . . ., z n are distinct.Let V ⊂ [u] be an n-dimensional vector space of Schubert type λ, such that Wr V = w n−1 .Then V = wV + E for some eigenspace E of n (z 1 , . . ., z n ), if and only if V belongs to the Schubert intersection Here X µ (z) for z ∈ is a Schubert variety inside Gr(n, 2n−1 [u]); we are following the notation and conventions of [5, §2.1].It should be possible to prove this by applying the machinery of [7] to the gl n ( [t])-representation (∧ n−1 n ) ⊗n , and using Schur-Weyl duality.The author has verified that this works up to n = 3, but a complete proof is beyond the intended scope of this paper.
If Conjecture 8.2 is correct, it still does not fully characterize ker( + E ).A more complete answer would describe the precise the relationship between ker( − E ) and ker( + E ), analogously to the way Theorem 7.4 describes the relationship between ker( − E ) and ker(⋆ − E ).One might hope that understanding this relationship could lead to a more conceptual proof of Theorem 1.2.
A natural question is whether there is a more general form of Theorem 1.2, for example, an identity inside the full Bethe algebra in U gl m ( [t]) , rather than just inside the Bethe subalgebra of [S n ].One problem with this notion is that in formulations of Theorem 1.1 using the full Bethe algebra, there is no uniform upper bound on the degrees of the polynomials involved; instead one has different bounds for different representations of gl m .By contrast, working in [S n ], we can say for any eigenspace E ⊂ M λ , ker( − E ) only involves polynomials of degree at most 2n, independent of λ.V. Tarasov has pointed out that the results of [15] provide a sort of analogue.In particular Section 6 therein describes a factorization of a fixed differential operator, of the same form as our Theorem 1.2.These results are based on the Bethe ansatz in the context of (gl k , gl m ) duality on the exterior algebra ∧ • ( k ⊗ m ).The authors show that the image of the Bethe algebras for gl k and gl m coincide on the representation ∧ • ( k ⊗ m ).A similar result holds for the Lie superalgebras gl k and gl m|n [3].This suggests that operators − n and + n should be regarded as elements of a quotient of this common image, but coming from the two different factors of gl n acting ∧ • ( n ⊗ n ).It would be great to see this worked out explicitly.
A related question is whether Theorem 1.2 has an analogue for the XXX model.In [8], Mukhin, Tarasov and Varchenko define a Bethe subalgebra of [S n ] of XXX type, which is a 1-parameter deformation of n (z 1 , . . ., z n ).A paper of Uvarov [16] generalizes the results of [15] in this direction, and it would be valuable to have versions of these results formulated concretely inside [S n ].
Finally, it would be nice to have a more explicit formula for the polynomials which express β + k,l in terms in terms of β − k ′ ,l ′ , or for the map Θ λ defined in the proof of Theorem 7.6.Since the elements β + k,l are (or are at least related to) "coefficients" of + n , this may shed some light on the aforementioned problem of describing the relationship between ker( + E ) and ker( − E ).

First(− 1 )
we compute p σ[n]  , .By definition this is a sum over σ[n]  , , the set of all factorizations of σ into two supported permutations δ X and ǫ Y where (X , Y ) is a partition of [n].The only way to obtain such a factorization is to partition the cycles of σ: we must have ǫ = i∈K γ i and δ = i / ∈K γ i for some subset K ⊆ [m].For this factorization, we have (−1) |Y | sgn(ǫ) = (−1) |K| , |Y | = ν K and |X | = n − ν K .Plugging this information into (4.1),we obtain p σ [n] , = K⊆[m] |K| s ν K t n−ν K = p ν .Next we compute p σ [n] ,{a} .Without loss of generality, we may assume that a appears in the last cycle σ m , say σ m = (a b 1 b 2 b ν m −1 ).Consider the following cycles: π i = (a b 1 . . .b i−1 ) and π ′ i = (a b i . . .b ν m −1 ).The factorizations of σ into δ X and ǫ Y such that X ∪ Y = [n] and X ∩ Y = {a} are of the form

Lemma 5 . 1 .Lemma 5 . 2 .Proposition 5 . 3 .
Let Z ⊆ [n].For every τ ∈ S n there exists permutation π ∈ S [n]\Z such that π τ and every cycle of πτ contains at most one element of [n] \ Z.Let Z ⊆ [n].Let τ ∈ S n be a permutation such that every cycle contains at most one element [n] \ Z. Then there exists an involution ξ ∈ S Z such that ξ τ and τ = ξτ −1 ξ.For every pair τ ∈ S n , Z ⊆ [n], choose a permutation π = π τ,Z ∈ S n as in Lemma 5.1 and let τ = π τ,Z τ; then choose an involution ξ = ξ τ,Z ∈ S n as in Lemma 5.2 and let ξτ,Z = π τ,Z ξ τ,Z .Note that ξ τ,Z commutes with π τ,Z , since π τ,Z ∈ S [n]\Z and ξ τ,Z ∈ S Z , Note also that ξτ,Z τ.Consider the map and therefore V E ∈ λ .Proof of Theorem 3.2.Suppose γ 1 , . . ., γ m ∈ Z( [S n ]), where γ i acts as the scalar γ λ i on M λ .The elements γ 1 , . . ., γ m generate Z( [S n ]) if and only if the tuples (γ λ 1 , . . ., γ λ m ) are distinct for distinct partitions λ ⊢ n.The proof of Theorem 6.2 establishes this for the elements β If z 1 , . . ., z n are generic, then there are exactly dim M λ distinct solutions to the inverse Wronskian problem in the Schubert cell λ , and there are exactly dim M λ eigenspaces of the Bethe algebra of type λ.The first statement is a computation in the Schubert calculus (see e.g.[5, §2.2]); the second statement follows from the fact that − n (z 1 , . . ., z n ) is a deformation of the Gelfand-Tsetlin algebra [8, Proposition 2.5].This numerical coincidence explains why every n-dimensional solution to the inverse Wronskian problem is of the form V E = ker( − E ).None of the theorems discussed in this section are new.Proposition 6.1 is from the classical theory of Fuchsian differential equations.Theorem 6.2 is implicitly part of the content of [8, Theorem 4.3]; the proof above is based on the same main idea, but avoids using Schur-Weyl duality.Theorem 3.2 is [8, Proposition 2.1], and the authors' assertion that this follows from [8, Proposition 3.5] is essentially the proof given above.
Proposition 8.1.If V + E = ker( + E ), where E is an eigenspace of the Bethe algebra, then wV + E = {wg | g ∈ V + E } is an n-dimensional vector space of polynomials, which is an ndimensional solution to the inverse Wronskian problem for w n−1 .Furthermore wV + E ∈ λ for some partition λ = (λ 1 , . .., λ n ) with λ 1 ≤ n.Proof.It also follows from Theorem 1.2 that ker( + E ) is contained in the image of − E restricted to ker(∂ 2n u ).In particular, ker( + E ) has a basis of the form (