Quantum double Schubert polynomials, quantum Schubert polynomials and

Vafa–Intriligator formula.

Anatol N. Kirillov^{∗} and Toshiaki Maeno^{†}

Department of Mathematical Sciences, University of Tokyo,

Komaba, Meguro-ku, Tokyo 153, Japan

Abstract

We study the algebraic aspects of equivariant quantum cohomology algebra of the flag manifold. We introduce and study the quantum double Schubert polynomials 𝔖̃_{w}(x, y), which are the Lascoux– Schützenberger type representatives of the equivariant quantum cohomology classes. Our approach is based on the quantum Cauchy identity. We define also quantum Schubert polynomials 𝔖̃_{w}(x) as the Gram–Schmidt orthogonalization of some set of monomials with respect to the scalar product, defined by the Grothendieck residue. Using quantum Cauchy identity, we prove that 𝔖̃_{w}(x) = 𝔖̃_{w}(x, y)|_{y=0} and as corollary obtain a simple formula for the quantum Schubert polynomials 𝔖̃_{w}(x) = ∂_{ww_{0}}^{(y)}𝔖̃_{w_{0}}(x, y)|_{y=0} . We also prove the higher genus analog of Vafa–Intriligator’s formula for the flag manifolds and study the quantum residues generating function. We introduce the extended Ehresman–Bruhat order on the symmetric group and formulate the equivariant quantum Pieri rule.

∗On leave from Steklov Mathematical Institute, Fontanka 27, St.Petersburg, 191011, Russia

†Supported by JSPS Research Fellowships for Young Scientists

1 Introduction.

The structure constants of the quantum cohomology ring are given by the third derivatives of the Gromov-Witten potential F. The Gromov-Witten potential F is a generating function of the Gromov-Witten invariants. The axioms of the tree level Gromov-Witten invariants

⟨I_{0,m,β}^{V}⟩ : H^{∗}(V, Q)^{⊗m} ⟶ Q,

β ∈ H_{2}(V, Z), for a target space V are given by Kontsevich and Manin [KM]. Let X_{1}, …, X_{m} be cycles on V and X_{1}^{∗}, …, X_{m}^{∗} their dual classes. Then the invariant ⟨I_{0,m,β}^{V}⟩(X_{1}^{∗} ⊗ ⋯ ⊗ X_{m}^{∗}) can be considered as the virtual number of the stable maps f from m-pointed rational curve (P^{1}; p_{1}, …, p_{m}) to V, such that the image of f represents the homology class β and f(p_{i}) ∈ X_{i}.

In case of flag variety F l_{n} := SL_{n}/B of type A_{n−1} the potential F is given as follows. Let Ω_{v} be the dual class of the Schubert cycle X_{v} corresponding to a permutation v ∈ S_{n}. Then the potential F_{ω}((t_{v})_{v∈S_{n}}) is defined by

F_{ω}((t_{v})_{v∈S_{n}}) = ∑_{β} ∑_{m=Σm_{v}≥3} exp(− ∫_{β} ω)\frac{⟨I_{0,m,β}^{V}⟩( ⨂_{v∈S_{n}} Ω_{v}^{⊗m_{v}})}{∏_{v∈S_{n}} m_{v}!} ∏_{v∈S_{n}} t^{m_{v}},

where ω is a Kähler form. For each point t ∈ H^{∗}(F l_{n}), the quantum multiplication law is given by

Ω_{u} ∗ Ω_{v} = ∑_{w∈S_{n}} \frac{∂^{3}F_{ω}}{∂t_{u}∂t_{v}∂t_{w}} (t)Ω_{ww_{0}},

where w_{0} is the permutation of maximal length. The algebra with this multiplication law is called a quantum cohomology ring, which is denoted by QH_{t}^{∗}(F l_{n}). The associativity of the quantum multiplication is equivalent to the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equation

∑_{v∈S_{n}} \frac{∂^{3}F_{ω}}{∂t_{u_{1}}∂t_{u_{2}}∂t_{v}} \frac{∂^{3}F_{ω}}{∂t_{vw_{0}}∂t_{u_{3}}∂t_{u_{4}}} = ∑_{v∈S_{n}} \frac{∂^{3}F_{ω}}{∂t_{u_{2}}∂t_{u_{3}}∂t_{v}} \frac{∂^{3}F_{ω}}{∂t_{vw_{0}}∂t_{u_{1}}∂t_{u_{4}}} ,

for any u_{1}, u_{2}, u_{3}, u_{4} ∈ S_{n}. ¿From [KM, Proposition 4.4], the potential F_{ω} satisfies

F_{ω}(t) = F_{ω−t^{(2)}}(t − t^{(2)}),

where t^{(2)} = ∑_{l(v)=1} t_{v}Ω_{v}. Hence, we may assume ω = 0. The potential F is

decomposed as a sum of the classical part f_{cl} and the quantum correction f, where

f_{cl} = \frac{1}{6}t_{id} (∑_{u,v} δ_{u,vw_{0}}t_{u}t_{v}) ,

and

f = ∑_{β} ∑_{m=Σm_{v}≥3} \frac{⟨I_{0,m,β}^{F l}⟩( ⨂_{l(v)≥1} Ω_{v}^{⊗m_{v}})}{∏_{l(v)≥1} m_{v}!} ∏_{l(v)≥1} t^{m_{v}}.

¿From the axioms of the Gromov-Witten invariants ([KM,(2.2.4)]), we have

⟨I_{0,m,β}^{F l}⟩( ⨂_{l(v)≥1} Ω_{v}^{⊗m_{v}}) = ⟨I_{0,m−Σ_{l(u)=1}m_{u},β}^{F l}⟩( ⨂_{l(v)>1} Ω_{v}^{⊗m_{v}}) ∏_{l(u)=1} (∫_{β} Ω_{u}) .

Hence, the quantum correction f is expressed as

f = ∑_{m_{v},b_{u}} N((m_{v})_{l(v)>1} | (b_{u})_{l(u)=1}) exp( ∑_{l(u)=1}(b_{u}t_{u})) ∏_{l(v)>1} \frac{t_{v}^{m_{v}}}{m_{v}!},

where

N((m_{v}) | (b_{u})) = ⟨I_{0,Σm_{v},Σb_{u}X_{u}}^{F l}⟩ ( ⨂_{l(v)>1} Ω_{v}^{⊗m_{v}} ) .

If n = 3, the WDVV equations with the initial condition

N(0, 0, 1 | 1, 0) = N(0, 0, 1 | 0, 1) = 1

or

N(2, 0, 0 | 1, 0) = N(0, 2, 0 | 0, 1) = 1

determine all the coefficients N(λ, μ, ν | a, b) uniquely. In fact, Di Francesco and Itzykson [FI] gave the coefficients N(λ, μ, ν | a, b) for a + b ≤ 10.

Let x_{1} = Ω_{(1,2)} and x_{i} = Ω_{(i,i+1)} − Ω_{(i−1,i)} for 2 ≤ i ≤ n − 1, where (i, j) is the transposition that interchanges i and j. Since the classical cohomology ring is generated by x_{1}, …, x_{n−1}, the quantum cohomology ring QH_{t}^{∗}(F l_{n}) is also generated by x_{1}, …, x_{n−1} in the neighborhood of the origin t = 0. Moreover, we can choose a neighborhood of the origin on which QH_{t}^{∗}(F l_{n}) is a complete intersection ring. Then, let I ⊂ C[x_{1}, …, x_{n−1}] be the defining ideal of the quantum cohomology ring QH_{t}^{∗}(F l_{n}). The Schubert class Ω_{v} is expressed by the Schubert polynomial 𝔖_{v}(x_{1}, …, x_{n−1}) in the classical cohomology ring. However, in the quantum cohomology ring, the class corresponding to the Schubert polynomial 𝔖_{v} is no longer the Schubert class Ω_{v}. Hence, the polynomial 𝔖̃_{v}^{t}(x_{1}, …, x_{n−1}) expressing Ω_{v} gives a deformation of the Schubert polynomial. We call it a big quantum Schubert polynomial. We identify the residue pairing defined by I with the intersection form on the cohomology ring, so the big quantum Schubert polynomials are obtained by orthogonalization the basis consisting of the classical Schubert polynomials.

It is difficult to describe the defining ideal I of the big quantum cohomology ring for generic t, so the big quantum Schubert polynomials are complicated in general. However, in the case where parameters t_{v} = 0 for all permutations v ∈ S_{n} such that l(v) > 1, the defining relations of the quantum cohomology ring are known by the results of A. Givental and B. Kim [GK] and I. Ciocan-Fontanine [C]. We call it the small quantum cohomology ring. The structure constants of the small quantum cohomology ring are given by

\frac{∂^{3}F}{∂t_{v_{1}}∂t_{v_{2}}∂t_{v_{3}}} (t^{(2)}) =

= ∑_{β} ∑_{m_{u}≥0} \frac{⟨I_{0,Σm_{u}+3,β}^{F l}⟩(Ω_{v_{1}} ⊗ Ω_{v_{2}} ⊗ Ω_{v_{3}} ⊗ ( ⨂_{l(u)=1} Ω_{u}^{⊗m_{u}}))}{∏_{l(u)=1} m_{u}!} ∏_{l(u)=1} t_{u}^{m_{u}}

= ∑_{β}⟨I_{0,3,β}^{F l}⟩(Ω_{v_{1}} ⊗ Ω_{v_{2}} ⊗ Ω_{v_{2}})e^{Σb_{i}t_{(i,i+1)}},

where the sum runs over

β = b_{1}X_{(1,2)} + ⋯ + b_{n−1}X_{(n−1,n)}

with b_{i} ∈ Z_{≥0}. Hence the small quantum cohomology ring is determined by the invariants ⟨I_{0,3,β}^{F l}⟩. For a monomial x_{i_{1}} ∗ ⋯ ∗ x_{i_{m}}, the m-point correlation function determined by the small quantum cohomology ring (the so-called small quantum cohomology ring correlation function) is defined to be

⟨x_{i_{1}} ⋯ x_{i_{m}}⟩ = ∫_{F l_{n}} x_{i_{1}} ∗ ⋯ ∗ x_{i_{m}}.

If m ≥ 4, the small quantum cohomology ring correlation function can be expressed as follows:

⟨x_{i_{1}} ⋯ x_{i_{m}}⟩ =

∑_{β=β_{1}+···+β_{m−3}} ∑_{v_{1},...,v_{m−3}} e^{− ∫_{β} ω}⟨I_{0,3,β_{1}}⟩(x_{i_{1}} ⊗ x_{i_{2}} ⊗ Ω_{v_{1}})⟨I_{0,3,β_{2}}⟩(Ω_{v_{1}w_{0}} ⊗ x_{i_{3}} ⊗ Ω_{v_{2}})

⋯ ⟨I_{0,3,β_{m−2}}⟩(Ω_{v_{m−3}w_{0}} ⊗ x_{i_{m−1}} ⊗ x_{i_{m}}).

Hence, if m ≥ 4, the Gromov-Witten invariants ⟨I_{0,m,β}⟩(x_{i_{1}} ⊗ ⋯ ⊗ x_{i_{m}}) do not appear as coefficients of small quantum cohomology ring correlation functions.

Let us explain briefly the main results obtained in our paper. Follow to A. Givental and B. Kim [GK], and I. Ciocan–Fontanine [C], we define the quantum elementary symmetric polynomials ẽ_{1}, …, ẽ_{n} by the formula

det ( \begin{matrix}x_{1} + t & q_{1} & 0 & … & … & … & 0 \\ −1 & x_{2} + t & q_{2} & 0 & … & … & 0 \\ 0 & −1 & x_{3} + t & q_{3} & 0 & … & 0 \\ ⋮ & ... & ... & ... & ... & ... & ⋮ \\ 0 & … & 0 & −1 & x_{n−2} + t & q_{n−2} & 0 \\ 0 & … & … & 0 & −1 & x_{n−1} + t & q_{n−1} \\ 0 & … & … & … & 0 & −1 & x_{n} + t \end{matrix} )

= t^{n} + ẽ_{1}t^{n−1} + ẽ_{2}t^{n−2} + ⋯ + ẽ_{n},

where q_{i} = e^{t_{(i,i+1)}}. The defining ideal Ĩ of the small quantum cohomology ring is generated by the quantum elementary symmetric polynomials, namely

QH^{∗}(F l_{n}, Z) := QH^{∗}(F l_{n}) = Z[x_{1}, …, x_{n}; q_{1}, …, q_{n−1}]/(ẽ_{1}, …, ẽ_{n}). (1)

In the classical case q_{1} = ⋯ = q_{n−1} = 0, on the quotient ring

𝔄 := Z[x_{1}, …, x_{n}]/(e_{1}(x), …, e_{n}(x)) ≃ H^{∗}(F l_{n}, Z)

there exists a natural pairing ⟨f, g⟩ = η(∂_{w_{0}}(fg)) which comes from the intersection pairing in the homology group H_{∗}(F l_{n}, Z) of the flag variety. We can interpret the pairing ⟨, ⟩ as the Grothendieck residue pairing with respect to the ideal I (see Subsection 2.5):

⟨f, g⟩ = Res_{I}(fg),

where I = Ĩ|_{q=0}.

Our first observation is that a natural residue pairing (we call it the quantum residue pairing)

⟨f.g⟩_{Q} = Res_{I}(fg)

on the quotient ring \overline{𝔄} := Z[x_{1}, …, x_{n}]/Ĩ corresponds to the intersection pairing in quantum cohomology QH^{∗}(F l_{n}, Z) under a natural isomorphism (??).

It is well-known (e.g. [LS2], [M]) that the classical Schubert polynomials form an orthonormal basis (with respect to the pairing ⟨, ⟩) in the cohomology ring of flag manifold and also give a linear basis in the quantum cohomology ring QH^{∗}(F l_{n}, Z), [GK], [MS]. However, the classical Schubert polynomials do not orthogonal with respect to the quantum pairing any more. Thus, it is natural to ask: what kind of polynomials one can obtain applying the Gram–Schmidt orthogonalization to the classical Schubert polynomials with respect to the quantum pairing ⟨, ⟩_{Q}? Omiting some details with ordering (see Definition 5), the answer is: quantum Schubert polynomials.

Our second observation is: to work with the equivariant quantum cohomology algebra ([GK], [K2]) is more convenient then with quantum cohomology ring itself. The main reason is that one can find Lascoux–Schützenberger’s type representative for any equivariant quantum cohomology class. In other words, each quantum double Schubert polynomial 𝔖̃_{w}(x, y) can be obtained from the top one by using the divided difference operators acting on the y variables.

Theorem-Definition A Let x = (x_{1}, …, x_{n}), y = (y_{1}, …, y_{n}) be two sets of variables, and

𝔖̃_{w_{0}}^{(q)}(x, y) := ∏_{i=1}^{n−1} Δ_{i}(y_{n−i} | x_{1}, …, x_{i}),

where Δ_{k}(t | x_{1}, …, x_{k}) := ∑_{j=0}^{k} t^{k−j}e_{j}(x_{1}, …, x_{k} | q_{1}, …, q_{k−1}) is the generat-

ing function for the quantum elementary symmetric functions in x_{1}, …, x_{k}. Then 𝔖̃_{w}^{(q)}(x, y) = ∂_{ww_{0}}^{(y)}𝔖̃_{w_{0}}^{(q)}(x, y). We define the quantum Schubert polynomials 𝔖̃_{w}^{(q)}(x) as Gram–Schmidt’s

orthogonalization of the set of lexicographically ordered monomials {x^{I} | I ⊂ (n − 1, n − 2, …, 1, 0)} with respect to the quantum pairing ⟨, ⟩_{Q}, see Definition 5. One of our main results is the quantum analog of Cauchy’s identity for (classical) Schubert polynomials, [M], (5.10).

Theorem B (Quantum Cauchy’s identity)

∑_{w∈S_{n}} 𝔖̃_{w}^{(q)}(x)𝔖_{ww_{0}}(y) = 𝔖̃_{w_{0}}^{(q)}(x, y).  (2)

We give a geometric proof of Theorem B in Section 7 using the arguments due to I. Ciocan-Fontanine [C]; more particularly, we reduce directly a proof of Theorem B to that of the following geometric statement:

Lemma Let I ⊂ δ = (n − 1, n − 2, …, 1, 0) and w ∈ S_{n} be a permutation, then

⟨ẽ_{I}(x), 𝔖̃_{w}(x)⟩_{Q} = ⟨e_{I}(x), 𝔖_{w}(x)⟩,  (3)

where e_{I}(x) := ∏_{k=1}^{n−1} e_{i_{k}}(x_{1}, …, x_{n−k})

(resp. ẽ_{I}(x) := ∏_{k=1}^{n−1} ẽ_{i_{k}}(x_{1}, …, x_{n−k} | q_{1}, …, q_{n−k−1}))

is the elementary polynomial (resp. quantum elementary polynomial), see Section 5.2.

It is the formula (3) that we prove in Section 7 using the geometrical arguments from [C] and [K2]. By product, it follows from our proof that quantum Schubert polynomials _{ 𝔖̂}_{w}(x) defined geometrically (see Section 6) coincide with those defined algebraically (see Definition 5):

_{𝔖̂}_{w}(x) ≡ 𝔖̃_{w^{−1}}(x) (mod Ĩ).

It is interesting to note, that the intersection numbers ⟨e_{I}(x), 𝔖_{w}(x)⟩ (which are nonnegative!) are precisely the coefficients of corresponding Schubert polynomial:

𝔖_{w}(x) = ∑_{I⊂δ}⟨e_{I}(x), 𝔖_{w}(x)⟩x^{δ−I}.

The quantum Cauchy formula (2) plays the important role in our approach to the quantum Schubert polynomials. As a direct consequence of (2), we obtain the Lascoux–Schützenberger type formula for quantum Schubert polynomials (cf. Theorem-Definition A).

Theorem C Let 𝔖̃_{w_{0}}(x, y) be as in Theorem-Definition A, then

𝔖̃_{w}(x) = ∂_{ww_{0}}^{(y)}𝔖̃_{w_{0}}(x, y)|_{y=0}.

In Section 5 we introduce a quantization map

P_{n} → \overline{P}_{n}, f ↦ f̃.

The quantization is a linear map which preserves the pairings, i.e.,

⟨f̃, g̃⟩_{Q} = ⟨f, g⟩, f, g ∈ P_{n}.

Using the quantum Cauchy formula (2), we prove that quantum double Schubert polynomials are the quantization of classical ones. Another class of polynomials having a nice quantization is the set of elementary polynomials

e_{I}(x) := ∏_{k=1}^{n−1} e_{i_{k}}(x_{1}, … x_{n−k}), I = (i_{1}, …, i_{n−1}) ⊂ δ.

It follows from Theorem B that quantization ẽ_{I}(x) of elementary polynomial e_{I}(x) is given by

ẽ_{I}(x) = ∏_{k=1}^{n−1} e_{i_{k}}(x_{1}, …, x_{n−k} | q_{1}, …, q_{n−k−1}).

More generally, we make a conjecture (”quantum Schur functions ”) that quantization of the flagged Schur function (see [M], (3.1), (4.9) and (6.16))

s_{λ/μ}(X_{1}, …, X_{n}) = det (h_{λ_{i}−μ_{j}−i+j}(X_{i}))_{1≤i,j≤n}

is given by

s̃_{λ/μ}(X_{1}, …, X_{n}) = det (h̃_{λ_{i}−μ_{j}−i+j}(X_{i}))_{1≤i,j≤n} ,

where h̃_{k}(X) is the quantum complete homogeneous symmetric function of degree k, and X_{1} ⊂ ⋯ ⊂ X_{n} are the flagged sets of variables (see Section 5).

In Section 5.2 we consider a problem how to quantize monomials. It seems to be difficult to find an explicit determinantal formula for a quantum monomial x̃^{I}, i.e., to find a quantum analog of the Billey-Jockusch-Stanley formula for Schubert polynomials in terms of compatible sequences [BJS]. We prove the following formulae for quantum monomials

x̃^{I} = ∑_{w∈S_{n}} η(∂_{w}x^{I})𝔖̃_{w}(x), I ⊂ δ,

𝔖̃_{w_{0}}(x, y) = ∑_{I⊂δ} x̃^{I}e_{δ−I}(y).

In section 8.1 we give a proof of the higher genus analog of the Vafa– Intriligator type formula for the flag manifold.

In Section 8.3 we study a problem how to compute the quantum residues. This is important for computation of small quantum cohomology ring correlation functions (or correlation functions, for short) and the Gromov–Witten invariants, see Introduction and Theorem 11. We introduce the generating function

Ψ(t) = ⟨ ∏_{i=1}^{n−1} \frac{t_{i}}{t_{i} − x_{i}} ⟩

for quantum residues and give a characterization of this function as the unique solution to some system of differential equations, see Proposition 14. In Appendix B we calculate the generating function Ψ(t) for the case n = 3 explicitly.

In Section 9 we introduce the extended Ehresman–Bruhat order and give a sketch of a proof of equivariant quantum Pieri rule. Details will appear elsewhere.

In Appendix A one can find a list of explicit expressions for the quantum double Schubert polynomials for the symmetric group S_{4}.

We would like to mention, that in the recent preprint “Quantum Schubert polynomials” by S. Fomin, S. Gelfand and A. Postnikov, [FGP], developed a different approach to the theory of quantum Schubert polynomials, based on the remarkable family of commuting operators X_{i} ([FGP], (3.2)). Among main results, obtained by S. Fomin, S. Gelfand and A. Postnikov, are definitions, orthogonality, quantum Monk’s formula and other properties of quantum Schubert polynomials; definition of quantization map and quantum multiplication.

Besides some overlap with the preprint of S. Fomin, S. Gelfand and A. Postnikov, our works were done independently and based on the different approaches, which allow to obtain the mutually complementary results.

2 Classical Schubert polynomials.

In this section we give a brief review of the theory of Schubert polynomials created by A. Lascoux and M.-P. Schützenberger. In exposition we follow to the I. Macdonald book [M1] where proofs and more details can be found.

2.1 Divided differences.

Let x_{1}, …, x_{n}, … be independent variables, and let

P_{n} := Z[x_{1}, …, x_{n}]

for each n ≥ 1, and

P_{∞} := Z[x_{1}, x_{2}, …] = ⋃_{n=1}^{∞} P_{n}.  (4)

Let us denote by Λ_{n} := Z[x_{1}, …, x_{n}]^{S_{n}} ⊂ P_{n} the ring of symmetric polynomi-

als in x_{1}, …, x_{n}, and by H_{n} := { ∑_{I=(i_{1},...,i_{n})} a_{I}x^{I} | a_{I} ∈ Z, 0 ≤ i_{k} ≤ n − k, ∀k}

the additive subgroup of P_{n} spanned by all monomials x^{I} := x_{1}^{i_{1}}x_{2}^{i_{2}} … x_{n}^{i_{n}} with I ⊂ δ := δ_{n} = (n − 1, n − 2, …, 1, 0). For 1 ≤ i ≤ n − 1 let us define a linear operator ∂_{i} acting on P_{n}

(∂_{i}f)(x) = \frac{f(x_{1}, …, x_{i}, x_{i+1}, …, x_{n}) − f(x_{1}, …, x_{i+1}, x_{i}, …, x_{n})}{x_{i} − x_{i+1}}. (5)

Divided difference operators ∂_{i} satisfy the following relations

∂_{i}^{2} = 0,
∂_{i}∂_{j} = ∂_{j}∂_{i}, if | i − j |> 1,  (6)
∂_{i}∂_{i+1}∂_{i} = ∂_{i+1}∂_{i}∂_{i+1},

and the Leibnitz rule

∂_{i}(fg) = ∂_{i}(f)g + s_{i}(f)∂_{i}(g).  (7)

It follows from (??) that ∂_{i} is a Λ_{n}-linear operator.

For any permutation w ∈ S_{n}, let us denote by R(w) the set of reduced words for w, i.e. sequences (a_{1}, …, a_{p}) such that w = s_{a_{1}} ⋯ s_{a_{p}}, where p = l(w) is the length of permutation w ∈ S_{n}, and s_{i} = (i, i+1) is the simple transposition that interchanges i and i + 1.

For any sequence a = (a_{1}, …, a_{p}) of positive integers, we define

∂_{a} = ∂_{a_{1}} ⋯ ∂_{a_{p}}.

Proposition 1 ([M1], (2.5),(2.6))

• If a, b ∈ R(w), then ∂_{a} = ∂_{b}.

• If a is not reduced, then ∂_{a} = 0.

¿From Proposition ?? it follows that an operator

∂_{w} = ∂_{a}

is well-defined, where a is any reduced word for w. By (??), the operators ∂_{w}, w ∈ S_{n}, are Λ_{n} linear, i.e. if f ∈ Λ_{n}, then

∂_{w}(fg) = f∂_{w}(g).

2.2 Schubert polynomials.

Let δ = δ_{n} = (n − 1, n − 2, …, 1, 0), so that x^{δ} = x_{1}^{n−1}x_{2}^{n−2} … x_{n−1}.

Definition 1 (Lascoux–Schützenberger [LS1]). For each permutation w ∈ S_{n} the Schubert polynomial 𝔖_{w} is defined to be

𝔖_{w}(x) = ∂_{w^{−1}w_{0}}(x^{δ}),

where w_{0} is the longest element of S_{n}.

Proposition 2 ([M1], (4.2),(4.5),(4.11),(4.15)).

• Let v, w ∈ S_{n}. Then

∂_{v}𝔖_{w} = \{ \begin{matrix}𝔖_{wv^{−1}}, & if l(wv^{−1}) = l(w) − l(v), \\ 0, & otherwise. \end{matrix}

• (Stability). Let m > n and let i : S_{n} ↪ S_{m} to be the natural embedding. Then

𝔖_{w} = 𝔖_{i(w)}.

• The Schubert polynomials 𝔖_{w}, w ∈ S_{n} form a Z–basis of H_{n}.

• (Monk’s formula). Let f = ∑_{i=1}^{n} α_{i}x_{i}, w ∈ S_{n}. Then

f𝔖_{w} = ∑(α_{i} − α_{j})𝔖_{wt_{ij}},
∂_{w}(fg) = w(f)∂_{w}g + ∑(α_{i} − α_{j})∂_{wt_{ij}}g,

where t_{ij} is the transposition that interchanges i and j, and both sums are over all pairs i < j such that l(wt_{ij}) = l(w) + 1.

2.3 Scalar product.

Let us define a scalar product on P_{n} with values in Λ_{n}, by the rule

⟨f, g⟩ = ∂_{w_{0}}(fg), f, g ∈ P_{n},  (8)

where w_{0} is the longest element of S_{n}.

The scalar product ⟨, ⟩ defines a non-degenerate pairing ⟨, ⟩_{0} on the quotient ring P_{n}/I_{n} ≅ H^{∗}(F l_{n}, Z), where I_{n} is the ideal in P_{n} generated by the elementary symmetric polynomials e_{1}(x), …, e_{n}(x).

Proposition 3 ([M1], (5.3),(5.4),(5.6),(4.13),(5.10)).

• If f ∈ Λ_{n}, then ⟨fh, g⟩ = f⟨h, g⟩;

• If f, g ∈ P_{n}, w ∈ S_{n}, then ⟨∂_{w}f, g⟩ = ⟨f, ∂_{w^{−1}}g⟩;

• (Orthogonality) If l(u) + l(v) = l(w_{0}), then ⟨𝔖_{u}, 𝔖_{v}⟩ = \{ \begin{matrix}1, & if u = w_{0}v, \\ 0, & otherwise. \end{matrix}

• The Schubert polynomials 𝔖_{w}, w ∈ S_{n}, form a Λ_{n}–basis of P_{n};

• The Schubert polynomials 𝔖_{w}, w ∈ S^{(n)}, form a Z–basis of P_{n}, where for each n ≥ 1, S^{(n)} is the set of all permutations w such that the code w has length ≤ n;

• (Cauchy’s formula)

∑_{w∈S_{n}} 𝔖_{w}(x)𝔖_{ww_{0}}(y) = ∏_{i+j≤n}(x_{i} + y_{j}).

Proposition 4 Schubert polynomials are uniquely characterized by the following properties

1. (Orthogonality) ⟨𝔖_{u}, 𝔖_{v}⟩_{0} = \{ \begin{matrix}1, & if u = w_{0}v, \\ 0, & otherwise. \end{matrix}

2. Let w be a permutation in S_{n} and c(w) = (c_{1}, c_{2}, …, c_{n}) its code, then

𝔖_{w}(x) = x^{c(w)} + ∑ α_{I}x^{I},

where I ⊂ δ, α_{I} > 0 and I lexicographically smaller then c(w).

Remark 1 1) (Definition of the code, [M1], p.9). For a permutation w ∈ S_{n}, we define

c_{i} = ♯{j | i < j, w(i) > w(j)}.

The sequence c(w) = (c_{1}, c_{2}, …, c_{n}) is called the code of w.

2) Schubert polynomials are obtained as Gram-Schmidt’s orthogonalization of the set of monomials {x^{I}}_{I⊂δ} ordered lexicographically.

2.4 Double Schubert polynomials.

Let x = (x_{1}, …, x_{n}), y = (y_{1}, …, y_{n}) be two sets of independent variables, and

𝔖_{w_{0}}(x, y) := ∏_{i+j≤n}(x_{i} + y_{j}).

Definition 2 (Lascoux–Schützenberger [LS2]). For each permutation w ∈ S_{n}, the double Schubert polynomial 𝔖_{w}(x, y) is defined to be

𝔖_{w}(x, y) = ∂_{w^{−1}w_{0}}^{(x)}𝔖_{w_{0}}(x, y),

where divided difference operator ∂_{w^{−1}w_{0}}^{(x)} acts on the x variables.

Proposition 5 ([M1], (6.3),(6.8)).

• 𝔖_{w}(x, y) = ∑_{u} 𝔖_{u}(x)𝔖_{uw^{−1}}(y), summed over all u ∈ S_{n}, such that

l(u) + l(uw^{−1}) = l(w);

• (Interpolation formula). For all f ∈ Z[x_{1}, …, x_{n}] we have

f(x) = ∑_{w} 𝔖_{w}(x, −y)∂_{w}^{(y)}f(y)

summed over all permutations w ∈ S^{(n)}.

Double Schubert polynomials appear in algebra and geometry as cohomology classes related to degeneracy loci of flagged vector bundles. If h : E → F is a map of rank n vector bundles on a smooth variety X,

E_{1} ⊂ E_{2} ⊂ ⋯ ⊂ E_{n} = E, F := F_{n} → F_{n−1} → ⋯ → F_{1}

are flags of subbundles and quotient bundles, then there is a degeneracy locus Ω_{w}(h) for each permutation w in the symmetric group S_{n}, described by the conditions

Ω_{w}(h) = {x ∈ X | rank(E_{p}(x) → F_{q}(x)) ≤ #{i ≤ q, w_{i} ≤ p}, ∀p, q}.

For generic h, Ω_{w}(h) is irreducible, codim Ω_{w}(h) = l(w), and the class [Ω_{w}(h)] of this locus in the Chow ring of X is equal to the double Schubert polynomial 𝔖_{w_{0}w}(x, −y), where

x_{i} = c_{1}(ker(F_{i} → F_{i−1})),
y_{i} = c_{1}(E_{i}/E_{i−1}), 1 ≤ i ≤ n.

It is well-known [F] that the Chow ring of flag variety F l_{n} admits the following description

CH^{∗}(F l_{n}) ≅ Z[x_{1}, …, x_{n}, y_{1}, …, y_{n}]/J,

where J is the ideal generated by

e_{i}(x_{1}, …, x_{n}) − e_{i}(y_{1}, …, y_{n}), 1 ≤ i ≤ n,

and e_{i}(x) is the i-th elementary symmetric function in the variables x_{1}, …, x_{n}.

• ([LS2], [KV]) The ring Z[x_{1}, …, x_{n}, y_{1}, …, y_{n}]/J is a free module of dimension n! over the ring R, with basis either 𝔖_{w}(x), or 𝔖_{w}(x, y), w ∈ S_{n}, where

R := \frac{Z[x_{1}, …, x_{n}] ⊗ Sym[y_{1}, …, y_{n}]}{J}.

2.5 Residue pairing.

Let I be an ideal in P̄_{n} = R[x_{1}, …, x_{n}], R ⊂ C, generated by a regular system of parameters φ_{1}, …, φ_{n}, and 𝔄 := P̄_{n}/I.

Proposition 6 ([GH], [EL]).

• dim_{R} 𝔄 < ∞.

• H := det (\frac{∂φ_{i}}{∂x_{j}} ) ∉ I.

Let d_{0} := deg H, where we assume that deg x_{i} = 1 for all 1 ≤ i ≤ n.

Proposition 7 ([EL])

• If f ∈ P̄_{n} and deg f = d_{0}, then there exists a non zero α ∈ R such that

f ≡ \frac{α}{n!}H (mod I).

• If f ∈ P̄_{n}, f ≠ 0, and deg f > d_{0}, then there exists g ∈ P̄_{n} such that deg g ≤ d_{0} and g ≡ f (mod I).

Definition 3 (Grothendieck residue with respect to the ideal I). Let f ∈ P̄_{n} and deg f < d_{0}, then we define

Res_{I}(f) = 0.

If deg f = d_{0}, then f ≡ \frac{α}{n!}H (mod I) and we define Res_{I}(f) := α.

Finally, if deg f > d_{0}, then choose g ∈ P̄_{n} such that g ≡ f (mod I) and deg g ≤ d_{0}, and define

Res_{I}(f) := Res_{I}(g).

We will use also notation ⟨f⟩_{I} instead of Res_{I}(f).

Finally, let us define a residue pairing ⟨, ⟩_{I} on \overline{P}_{n} using the Grothendieck residue

⟨f, g⟩_{I} = Res_{I}(f, g), f, g ∈ \overline{P}_{n}.

Proposition 8 ([GH]).

• If f ∈ I, then Res_{I}(f) = 0.

• The residue pairing ⟨, ⟩_{I} induces a non-degenerate pairing on 𝔄 = \overline{P} /I.

We will use this general construction of residue pairing in the following two cases:

i) R = Z, I_{n} ⊂ P_{n} is an ideal generated by elementary symmetric polynomials e_{1}(x), …, e_{n}(x). It is well-known that if F l_{n} := SL(n)/B is the flag variety of type A_{n−1}, then

H^{∗}(F l_{n}, Z) ≃ P_{n}/I_{n},

and residue pairing ⟨, ⟩ on P_{n}/I_{n} coincides with the scalar product on P_{n}/I_{n} induced by (??).

ii) R = Z[q_{1}, …, q_{n−1}], Ĩ_{n} ⊂ \overline{P}_{n} is an ideal generated by the quantum elementary symmetric functions ẽ_{1}(x), …, ẽ_{n}(x). It is a result of A. Grivental and B. Kim, and I. Ciocan–Fontanine, that

QH^{∗}(F l_{n}) ≃ \overline{P}_{n}/Ĩ_{n},

and the residue pairing defined by Ĩ_{n} may be naturally identified with the intersection form on the quantum cohomology ring. We will call this residue pairing as quantum pairing on \overline{P}_{n}/Ĩ_{n} and denote it by ⟨, ⟩_{Q}.

3 Quantum double Schubert polynomials.

Quantum double Schubert polynomials are closely related with the equivariant quantum cohomology. Let us remind the result of A. Givental and B. Kim [GK] (see also [K2]) on the structure of the equivariant quantum cohomology algebra of the flag variety F l_{n}:

QH_{U_{n}}^{∗}(F l_{n}) ≅ Z[x_{1}, …, x_{n}, y_{1}, …, y_{n}, q_{1}, …, q_{n−1}]/J̃,

where the ideal J̃ generated by

e_{i}(x_{1}, …, x_{n} | q_{1}, …, q_{n−1}) − e_{i}(y_{1}, …, y_{n}), 1 ≤ i ≤ n.

In classical case q = 0, the double Schubert polynomials 𝔖_{w}(x, y) represent the equivariant cohomology classes [F]. Quantum double Schubert polynomials have to play the similar role for the quantum equivariant cohomology ring. Let us define at first the “top” quantum double Schubert polynomial 𝔖̃_{w_{0}}(x, y).

Let x = (x_{1}, …, x_{n}), y = (y_{1}, …, y_{n}) be two sets of variables, put

𝔖̃_{w_{0}}(x, y) := 𝔖̃_{w_{0}}^{(q)}(x, y) = ∏_{i=1}^{n−1} Δ_{i}(y_{n−i} | x_{1}, …, x_{i}),

where Δ_{k}(t | x_{1}, …, x_{k}) := ∑_{j=0}^{k} t^{k−j}e_{j}(x_{1}, …, x_{k} | q_{1}, …, q_{k−1}) is the generat-

ing function for the quantum elementary symmetric polynomials in x_{1}, …, x_{k},

i.e. Δ_{k}(t|x) := ∑_{i=1}^{k} e_{i}(x|q)t^{i} =

det ( \begin{matrix}x_{1} + t & q_{1} & 0 & … & … & … & 0 \\ −1 & x_{2} + t & q_{2} & 0 & … & … & 0 \\ 0 & −1 & x_{3} + t & q_{3} & 0 & … & 0 \\ ⋮ & ... & ... & ... & ... & ... & ⋮ \\ 0 & … & 0 & −1 & x_{k−2} + t & q_{k−2} & 0 \\ 0 & … & … & 0 & −1 & x_{k−1} + t & q_{k−1} \\ 0 & … & … & … & 0 & −1 & x_{k} + t \end{matrix} ) . (9)

Definition 4 For each permutation w ∈ S_{n}, the quantum double Schubert polynomial 𝔖̃_{w}(x, y) is defined to be

𝔖̃_{w}(x, y) = ∂_{ww_{0}}^{(y)}𝔖̃_{w_{0}}(x, y),

where divided difference operator ∂_{ww_{0}}^{(y)} acts on the y variables.

Remark 2 i) In the ”classical limit” q_{1} = ⋯ = q_{n−1} = 0,

𝔖̃_{w}(x, y)|_{q=0} = ∂_{ww_{0}}^{(y)}𝔖_{w_{0}}(x, y) = 𝔖_{w^{−1}}(y, x) = 𝔖_{w}(x, y),

i.e. 𝔖̃_{w}(x, y)|_{q=0} = 𝔖_{w}(x, y).

ii) (Stability) Let m > n and let i : S_{n} ↪ S_{m} be the embedding. Then

𝔖̃_{w}(x, y) = 𝔖̃_{i(w)}(x, y).

iii) One can check that the ring

Z[x_{1}, …, x_{n}, y_{1}, …, y_{n}, q_{1}, …, q_{n−1}]/J̃

is a free module of dimension n! over the quotient ring R̃ with basis either 𝔖̃_{w}(x) or 𝔖̃_{w}(x, y), w ∈ S_{n}, where

R̃ := \frac{Z[x_{1}, …, x_{n}, q_{1}, …, q_{n−1}] ⊗ Sym[y_{1}, …, y_{n}]}{J̃}.

Example. Quantum double Schubert polynomials for S_{3}:

𝔖̃_{s_{1}s_{2}s_{1}}(x, y) = (x_{1} + y_{2})(x_{1} + y_{1})(x_{2} + y_{1}) + q_{1}(x_{1} + y_{2}),
𝔖̃_{s_{2}s_{1}}(x, y) = (x_{1} + y_{1})(x_{1} + y_{2}) − q_{1},
𝔖̃_{s_{1}s_{2}}(x, y) = (x_{1} + y_{1})(x_{2} + y_{1}) + q_{1},
𝔖̃_{s_{1}}(x, y) = x_{1} + y_{1},
𝔖̃_{s_{2}}(x, y) = x_{1} + x_{2} + y_{1} + y_{2},
𝔖̃_{id}(x, y) = 1.

For the list of the quantum double Schubert polynomials corresponding to the symmetric group S_{4}, see appendix A.

Theorem 1 Let z = (z_{1}, …, z_{n}) be a third set of variables. Then

⟨𝔖̃_{w_{0}}(x, y), 𝔖̃_{w_{0}}(x, z)⟩_{Q}^{(x)} = C(y, z),  (10)

where the upper index x means that the quantum pairing is taken in the x variables, and

C(x, y) = ∑_{w∈S_{n}} 𝔖_{w}(x)𝔖_{w_{0}w}(y)

is the ”canonical” element in the tensor product H^{∗}(F l_{n}) ⊗ H^{∗}(F l_{n}).

Theorem 1 plays the important role in our approach to the quantum Schubert polynomials. We will give a proof later, and now let us consider some applications of the formula (10).

4 Quantum Schubert polynomials.

4.1 Definition.

Let us remind the result of A. Givental and B. Kim, and I. Ciocan-Fontanine on the structure of the small quantum cohomology ring of flag variety F l_{n}

QH^{∗}(F l_{n}) ≅ Z[x_{1}, …, x_{n}, q_{1}, …, q_{n−1}]/Ĩ,

where the ideal Ĩ is generated by the quantum elementary symmetric polynomials ẽ_{i}(x) := e_{i}(x_{1}, …, x_{n}|q_{1}, …, q_{n−1}), 1 ≤ i ≤ n with generating function Δ_{n}(t|x), see (9).

We define a pairing on the ring of polynomials Z[x; q] and the quantum cohomology ring QH^{∗}(F l_{n}) ≃ Z[x; q]/Ĩ using the Grothendieck residue

⟨f, g⟩_{Q} = Res_{I}(fg), f, g ∈ Z[x_{1}, …, x_{n}, q_{1}, …, q_{n−1}].

Then 1) ⟨f, g⟩_{Q} = 0 if f ∈ Ĩ;

2) ⟨f, g⟩_{Q} defines a nondegenerate pairing in QH^{∗}(F l_{n}).

Definition 5 Define the quantum Schubert polynomials 𝔖̃_{w} := 𝔖̃_{w}(x) as Gram–Schmidt’s orthogonalization of the set of lexicographically ordered monomials {x^{I} | I ⊂ δ} with respect to the quantum residue pairing ⟨f, g⟩_{Q}:

1) ⟨𝔖̃_{u}, 𝔖̃_{v}⟩_{Q} = ⟨𝔖_{u}, 𝔖_{v}⟩ = \{ \begin{matrix}1, & if v = w_{0}u \\ 0, & otherwise \end{matrix}

2) 𝔖̃_{w}(x) = x^{c(w)} + ∑_{I<c(w)} a_{I}(q)x^{I}, where a_{I}(q) ∈ Z[q_{1}, …, q_{n−1}] and

I < c(w) means the lexicographic order.

Here c(w) is the code of a permutation w ∈ S_{n}, [M1], p.9.

Remark 3 This definition is the analogue of the characterization of Schubert polynomials from Proposition 4. Example. For the symmetric group S_{3}, we have

⟨x_{1}^{2}x_{2}, x_{1}^{2}⟩_{Q} = q_{1}, ⟨x_{1}^{2}x_{2}, x_{1}x_{2}⟩_{Q} = −2q_{1}.

Consequently,

𝔖̃_{1} = x_{1}, 𝔖̃_{2} = x_{1} + x_{2}, 𝔖̃_{12} = x_{1}x_{2} + q_{1}, 𝔖̃_{21} = x_{1}^{2} − q_{1}, 𝔖̃_{121} = x_{1}^{2}x_{2} + q_{1}x_{1}.

Let us remark that in our example (n = 3) 𝔖̃_{121} = 𝔖̃_{w_{0}^{(3)}}(x) = ẽ_{1}(x_{1})ẽ_{2}(x_{1}, x_{2}). More generally, we have

Proposition 9 Let w_{0} ∈ S_{n} be the longest element. Then

𝔖̃_{w_{0}}(x) = ẽ_{1}(x_{1})ẽ_{2}(x_{1}, x_{2}) … ẽ_{n−1}(x_{1}, … x_{n−1}).

In other words, the quantum Schubert polynomial corresponding to the longest element of the symmetric group S_{n}, is equal to the product of all prin-

cipal minors of the Jacobi matrix (\frac{∂e_{i}(x|q)}{∂x_{j}})_{1≤i,j≤n}. We can also compute

the Grothendieck residue w.r.t. ideal Ĩ of the Jacobian det (\frac{∂e_{i}(x|q)}{∂x_{j}})_{1≤i,j≤n}.

Proposition 10 (cf. [EL])

det (\frac{∂e_{i}(x|q)}{∂x_{j}}) ≡ n!𝔖̃_{w_{0}}(x) (mod Ĩ),

where e_{i}(x|q) = e_{i}(x_{1}, …, x_{n} | q_{1}, …, q_{n−1}), 1 ≤ i ≤ n, are the quantum elementary symmetric functions.

Remind that n! is equal to the Euler number of F l_{n}.

4.2 Orthogonality.

We use the Jack–Macdonald type definition ([M2], Chapter VI) of the quantum Schubert polynomials, see Definition 5. On this way the orthogonality of quantum Schubert polynomials is valid by “definition”. We are going to prove that the y = 0 specialization of quantum double Schubert polynomials 𝔖̃_{w}(x, 0) also satisfies the conditions 1) and 2) of Definition 5. As a corollary, we obtain that the specialization 𝔖̃_{w}(x, 0) coincides with the quantum Schubert polynomial 𝔖̃_{w}(x) from Definition 5.

Theorem 2 Let v, w ∈ S_{n}. Then

⟨𝔖̃_{v}(x, 0), 𝔖̃_{w}(x, 0)⟩_{Q} = \{ \begin{matrix}1, & if w = w_{0}v, \\ 0, & otherwise. \end{matrix}

Proof. Let us apply the operator ∂_{vw_{0}}^{(y)}∂_{ww_{0}}^{(z)} to the both sides of (??). The LHS gives

∂_{vw_{0}}^{(y)}∂_{ww_{0}}^{(z)}⟨𝔖̃_{w_{0}}(x, y), 𝔖̃_{w_{0}}(x, z)⟩_{Q}^{(x)} = ⟨𝔖̃_{v}(x, y), 𝔖̃_{w}(x, z)⟩_{Q}^{(x)}.

The RHS transforms to the following form ∑_{u∈S_{n}} ∂_{vw_{0}}^{(y)}𝔖_{u}(y)∂_{ww_{0}}^{(z)}𝔖_{w_{0}u}(z). Now

taking y = z = 0 we obtain an equality

⟨𝔖̃_{v}(x, 0)𝔖̃_{w}(x, 0)⟩_{Q} = ∑_{u∈S_{n}} η(∂_{vw_{0}}𝔖_{u})η(∂_{ww_{0}}𝔖_{w_{0}u}),  (11)

where η : P_{n} → Z is the homomorphism defined by η(x_{i}) = 0 (1 ≤ i ≤ n). It is clear that

η(∂_{v}𝔖_{u}) = \{ \begin{matrix}1, & if v = u, \\ 0, & otherwise. \end{matrix}

Thus, the RHS of (??) is equal to 1 if w_{0}ww_{0} = vw_{0} and is equal to 0 otherwise.

Remark 4 Orthogonality of quantum Schubert polynomials was proven in [FGP], using a combinatorial definition, see ibid, Section 5; the proof is highly non trivial.

4.3 Quantum Cauchy formula.

Theorem 3 Let 𝔖̃_{w}(x) := 𝔖̃_{w}(x, 0), then

∑_{w∈S_{n}} 𝔖̃_{w}(x)𝔖_{ww_{0}}(y) = 𝔖̃_{w_{0}}(x, y).  (12)

Proof. Let us apply the divided difference operator ∂_{ww_{0}}^{(z)} to the both sides of (??) and then take z = 0. The right hand side transforms to the following form

∑_{u∈S_{n}} 𝔖_{u}(y)∂_{ww_{0}}^{(z)}𝔖_{w_{0}u}(z)|_{z=0} = 𝔖_{w_{0}ww_{0}}(y).

As for the LHS, it takes the form ⟨𝔖̃_{w_{0}}(x, y), 𝔖̃_{w}(x)⟩_{Q}. Hence,

⟨𝔖̃_{w_{0}}(x, y), 𝔖̃_{w}(x)⟩_{Q} = 𝔖_{w_{0}ww_{0}}(y).

The last identity is equivalent to (??).

More generally, we have

Proposition 11

∑_{w∈S_{n}} 𝔖̃_{w}(x, z)𝔖_{ww_{0}}(y, −z) = 𝔖̃_{w_{0}}(x, y),  (13)
∑_{u∈S_{n}, l(u)+l(uw^{−1})=l(w)} 𝔖̃_{u}(x, z)𝔖_{uw^{−1}}(y, −z) = 𝔖̃_{w}(x, y).  (14)

Proof. Let us apply the Interpolation formula to f(x) = 𝔖̃_{w_{0}}(x, y) and then divided difference operator ∂_{ww_{0}}^{(y)}.

Corollary 1

C^{(q,q′)}(x, y) := ∑_{w∈S_{n}} 𝔖̃_{w}^{(q)}(x)𝔖̃_{w_{0}w}^{(q′)}(y) = ⟨𝔖̃_{w_{0}}^{(q)}(x, z), 𝔖̃_{w_{0}}^{(q′)}(y, z)⟩^{(z)},

where the upper index z means that the scalar product is taken in the z variables.

Corollary 2

∑_{w∈S_{n}} Ω_{w}Ω_{w}^{∗} = C^{(q,q)}(x, x).

One can show that

C^{(q,q)}(x, x) = ∑_{w∈S_{n}} 𝔖̃_{w}(x)𝔖̃_{ww_{0}}(x) ≡ n! 𝔖̃_{w_{0}}(x) (mod Ĩ).

Let us summarize our results. It follows from Theorem ?? that polynomials 𝔖̃_{w}(x, 0) are orthogonal with respect to the quantum pairing ⟨, ⟩_{Q}. It is also clear that 𝔖̃_{w}(x, 0)|_{q=0} = 𝔖_{w}(x) and 𝔖̃_{w}(x) = x^{c(w)}+lower degree terms w.r.t. lexicographic order on the set of monomials. These two properties characterize the polynomials 𝔖̃_{w}(x, 0) uniquely, consequently, the polynomials 𝔖̃_{w}(x, 0) coincide with the quantum Schubert polynomials 𝔖̃_{w}(x) from Definition ??. As a matter of fact, we obtain the Lascoux-Schützenberger type formula for quantum Schubert polynomials.

Theorem 4 Let w ∈ S_{n}, then

𝔖̃_{w}(x) = ∂_{ww_{0}}^{(y)}𝔖̃_{w_{0}}(x, y)|_{y=0}.

5 Quantization.

5.1 Definition.

Let f ∈ P_{n} = Z[x_{1}, …, x_{n}] be a polynomial. According to the Interpolation formula,

f(x) = ∑_{w∈S^{(n)}} ∂_{w}^{(y)}f(y)𝔖_{w}(x, y).

We define a quantization f̃ of the function f by the rule

f̃(x) = ∑_{w∈S^{(n)}} ∂_{w}^{(y)}f(y)𝔖̃_{w}(x, y)|_{\overline{P}_{n}},  (15)

where for a polynomial f ∈ \overline{P}_{∞}, the symbol f |_{\overline{P}_{m}} means the restriction of f to the ring of polynomials \overline{P}_{m}, i.e. the specialization x_{m+1} = x_{m+2} = ⋯ = 0 and q_{m} = q_{m+1} = ⋯ = 0.

Hence, the quantization is a Z[q_{1}, …, q_{n−1}]–linear map P_{n} → \overline{P}_{n}.

The main property of quantization is that it preserves the pairings, i.e.

⟨f̃, g̃⟩_{Q} = ⟨f, g⟩, f, g ∈ P_{n}.  (16)

It follows from (??) that the quantization map maps the ideal I_{n} ⊂ \overline{P}_{n} into ideal Ĩ_{n} ⊂ \overline{P}_{n}. Remark 5 i) Quantization does not preserve multiplication, i.e. in general

f̃ · g̃ ≠ f̃g. For example, if f = ∑_{i=1}^{n} α_{i}x_{i} is a linear form, then (quantum

Monk’s formula, see [FGP] and our Section 9)

f̃𝔖̃_{w} − f̃𝔖_{w} = ∑(λ_{i} − λ_{j})q_{ij}𝔖̃_{wt_{ij}},

summed over i < j such that l(w) = l(wt_{ij})+l(t_{ij}). Here q_{ij} = q_{i}q_{i+1} … q_{j−1}. ii) It is clear that if f ∈ H_{n}, then f̃ ∈ \overline{H}_{n}. iii) It follows from Proposition ??, that the quantum double Schubert

polynomials 𝔖̃_{w}(x, y) are the quantization of classical ones.

iv) It follows from Interpolation formula and quantization procedure, that • Quantum Schubert polynomials 𝔖̃_{w}(x), w ∈ S_{n} form a \overline{I}-basis in \overline{P}_{n}.

• Quantum Schubert polynomials 𝔖̃_{w}(x), w ∈ S^{(n)} form a Z[q_{1}, …, q_{n−1}]– basis of \overline{P}_{n}.

• Quantum Schubert polynomials 𝔖̃_{w}(x), w ∈ S_{n} form a Z[q_{1}, …, q_{n−1}]– basis of \overline{H}_{n} = H_{n} ⊗ Z[q_{1}, …, q_{n−1}].

The proof of the statement iv) can be found in [FGP].

Now we are going to describe another families of polynomials having a nice quantization.

5.2 Elementary and complete polynomials.

Let δ := δ_{n} = (n − 1, n − 2, …, 1, 0), and consider the set 𝔗 of sequences I = (i_{1}, …, i_{n}) ∈ Z^{n} such that 0 ≤ i_{j} ≤ n − j for all j = 1, …, n. It is clear that |𝔗| = n!, and

• P_{n} is a free Λ_{n}-module of rank n! with basis {x^{I} = x_{1}^{i_{1}}x_{2}^{i_{2}} … x_{n}^{i_{n}} | I ∈ 𝔗}. Follow to [LS2], for each I ∈ 𝔗 let us define the elementary polynomial

e_{I}(x) as the following product

∏_{k=1}^{n−1} e_{i_{k}}(x_{1}, …, x_{n−k}).

• (Lascoux-Schützenberger [LS2]) P_{n} is a free Λ_{n}-module of rank n! with basis {e_{I}(x) | I ∈ 𝔗}.

Definition 6 For each sequence I ∈ 𝔗 the quantum elementary polynomial ẽ_{I}(x) is defined to be

ẽ_{I}(x) = ∏_{k=1}^{n−1} ẽ_{i_{k}}(x_{1}, …, x_{n−k}),

where ẽ_{k}(x_{1}, …, x_{m}) := e_{k}(x_{1}, …, x_{m} | q_{1}, …, q_{m−1}) are the quantum elementary symmetric functions.

Theorem 5 Assume that I ⊂ δ. Then ẽ_{I}(x) is the quantization of elementary polynomial e_{I}(x).

Proof. It is enough to prove the following

Proposition 12 If I ⊂ δ, then

ẽ_{I}(x) = ∑_{w∈S_{n}} 𝔖̃_{w}(x)η(∂_{w}e_{I}).  (17)

We are going to show that Proposition ?? follows from the quantum Cauchy formula. First of all, let us remark that

𝔖̃_{w_{0}}(x, y) = ∑_{I⊂δ} ẽ_{I}(x)y^{δ−I},
𝔖̃_{w_{0}}(x, z) = ∑_{w∈S_{n}} 𝔖̃_{w_{0}ww_{0}}(x)𝔖_{w_{0}w}(z).

Substituting these two expressions in (??), we obtain a formula for the classical Schubert polynomials

𝔖_{w}(y) = ∑_{I⊂δ}⟨ẽ_{I}(x), 𝔖̃_{w_{0}ww_{0}}(x)⟩_{Q}y^{δ−I}.  (18)

It follows from (??) that

⟨ẽ_{I}(x), 𝔖̃_{w_{0}ww_{0}}(x)⟩_{Q} = ⟨e_{I}(x), 𝔖_{w_{0}ww_{0}}(x)⟩.  (19)

Conversely, the quantum Cauchy formula (??) follows from the classical one and (??). To continue, let us remark that

⟨e_{I}(x), 𝔖_{w_{0}ww_{0}}(x)⟩ = ⟨e_{I}(x), ∂_{w_{0}w^{−1}}𝔖_{w_{0}}(x)⟩ =
⟨∂_{ww_{0}}e_{I}(x), 𝔖_{w_{0}}(x)⟩ = η(∂_{ww_{0}}e_{I}(x)).

As a corollary we obtain a formula for Schubert polynomials, which seems to be new,

𝔖_{w}(x) = ∑_{I⊂δ} η(∂_{ww_{0}}e_{I}(x))x^{δ−I}.  (20)

Formula (??) gives a geometric interpretation of the coefficients a_{I,w} of the Schubert polynomial 𝔖_{w}(x) = ∑_{I⊂δ} a_{I,w}x^{δ−I} as the intersection numbers

a_{I,w} = η(∂_{ww_{0}}e_{I}(x)) = ⟨e_{I}(x), 𝔖_{w_{0}ww_{0}}(x)⟩ ≥ 0.

Finally, let us finish a proof of Theorem ??. We have

∑_{I⊂δ} RHS(??)y^{δ−I} = ∑_{w∈S_{n}} 𝔖̃_{w}(x) ∑_{I⊂δ} η(∂_{w}e_{I})y^{δ−I}
= ∑_{w∈S_{n}} 𝔖̃_{w}(x)𝔖_{ww_{0}}(y) = 𝔖̃_{w_{0}}(x, y) = ∑_{I⊂δ} ẽ_{I}(x)y^{δ−I}.

Hence, RHS(??) = ẽ_{I}(x).

Corollary 3 If I and J belong to 𝔗, then

⟨ẽ_{I}(x), ẽ_{J}(x)⟩_{Q} = ⟨e_{I}(x), e_{J}(x)⟩.

Remark 6 In the next section we will give a proof of Corollary 3 using a geometric technique due to I. Ciocan–Fontanine [C] (see also [K1]). Repeating our arguments in the reverse order, we see that the quantum Cauchy formula (12), as well as Theorems 1 and 3, follow directly from Corollary 3.

Using quantum Cauchy formula (??), we can describe a transition matrix between quantum Schubert polynomials and quantum elementary polynomials.

Theorem 6

𝔖̃_{w}(x) = ∑_{I⊂δ} ẽ_{I}(x)η(∂_{ww_{0}}x^{δ−I}).

Proof. It follows from Cauchy’s formula that

∑_{w∈S_{n}} 𝔖̃_{w}(x)𝔖_{ww_{0}}(y) = ∑_{I⊂δ} ẽ_{I}(x)y^{δ−I}.

Consequently,

𝔖̃_{w_{0}ww_{0}}(x) = ∑_{I⊂δ} ẽ_{I}(x)⟨y^{δ−I}, 𝔖_{w}(y)⟩.

Now we have

⟨y^{δ−I}, 𝔖_{w}(y)⟩ = ⟨y^{δ−I}, ∂_{w^{−1}w_{0}}𝔖_{w_{0}}(y)⟩ = ⟨∂_{w_{0}w}y^{δ−I}, 𝔖_{w_{0}}(y)⟩ = η(∂_{w_{0}w}y^{δ−I}).

Example. Take the permutation w = 24531 ∈ S_{5}. It is easy to check that ww_{0} = 42135 = s_{1}s_{2}s_{1}s_{3} and there exists 6 monomials x^{I} such that I ⊂ (43210) and η(∂_{1213}x^{I}) ≠ 0. They are

\begin{matrix}x^{I} : & x_{1}^{2}x_{2}x_{3}, & x_{1}^{2}x_{2}x_{4}, & x_{1}^{2}x_{3}^{2}, & x_{1}x_{2}^{2}x_{3}, & x_{1}x_{2}^{2}x_{4}, & x_{2}^{2}x_{3}^{2} \\ η(∂_{1213}x^{I}) : & +1 & −1 & −1 & −1 & +1 & +1 \end{matrix}

We can check using Theorem ??, that

𝔖̃_{24531}(x) = ẽ_{2211}(x) − ẽ_{2220}(x) − ẽ_{2301}(x) − ẽ_{3111}(x) + ẽ_{3120}(x) + ẽ_{4101}(x).

Now let us consider a problem how to quantize monomials.

Proposition 13 Let I ∈ 𝔗, then

x^{I} = ∑_{w∈S_{n}, l(w)=|I|} η(∂_{w}x^{I})𝔖_{w}(x),
x̃^{I} = ∑_{w∈S_{n}, l(w)=|I|} η(∂_{w}x^{I})𝔖̃_{w}(x).

Corollary 4 Let v, w ∈ S_{n}, then

∑_{I⊂δ} η(∂_{v}x^{I})η(∂_{ww_{0}}e_{δ−I}(x)) = δ_{v,w}.

Corollary 5

𝔖̃_{w}(x) = ∑_{I⊂δ} η(∂_{ww_{0}}e_{δ−I}(x))x̃^{I}.

Corollary 6

𝔖̃_{w_{0}}(x, y) = ∑_{I⊂δ} x̃^{I}e_{δ−I}(y).

Now let us consider a problem how to quantize the complete homogeneous symmetric functions

h_{k}^{m} := ∑_{i_{1}+···+i_{m}=k} x_{1}^{i_{1}} ⋯ x_{m}^{i_{m}}.

We define the quantum complete homogeneous symmetric function h̃_{k}^{m} = h̃_{k}(x_{1}, …, x_{m}) of degree k using the generating function

H_{m}(t | x) := H_{m}(t | x_{1}, … x_{m}) = (t^{m}Δ_{m}(t^{−1} | −x))^{−1} = ∑_{k≥0} t^{k}h̃_{k}^{m}.

It is clear that

t^{m}Δ_{m}(t^{−1} | x) = ∑_{k≥0} t^{k}ẽ_{k}^{m},

and

Δ_{m}(t^{−1} | x) · H_{m}(t | −x) = 1.

The last relation gives possibility to express the quantum complete homogeneous symmetric functions in terms of quantum elementary ones:

h̃_{k}^{m} = det \binom{}{e_{j−i+1}^{m}}_{1≤i,j≤k}
= (det \binom{}{e_{j−i+1}^{m+k−i}}_{1≤i,j≤k}) |_{\overline{P}_{m}} .  (21)

The equality (20) follows from the recurrence relation for ẽ_{k}^{N} (see [C], [GK]):

ẽ_{k}^{N} = ẽ_{k}^{N−1} + x_{N} ẽ_{k−1}^{N−1} + q_{N−1}ẽ_{k−2}^{N−2}.

But each term in the expansion of the determinant (20) is a quantum elementary polynomial in \overline{P}_{m+k}. Hence, it follows from Theorem 7 that the quantum complete homogeneous symmetric function h̃_{k}^{m} is the quantization of classical one.

Finally, let us define the complete and quantum complete polynomials.

Definition 7 For each sequence I ⊂ δ_{n} the complete and quantum complete polynomials h_{I}(x) and h̃_{I}(x) are defined to be

h_{I}(x) = ∏_{k=1}^{n−1} h_{i_{k}}(x_{1}, …, x_{k}),  (22)

h̃_{I}(x) = ∏_{k=1}^{n−1} h̃_{i_{k}}(x_{1}, …, x_{k}).  (23)

Remark 8 It follows from (20) that if m + k > n then h̃_{k}^{m} ∈ Ĩ_{n}. Hence, if I ⊄ δ, then h_{I}(x) ∈ I_{n} and h̃_{I}(x) ∈ Ĩ_{n}.

It is not difficult to see that P_{n} is a free Λ_{n}-module of rank n! with basis {h_{I}(x) | I ∈ 𝔗}. Theorem 5’ If I ⊂ δ, then h̃_{I}(x) is the quantization of complete polynomial h_{I}(x).

Proof. See Remark 10 in Section 7.

5.3 Canonical involution ω.

There exists an involution ω of the ring \overline{P}_{n}[y] given by ω(x) = x^{←}, ω(y) = y^{←}, ω(q) = q^{←}, where for any sequences z = (z_{1}, …, z_{m}) we define z^{←} to be equal to (z_{m}, z_{m−1}, …, z_{1}) :=z^{←}. It is clear from the definition of quantum elementary symmetric functions e_{i}(x|q), see (9), Section 3, that

ω(e_{i}(x|q)) = e_{i}(x|q)

and thus the involution ω preserves the ideal Ĩ_{n} (as well as the ideals I_{n}, J_{n} and J̃_{n}).

ω(𝔖̃_{u}^{(q)}(x, y)) ≡ ϵ(u)𝔖̃_{w_{0}uw_{0}}^{(q)}(x, y) mod J̃_{n},

where ϵ(u) = (−1)^{l(u)}.

Proof. First of all, if u = w_{0}, then

ω(𝔖̃_{w_{0}}^{(q)}(x, y)) ≡ ϵ(w_{0})𝔖̃_{w_{0}}^{( q^{←})}(x^{←}, y^{←}) mod J̃_{n}.  (24)

But ω∂_{u} = ϵ(u)∂_{w_{0}uw_{0}}ω (see [M1], (2.12)). Thus, applying the divided difference operator ϵ(u)∂_{w_{0}uw_{0}}^{(y)} to the both sides of (??), we obtain

ω(𝔖̃_{uw_{0}}^{(q)}(x, y)) ≡ ϵ(w_{0})𝔖̃_{w_{0}u}^{( q^{←})}(x^{←}, y^{←}) mod J̃_{n}.

Finally, let us describe the action of involution ω on the elementary polynomials.

Proposition 15

ω(ẽ_{I}(x)) ≡ (−1)^{|I|}h̃_{I^{←}}(x) mod Ĩ_{n}.

Remark 9 To our knowledge, originaly, construction of the quantization map, using a remarkable family of commuting operators X_{i}, appeared in [FGP]. We use a different definition of quantization map, but it can be shown that two forms of quantizations are equivalent. For original proofs of Theorem 5 and 5’, and Proposition 15, see Corollary 4.6, Corollary 7.16 and Proposition 7.13 in [FGP].

6 Quantum cohomology ring of flag variety.

Quantum cohomology ring of the flag variety F l_{n} is a deformed ring of the ordinary cohomology ring H^{∗}(F l_{n}, Z). The structure constants of the quantum cohomology ring are given by the Gromov–Witten invariants. Let Ω_{w_{1}}, …, Ω_{w_{m}} (w_{i} ∈ S_{n}) be Schubert cycles. We denote by Md̄(P^{1}, F l_{n}) the moduli space of morphisms from P^{1} to F l_{n} of multidegree d̄= (d_{1}, …, d_{n−1}). We consider the restriction of the universal map for t ∈ P^{1}:

ev_{t} : Md̄(P^{1}, F l_{n}) × {t} ↪ Md̄(P^{1}, F l_{n}) × P^{1}→^{ev}F l_{n}, (f, p) ↦ f(p).

Let Ω_{w}(t) = ev_{t}^{−1}(Ω_{w}).

Theorem 7 (I. Ciocan-Fontanine). If ∑_{i=1}^{m} l(w_{i}) = \frac{n(n − 1)}{2} + 2 ∑ d_{i} and

t_{1}, …, t_{m} ∈ P^{1} are distinct, then for general translates of Ω_{w_{i}}, the number

of points in ⋂_{i=1}^{m} Ω_{w_{i}}(t_{i}) is finite and independent of t_{i} and the translates of

Ω_{w_{i}}.

Definition 8 The Gromov–Witten invariant is defined as an intersection number

⟨Ω_{w_{1}} … Ω_{w_{m}}⟩d̄ = \{ \begin{matrix}# ⋂_{i} Ω_{w_{i}}(t_{i}), & if ∑ l(w_{i}) = \frac{n(n−1)}{2} + 2 ∑ d_{i} \\ 0, & otherwise \end{matrix} .

Now we can define the quantum multiplication as a linear map

m_{q} : Sym (H^{∗}(F l_{n}, Z)[q_{1}, …, q_{n−1}]) → H^{∗}(F l_{n}, Z)[q_{1}, …, q_{n−1}]

given by

m_{q}(∏_{i=1}^{m} Ω_{w_{i}}) = ∑_{d̄} qd̄ ∑_{w}⟨Ω_{w}Ω_{w_{1}} ⋯ Ω_{w_{m}}⟩d̄Ω_{w}^{∗},

where qd̄ = q_{1}^{d_{1}} ⋯ q_{n−1}^{d_{n−1}} and (Ω_{w}^{∗}) is the dual basis of (Ω_{w}).

Then the quantum cohomology ring QH^{∗}(F l_{n}) is a commutative and associative Z[q_{1}, …, q_{n−1}] – algebra.

Let 0 = E_{0} ⊂ E_{1} ⊂ ⋯ ⊂ E_{n} = C^{n} ⊗ O_{F} be the universal flag of subbundles on F l_{n}.

Theorem 8 (A. Givental and B. Kim, I. Ciocan–Fontanine). The small quantum cohomology ring is generated by x_{i} = c_{1}(E_{n−i+1}/E_{n−i}), i = 1, …, n, as a Z[q_{1}, …, q_{n−1}]-algebra and

QH^{∗}(F l_{n}) ≅ Z[x_{1}, …, x_{n}, q_{1}, …, q_{n−1}]/(e_{1}(x|q), …, e_{n}(x|q)),

where e_{i}(x|q) is given by the expansion of the following determinant

det ( \begin{matrix}x_{1} + t & q_{1} & 0 & … & … & … & 0 \\ −1 & x_{2} + t & q_{2} & 0 & … & … & 0 \\ 0 & −1 & x_{3} + t & q_{3} & 0 & … & 0 \\ ⋮ & ... & ... & ... & ... & ... & ⋮ \\ 0 & … & 0 & −1 & x_{n−2} + t & q_{n−2} & 0 \\ 0 & … & … & 0 & −1 & x_{n−1} + t & q_{n−1} \\ 0 & … & … & … & 0 & −1 & x_{n} + t \end{matrix} )

= t^{n} + e_{1}(x|q)t^{n−1} + ⋯ + e_{n}(x|q).

It follows from Theorem ?? that any Schubert cycle Ω_{w} may be expressed as a polynomial 𝔖̂_{w}(x, q) in QH^{∗}(F l_{n}). The polynomial 𝔖̂_{w}(x, q) is a deformation of the Schubert polynomial 𝔖_{w}(x) and 𝔖̂_{w}(x, 0) = 𝔖_{w}(x). Consider the correlation function

⟨Ω_{w_{1}} … Ω_{w_{m}}⟩ = ∑_{d̄} qd̄⟨Ω_{w_{1}} … Ω_{w_{m}}⟩d̄.

Then 𝔖̂_{w}(x; q) is characterized by the condition

⟨Ω_{w}Ω_{w_{1}} … Ω_{w_{m}}⟩ = ⟨𝔖̂_{w}(x, q)Ω_{w_{1}} … Ω_{w_{m}}⟩

for any w_{1}, …, w_{m} ∈ S_{n}.

𝔖̂_{w}(x; q) is called a geometric quantum Schubert polynomial. By definition 𝔖̂_{w}(x; q) ∈ QH^{2l(w)}(F l_{n}).

7 Proofs of Theorem 3 and quantum Cauchy formula.

Theorem 9 Let I ∈ 𝔗. Then

⟨ẽ_{I}(x), 𝔖̃_{w}(x)⟩_{Q} = ⟨e_{I}(x), 𝔖_{w}(x)⟩,

for any permutation w ∈ S_{n}.

Proof. The proof is based on the arguments due to I. Ciocan-Fontanine [C]. To begin with, let us remind his results. We consider the hyper-quot scheme HQd̄(P^{1}, F l_{n}) associated to P^{1} with multidegree d̄= (d_{1}, …, d_{n−1}). Let

C^{n} ⊗ O → T_{n−1} → ⋯ → T_{2} → T_{1} → 0

be the universal sequence of quotients on P^{1} × HQd̄(P^{1}, F l_{n}) and

S_{i} = Ker{C^{n} ⊗ O → T_{n−i}}.

We also consider the dual sequence

C^{n} ⊗ O → S_{n−1}^{∗} → ⋯ → S_{1}^{∗}.

We fix a flag

0 = V_{0} ⊂ V_{1} ⊂ ⋯ ⊂ V_{n} = C^{n}

and define the subschema D_{w}^{p,q} of P^{1} × HQd̄(P^{1}, F l_{n}) as the locus where rank(V_{p} ⊗ O → S_{q}^{∗}) ≤ r_{w}(q, p), and r_{w}(q, p) := ♯{i | i ≤ q, w_{i} ≤ p}. Let

D_{w}^{p,q}(t) = D_{w}^{p,q} ⋂{{t} × HQd̄(P^{1}, F l_{n})}

and

\overline{Ω}_{w}(t) = ⋂_{p,q=1}^{n−1} D_{w}^{p,q}(t).

Then the class of Ω̄_{w}(t) in the Chow ring CH^{l(w)}(HQd̄(P^{1}, F l_{n})) is independent of t ∈ P^{1} and the flag V_{0} ⊂ ⋯ ⊂ V_{n}.

The boundary HQd̄(P^{1}, F l_{n}) \ M\overline{d}(P^{1}, F l_{n}) consists of n − 1 divisors D_{1}, …, D_{n−1}, which are birational to

P^{1} × HQd̄_{1}(P^{1}, F l_{n}), …, P^{1} × HQd̄_{n−1}(P^{1}, F l_{n})

respectively, where d̄_{i} = (d_{1}, …, d_{i}−1, …, d_{n−1}). Let x_{i}(t) = \overline{Ω}_{s_{i}}(t)−Ω̄_{s_{i−1}}(t), then for any permutation w ∈ S_{n} there exists an element

G_{w}(t) ∈ CH_{∗}( ⋃_{i=1}^{n−2} D_{i})

such that

Ω̄_{w^{−1}}(t) − 𝔖_{w}(x_{1}(t), …, x_{n−1}(t)) = j_{∗}(G_{w^{−1}}(t)),

where

j : ⋃_{i=1}^{n−2} D_{i} → HQd̄(P^{1}, F l_{n})

is the inclusion.

Let [m, k] ∈ S_{n} be the permutation

( \begin{matrix}1 & 2 & … & m − k − 1 & m − k & m − k + 1 & … & m & m + 1 & … & n \\ 1 & 2 & … & m − k − 1 & m & m − k & … & m − 1 & m + 1 & … & n \end{matrix} ) .

Then the geometric Schubert polynomial _{ 𝔖̂}_{[m,k]^{−1}}(x) is the elementary symmetric function in x_{1}, …, x_{m−1} of degree k. Let I = (i_{1}, …, i_{n}). We have to calculate in CH^{∗}(HQd̄(P^{1}, F l_{n}))

(\overline{⋂_{ν=1}^{n−1} Ω_{[n−ν+1,i_{ν}]^{−1}}} ) (t) · ∏_{j=1}^{N} \overline{Ω}_{w_{j}}(t_{j}) − ∏_{ν=1}^{n−1} 𝔖_{[n−ν+1,i_{ν}]}(x(t)) · ∏_{j=1}^{N} \overline{Ω}_{w_{j}}(t_{j}), (25)

for distinct t, t_{1}, …, t_{N} and w_{1}, …, w_{N} ∈ S_{n} such that

∑_{ν=1}^{n−1} i_{ν} + ∑_{j=1}^{N} l(w_{j}) = n(n − 1)/2 + 2 ∑_{k=1}^{n−1} d_{k},

where ∩ is the classical intersection product and

(\overline{⋂_{ν=1}^{n−1} Ω_{[n−ν+1,i_{ν}]^{−1}}} ) (t)

is the corresponding degeneracy locus on HQd̄(P^{1}, F l_{n}). In order to calculate the expression (??), we are going to prove at first that

(\overline{⋂_{ν=1}^{n−1} Ω_{[n−ν+1,i_{ν}]^{−1}}} ) (t) · ∏_{j=1}^{N} Ω̄_{w_{j}}(t_{j}) − ∏_{ν=1}^{n−1} Ω̄_{[n−ν+1,i_{ν}]^{−1}}(t) · ∏_{j=1}^{N} Ω̄_{w_{j}}(t_{j}) = 0.

The LHS of the last expression can be computed as the number of points in

∏_{ν=1}^{n−1} Ω̄_{[n−ν+1,i_{ν}]^{−1}}(t) · ∏_{j=1}^{N} Ω̄_{w_{j}}(t_{j})

suppoted on ∪_{i=1}^{n−2}D_{i}. Let jd̄_{k} be the natural rational map

P^{1} × HQd̄_{k}(P^{1}, F l_{n})− → HQd̄(P^{1}, F l_{n}).

¿From Remark 3 in [C],

j−1 ; d̄_{k}(Ω̄_{[n−ν+1,i_{ν}]^{−1}}(t)) =

\{ \begin{matrix}P^{1} × Ω̄_{[n−ν+1,i_{ν}]^{−1}}(t) ∪ ⋂_{n−ν−i_{ν}+1≤p≤n−ν−1} D_{[n−ν+1,i_{ν}]^{−1}}^{p,n−ν−1}(t), & if k = n − ν, \\ P^{1} × Ω̄_{[n−ν+1,i_{ν}]^{−1}}(t) & otherwise, \end{matrix}

where

⋂_{n−ν−i_{ν}+1≤p≤n−ν−1} D_{[n−ν+1,i_{ν}]^{−1}}^{p,n−ν−1}(t) = {t} × Ω̄_{[n−ν,i_{ν}−1]^{−1}}(t).

Because, by assumption,

∑_{ν=1}^{n−1} i_{ν} + ∑_{ν=1}^{n−1} l(w_{j}) = \frac{n(n − 1)}{2} + 2 ∑_{k=1}^{n−1} d_{k},

we have

Ω̄_{[k,i_{n−k}−1]^{−1}}(t) · ∏_{ν≠n−k} Ω̄_{[n−ν+1,i_{ν}]^{−1}}(t) · ∏_{j=1}^{N} Ω̄_{w_{j}}(t_{j}) = 0

on the hyper-quot scheme HQd̄_{k}(P^{1}, F l_{n}). Hence, we have equality

(\overline{⋂_{ν=1}^{n−1} Ω_{[n−ν+1,i_{ν}]^{−1}}} ) (t) · ∏_{j=1}^{N} Ω̄_{w_{j}}(t_{j}) = ∏_{ν=1}^{n−1} Ω̄_{[n−ν+1,i_{ν}]^{−1}}(t) · ∏_{j=1}^{N} Ω̄_{w_{j}}(t_{j}).

Our next observation is that the intersection number in the RHS of the last equality is equal to

∏_{ν=1}^{n−1} \overline{Ω}_{[n−ν+1,i_{ν}]^{−1}}(s_{ν}) · ∏_{j=1}^{N} \overline{Ω}_{w_{j}}(t_{j}),

where we can chose s_{1}, …, s_{n−1}, t_{1}, …, t_{N} ∈ P^{1} to be the pairwise distinct points, since the class [Ω_{w}(t′)] in the Chow ring CH^{∗}(HQ\overline{d}(P^{1}, F l_{n})) does not depends on the chose of t′ ∈ P^{1}.

Now we are going to use the following identity

∏_{k=1}^{m} a_{k} − ∏_{k=1}^{m} b_{k} = ∑_{k=1}^{m} ∏_{j=1}^{k−1} b_{j}(a_{k} − b_{k}) ∏_{j=k+1}^{m} a_{j}.

Let us take in the last equality m = n − 1

a_{k} := \overline{Ω}_{[n−k+1,i_{k}]^{−1}}(s_{k}),
b_{k} := 𝔖_{[n−k+1,i_{k}]}(x(s_{k})).

Then we obtain the following equality

∏_{ν=1}^{n−1} \overline{Ω}_{[n−ν+1,i_{ν}]^{−1}}(s_{ν}) ∏_{j=1}^{N} \overline{Ω}_{w_{j}}(t_{j}) − ∏_{ν=1}^{n−1} 𝔖_{[n−ν+1,i_{ν}]}(x(s_{ν})) ∏_{j=1}^{N} \overline{Ω}_{w_{j}}(t_{j})

= \{∑_{k=1}^{n−1} ∏_{j=1}^{k−1} \overline{Ω}_{[n−j+1,i_{j}]^{−1}}(s_{j}) · j_{∗}(G_{[n,i_{k}]^{−1}}(t)) ∏_{j=k+1}^{n−1} 𝔖_{[n−j+1,i_{j}]}(x(s_{j}))\} · B,

where B := ∏_{j=1}^{N} \overline{Ω}_{w_{j}}(t_{j}).

The contributions from j_{∗}((G_{[n,i_{k}]^{−1}}(t)), 1 ≤ k ≤ n − 1, can be computed by using the arguments in [C]. Inded, as in the proof of Theorem 4 in [C], the intersection number (1 ≤ k ≤ n − 1)

∏_{l=1}^{k−1} \overline{Ω}_{[n−l+1,i_{l}]^{−1}}(s_{l}) · j_{∗}(G_{[n,i_{k}]^{−1}}(t)) ∏_{l=k+1}^{n−1} 𝔖_{[n−l+1,i_{l}]}(x(s_{l})) · B

is the number of points in

∏_{l=1}^{k−1} \overline{Ω}_{[n−l+1,i_{l}]^{−1}}(s_{j}) ∏_{l=k}^{n−1} 𝔖_{[n−l+1,i_{l}]}(x(s_{l})) · B

supported on ⋃_{j=1}^{n−2} D_{i}. Hence, by induction, we have the following identity for correlation functions

⟨( ⋂_{ν=1}^{n−1} Ω_{[n−ν+1,i_{ν}]^{−1}}) · ∏_{j=1}^{N} \overline{Ω}_{w_{j}}⟩ = ⟨ ∏_{ν=1}^{n−1} 𝔖̃_{[n−ν+1,i_{ν}]} · ∏_{j=1}^{N} \overline{Ω}_{w_{j}}⟩,

where ⋂ is the classical intersection product and · is the product in

Sym H^{∗}(F l_{n}, Z)[q_{1}, …, q_{n−1}].

The last equality for correlation functions is equivalent to the following one

m_{q}(Ω_{[n,i_{1}]^{−1}} ∩ Ω_{[n−1,i_{2}]^{−1}}, ∩ … ∩ Ω_{[l,i_{n−1}]^{−1}}, ∗) = m_{q}(ẽ_{i_{1}} · ẽ_{i_{2}} ⋯ ẽ_{i_{n−1}}, ∗).

This completes the proof.

Remark 10 Using the similar geometrical arguments we can prove an analog of Theorem 9 for the quantum complete polynomials: Theorem 9’ Let I ⊂ δ_{n}. Then

⟨h̃_{I}(x), 𝔖̃_{w}(x)⟩_{Q} = ⟨h_{I}(x), 𝔖_{w}(x)⟩,

for any permutation ω ∈ S_{n}.

It is easy to see that Theorem 5’ is a corollary of Theorem 9’.

8 Correlation functions.

8.1 Higher genus correlation function and the Vafa– Intriligator type formula.

Fix a Riemann surface C of genus g. We denote by M_{d}(C, F) the moduli space of morphism from C to F l_{n}. One can define the higher genus Gromov– Witten invariants by method which is similar to that in the case of genus zero, [RT].

We have the following recursion relation for higher genus correlation function corresponding to the generating function for higher genus Gromov– Witten invariants

⟨Ω_{w_{1}} … Ω_{w_{N}} ⟩_{g} = ∑_{v∈S_{n}}⟨Ω_{w_{1}} … Ω_{w_{N}} Ω_{v}Ω_{v}^{∗}⟩_{g−1}

(cf. Ruan–Tian [RT]).

¿From Corollary ?? and Theorem ?? we can deduce the Vafa–Intriligator type formula for higher genus correlation functions, namely, let ⟨P (x_{1}, …, x_{n})⟩_{g} be the genus g correlation function corresponding to a polynomial P, then

⟨P (x_{1}, …, x_{n})⟩_{g} = Res_{I}(P Φ^{g}) =

= ∑_{_{e_{1}}=···=_{e_{n}}=0} P (x_{1}, …, x_{n}) det (\frac{∂ẽ_{i}}{∂x_{j}} )^{−1} (Φ(x_{1}, …, x_{n}))^{g} ,

where Φ(x) = ⟨𝔖̃_{w_{0}}(x, y), 𝔖̃_{w_{0}}(x, y)⟩^{(y)} = ∑_{w∈S_{n}} 𝔖̃_{w}(x)𝔖̃_{w_{0}w}(x) = C^{(q,q)}(x, x).

To simplify the formula above, we use the following observations

• det (\frac{∂ẽ_{i}}{∂x_{j}} ) ≡ n! 𝔖̃_{w_{0}}(x) (mod Ĩ);

• Φ(x) := C^{(q,q)}(x, x) ≡ n! 𝔖̃_{w_{0}}(x) (mod Ĩ).

Hence, we obtain

Theorem 10 (Higher genus Vafa-Intriligator formula)

⟨P (x_{1}, …, x_{n})⟩_{g} = Res_{I}(P Φ^{g}) = ∑_{_{e_{1}}=···=_{e_{n}}=0} P (x_{1}, …, x_{n}) (𝔖̃_{w_{0}}(x))^{g−1} ,

where 𝔖̃_{w_{0}}(x) = ẽ_{δ}(x) := ẽ_{1}(x_{1})ẽ_{2}(x_{1}, x_{2}) … ẽ_{n−1}(x_{1}, …, x_{n−1}), and ẽ_{i}(z) is the quantum elementary polynomial of degree i in the variable z = (z_{1}, …, z_{m}), see Section 5.2.

Remark 11 The polynomial

C^{(q,q′)}(x, y) := ∑_{w∈S_{n}} 𝔖̃_{w}^{(q)}(x)𝔖̃_{w_{0}w}^{(q′)}(y)

corresponds to the dual class of the diagonal in the quantum cohomology ring QH^{∗}(F l_{n} × F l_{n}, (q, q′)) = QH^{∗}(F l_{n}, q) ⊗ QH^{∗}(F l_{n}, q′).

8.2 Witten–Dijkgraaf–Verlinde–Verlinde equations for symmetric group.

The Witten–Dijkgraaf–Verlinde–Verlinde equations (WDVV-equations) are equations on the correlation functions ⟨𝔖̃_{u}𝔖̃_{v}𝔖̃_{w}⟩ ∈ Z[q_{1}, …, q_{n−1}], where u, v, w ∈ S_{n}. The correlation functions satisfy the following conditions

1) Normalization:

⟨1𝔖̃_{v}𝔖̃_{w}⟩ = ⟨𝔖_{v}, 𝔖_{w}⟩.

2) Initial data:

⟨𝔖̃_{s_{k}} 𝔖̃_{s_{k}} 𝔖̃_{w_{0}}⟩ = q_{k}.

3) Degree conditions:

⟨𝔖̃_{u}𝔖̃_{v}𝔖̃_{w}⟩ = 0

if either l(u) + l(v) + l(w) < l(w_{0}), or difference l(u) + l(v) + l(w) − l(w_{0}) is an odd positive integer.

4) WDVV-equations:

∑_{v}⟨𝔖̃_{w_{1}} 𝔖̃_{w_{2}} 𝔖̃_{v}⟩⟨𝔖̃_{w_{0}v}𝔖̃_{w_{3}} 𝔖̃_{w_{4}}⟩ = ∑_{v}⟨𝔖̃_{w_{2}} 𝔖̃_{w_{3}} 𝔖̃_{v}⟩⟨𝔖̃_{w_{0}v}𝔖̃_{w_{1}} 𝔖̃_{w_{4}}⟩.

for any w_{1}, w_{2}, w_{3}, w_{4} ∈ S_{n}. Conjecture 1 Conditions 1)–4) uniquely determine the correlation functions ⟨𝔖̃_{u}𝔖̃_{v}𝔖̃_{w}⟩. Remark 12 1) Correlation function ⟨𝔖̃_{w_{1}} 𝔖̃_{w_{2}} 𝔖̃_{w_{3}}⟩ is a generating function for the Gromov–Witten invariants (see Definition 4):

⟨𝔖̃_{w_{1}} 𝔖̃_{w_{2}} 𝔖̃_{w_{3}}⟩ := ∑_{\overline{d}} q\overline{d}⟨𝔖̃_{w_{1}} 𝔖̃_{w}𝔖̃_{w_{3}}⟩\overline{d}.

2) More generally,

⟨𝔖̃_{s_{k}} 𝔖̃_{s_{ij}} 𝔖̃_{w_{0}}⟩ = \{ \begin{matrix}q_{i} … q_{j−1}, & if 1 ≤ i ≤ k < j ≤ n, \\ 0, & otherwise. \end{matrix}

8.3 Residue formula.

Theorem 11 Correlation function ⟨P (x_{1}, …, x_{n})⟩ is given by the formula

⟨P (x_{1}, …, x_{n})⟩ = ∑_{_{e_{1}}(p)=···=_{e_{n}}(p)=0} Res_{p} (\frac{P (x_{1}, …, x_{n})dx_{1} ∧ ⋯ ∧ dx_{n}}{ẽ_{1} ⋯ ẽ_{n}} ) .

Proof. If the polynomial P (x_{1}, …, x_{n}) is in the ideal generated by ẽ_{1}, …, ẽ_{n}, then the left and right hand sides of the formula are zero. Hence, it is enough to prove that

∑ Res_{p} (\frac{x_{1}^{ν_{1}} ⋯ x_{n−1}^{ν_{n−1}}dx_{1} ∧ ⋯ ∧ dx_{n}}{ẽ_{1} ⋯ ẽ_{n}} )

= \{ \begin{matrix}1, & if (ν_{1}, …, ν_{n−1}) = (n − 1, n − 2, …, 1) \\ 0, & if 0 ≤ ν_{1} + ⋯ + ν_{n−1} < n(n − 1)/2, 0 ≤ ν_{i} ≤ n − i. \end{matrix}

We can extend the meromorphic form

ω = \frac{x_{1}^{ν_{1}} ⋯ x_{n−1}^{ν_{n−1}}dx_{1} ∧ ⋯ ∧ dx_{n}}{ẽ_{1} ⋯ ẽ_{n}}

on the affine space A_{(x_{1},...,x_{n})}^{n} to (P^{1})^{n}. For each subset I ⊂ {1, …, n}, we consider the coordinate chart U_{J} = A_{(z_{1}^{J} ,...,z_{n}^{J})}^{n}, where

z_{i}^{J} = \{ \begin{matrix}x_{i}, & if i ∉ J, \\ 1/x_{i}, & otherwise. \end{matrix}

Then

(P^{1})^{n} = ⋃_{J} U_{J}.

Let

ē_{j}^{J}(z_{1}^{J}, …, z_{n}^{J}) = (∏_{i∈J} z_{i}^{J})ẽ_{j}(x_{1}, …, x_{n}).

Then

ω = (−1)^{♯J} \frac{x_{1}^{ν_{1}} ⋯ x_{n−1}^{ν_{n−1}}(∏_{i∈J} z_{i}^{J})^{n−2}dz_{1}^{J} ∧ ⋯ ∧ dz_{n}^{J}}{ē_{1}^{J} ⋯ ē_{n}^{J}}

on U_{ϕ} ∩ U_{J}. If ♯J = j, then there exists a polynomial Q_{i}(z_{1}^{J}, …, z_{n}^{J}) for i ∈ J such that

ē_{j}^{I}(z_{1}^{J}, …, z_{n}^{J}) = 1 + ∑_{i∈J} z_{i}^{J}Q_{i}.

This follows from

ẽ_{j}(x_{1}, …, x_{n}) = e_{j}(x_{1}, …, x_{n}) + (terms of lower degree).

Therefore ē_{1}^{J}, … ē_{n}^{J} do not have common zero on

B_{J} = {(z_{1}^{J}, …, z_{n}^{J}) ∈ U_{J} | z_{i}^{J} = 0, i ∈ J}.

¿From the residue theorem, if 0 ≤ ν_{1} + ⋯ + ν_{n} < n(n − 1)/2, 0 ≤ ν_{i} ≤ n − i, then

∑ Res_{p} (\frac{x_{1}^{ν_{1}} ⋯ x_{n−1}^{ν_{n−1}}dx_{1} ∧ ⋯ ∧ dx_{n}}{ẽ_{1} ⋯ ẽ_{n}} ) = 0.

On the other hand,

∑_{_{e_{i}}(p)=0} Res_{p} (\frac{x_{1}^{n−1} ⋯ x_{n−1}dx_{1} ∧ ⋯ ∧ dx_{n}}{ẽ_{1} ⋯ ẽ_{n}} ) = − ∑_{p} Res_{p}ω,

where p runs over the common zeros of ē_{1}^{J}, …, ē_{n}^{J} in ⋃_{1∈J} B_{J}. Let y_{1} = 1/x_{1},

z = (y_{1}, x_{2}, …, x_{n}) and

ē_{1}^{∗}(z) = y_{1}(1 + y_{1}(x_{2} + ⋯ + x_{n})).

Then we have − ∑_{p} Res_{p}ω =

∑_{ē_{1}^{∗} = ē_{2}^{\{1\}} = ⋯ = ē_{n}^{\{1\}} = 0 in the locus \{y_{1} = 0\}} Res_{p} (\frac{x_{2}^{n−2} ⋯ x_{n−1}dy_{1} ∧ ⋯ ∧ dx_{n}}{ē_{1}^{∗}(z) · ē_{2}^{\{1\}}(z) ⋯ ē_{N}^{\{1\}}(z)} )

= ∑_{p} Res_{p} (\frac{x_{2}^{n−2} ⋯ x_{n−1}dx_{2} ∧ ⋯ ∧ dx_{n}}{ẽ_{1}(x_{2}, …, x_{n}) ⋯ ẽ_{n−1}(x_{2}, …, x_{n})}) = 1,

by induction.

¿From the residue formula, the correlation function is given by the quantum residue Res_{I}, namely,

⟨P (x_{1}, …, x_{n−1})⟩ = Res_{I}P (t_{1}, …, t_{n−1}).

In order to relate the quantum residue with the classical one, we consider the quantum residue generating function

Ψ(t) = ⟨ ∏_{i=1}^{n−1} \frac{t_{i}}{t_{i} − x_{i}} ⟩ = ∑_{ν∈(Z_{≥0})^{n−1}}⟨x^{ν}⟩t^{−ν}.

Then, we have

Res_{I}P (t_{1}, …, t_{n−1}) = Res_{I} (P (x_{1}, …, x_{n−1})Ψ(x)) .

Hence, it is important to determine the generating function Ψ(t). Let

f_{i}(t) = t^{n} + ∑_{j=0}^{n−1} γ_{n−j}^{(i)}t^{j}

be the characteristic polynomial of the quantum multiplication by x_{i} with respect to the basis consisting of the quantum Schubert polynomials. Let us consider the (n! + 1) × (n! + 1)-matrix C_{n}(t) such that

(C_{n}(t))_{1,j} = \frac{(−1)^{j−1}t^{n−j+2}}{(j − 1)!} ;

(C_{n}(t))_{i,j} = \{ \begin{matrix}\frac{(−1)^{n}}{(j − 1)!}\binom{i − 2}{n − j + 1}t^{j−2}, & if i ≥ 2, i + j ≥ n + 2, \\ 0, & otherwise. \end{matrix}

We define the differential operator D_{i} by

D_{i} = (γ_{n!}^{(i)}, γ_{n!−1}^{(i)}, …, 1) · C_{n}(t_{i}) ·^{t}(1, ∂/∂t_{i}, …, (∂/∂t_{i})^{n!}).

Proposition 16 The generating function Ψ(t) satisfies the system of differential equations

D_{i}Ψ(t) = 0, 1 ≤ i ≤ n − 1.

Conversely, these differential equations and the initial values ⟨x_{1}^{ν_{1}} ⋯ x_{n−1}^{ν_{n−1}}⟩ for 0 ≤ ν_{i} ≤ n! − 1 determine the generating function uniquely.

Proof. Let x be a variable. Since

( 1 ; x ; x^{2} ; x^{3} ; ⋮ ; x^{n!} ) = C_{n} · ( t(t − x)^{n!} ; −x(t − x)^{n!−1} ; 2!x(t − x)^{n!−2} ; −3!x(t − x)^{n!−3} ; ⋮ ; (−1)^{n!}(n!)!x ) ,

we have

D_{i}Ψ(t) = ⟨\frac{f_{i}(x_{i})}{(t_{i} − x_{i})^{n!+1}} ∏_{j≠i} \frac{t_{j}}{t_{j} − x_{j}} ⟩ = 0.

On the other hand, the recursive relations

⟨f_{i}(x_{i})P (x_{1}, …, x_{n−1})⟩ = 0

with the initial values ⟨x_{1}^{ν_{1}} ⋯ x_{n−1}^{ν_{n−1}}⟩ for 0 ≤ ν_{i} ≤ n! − 1 determine the correlation function uniquely.

Remark 13 We can also consider another generating function

⟨exp(x_{1}t_{1} + ⋯ + x_{n}t_{n})⟩.

This is the generating volume function in [GK]. This generating function satisfies

ẽ_{i} (\frac{∂}{∂t_{1}} , ⋯, \frac{∂}{∂t_{n}} ) ⟨exp(x_{1}t_{1} + ⋯ + x_{n}t_{n})⟩ = 0,

for 1 ≤ i ≤ n.

In the case of n = 3, we can calculate the generating function Ψ(t) explicitly. The results of calculation one can find in the Appendix B.

9 Extended Ehresman–Bruhat order and quantum Pieri rule.

Let us remind that the Ehresman–Bruhat order denoted by ≤, is the partial order on S_{n} that is the transitive closure of the relation →. Relation v → w means that

1) l(w) = l(v) + 1,

2) w = v · t where t is a transposition.

In other words, if v and w are permutations, v ≤ w means that there exists r ≥ 0 and v_{0}, v_{1}, …, v_{r} in S_{n} such that

v = v_{0} → v_{1} → ⋯ → v_{r} = w.

Now let us define the extended Ehresman–Bruhat order v ⇐ w on S_{n}. First of all, we define a relation v ← w (see, also, [FGP]). Relation v ← w means that

1) w = v · t, where t is a transposition,

2) l(w) ≥ l(v) + l(v^{−1}w). Remark 14 i) It follows from [M], (1.10), that condition 2) is equivalent to the following one 2′) w(i) < w(j) and for all k such that i < k < j we have w(i) < w(k) < w(j).

ii) If w = vt_{i,i+1} and l(w) = l(v) + 1 (i.e. v → w in the Bruhat order), then we have also an arrow v ← w. This is clear because in our case we have l(w) = l(v) + l(t_{i,i+1}).

[figure: id; 2; 21; 121; 12; 1]

Figure 1: Extended Ehresman-Bruhat order for S_{3}.

Example. Symmetric group S_{3} (see Figure 1).

We define a weight of an arrow v ← w, denoted by wt(v ← w), to be equal to the product q_{i} … q_{i+s−1}, if t = t_{ij} and 2s := l(w) + 1 − l(v). We assume that weight of any arrow v → w is equal to 1 (see, also, [FGP]).

Let us say that an arrow v ← w (resp. v → w) has a color k if w = vt_{ij} and 1 ≤ i ≤ k < j ≤ n.

Extended Ehresman–Bruhat order on S_{n} (notation v ⇐ w) is the transitive closure of the relations ←, and →. In other words, there exists r ≥ 0 and v_{0}, v_{1}, …, v_{r} in S_{n} such that

v = v_{0} ⇀ ↽ v_{1} ⇀ ↽ v_{2} ⇀ ↽ ⋯ ⇀ ↽ v_{r} = w,  (26)

where symbol v_{i} ⇀ ↽ v_{i+1} means either v_{i} → v_{i+1} or v_{i} ← v_{i+1}.

For given pair v ⇐ w, we consider a sequence of arrows (??) as a path between v and w (notation v ↦ w) in the extended Ehresman–Bruhat order and call it as a BE–path (Bruhat–Ehresman path). We denote the number r in a representation (??) by l(v ↦ w).

Let us define a weight of a BE–path v ↦ w as follows

wt(v ↦ w) = ∏_{i=0}^{r−1} wt(v_{i} ⇀ ↽ v_{i+1}).

We will say that BE–path v ↦ w has a color k, notation v ↦^{k} w, if in the representation (??) all arrows v_{i} ⇀ ↽ v_{i+1} (i = 0, …, r − 1) have the same color k.

Theorem 12 (Quantum Pieri’s rule). Let us consider the Grassmanian permutation [b, d] = (1, 2, …, b − d − 1, b, b − d, b − d + 1 …, b − 1, b + 1, …, n), for 2 ≤ b ≤ n, 1 ≤ d ≤ b. Then

𝔖̃_{[b,d]} · 𝔖̃_{v} ≡ ∑ wt(v ↦^{b} w)𝔖̃_{w} (mod Ĩ_{n}),

where the sum runs over all BE–paths v ↦^{b} w, s.t.

1) l(v ↦ w) = d;

2) if v_{l} = v_{l+1}(i_{l}j_{l}) (l = 0, …, d − 1), then all i_{l} are different.

(Note that 𝔖_{[b,d]} = e_{d}(x_{1}, …, x_{b−1})).

Sketch of the proof. It is enough to consider the case d = 1 (induction!). In the case d = 1, we use a quantum analog of Kohnert–Veigneau’s method [KV]. Namely, at first we prove the quantum Pieri rule (for d = 1) for double quantum Schubert polynomials and then take y = 0 (see Theorem 4).

Proposition 17 (Quantum Pieri’s rule for 𝔖̃_{w_{0}}(x, y)).

(x_{j}+y_{n+1−j})𝔖̃_{w_{0}}(x, y) ≡ ∑_{i<j} q_{ij}𝔖̃_{w_{0}t_{ij}}(x, y)−∑_{j<k} q_{jk}𝔖̃_{w_{0}t_{jk}}(x, y) (mod J̃), (27)

where q_{ij} := q_{i}q_{i+1} … q_{j−1}, if i < j; J̃ is the ideal in the ring Z[x_{1}, x_{2}, …, x_{n}, y_{1}, …, y_{n}, q_{1}, …, q_{n−1}] generated by

e_{i}(x_{1}, … x_{n} | q_{1}, …, q_{n−1}) + (−1)^{i−1}e_{i}(y_{1}, …, y_{n}), 1 ≤ i ≤ n,

and e_{k}(x_{1}, …, x_{n} | q_{1}, …, q_{n−1}) is the k-th quantum elementary symmetric function.

Applying to (??) the generalized Monk formula (see Section 2.2), we obtain Corollary 7 (Equivariant quantum Pieri’s rule)

x_{j}𝔖̃_{w}(x, y) + y_{w_{j}} 𝔖̃_{w}(x, y) ≡
≡ ∑_{j<k, l(wt_{jk})=l(w)+1} 𝔖̃_{wt_{jk}}(x, y) + ∑_{j<k, l(w)=l(wt_{jk})+l(t_{jk})} q_{jk}𝔖̃_{wt_{jk}}(x, y)
− ∑_{i<j, l(wt_{ij})=l(w)+1} 𝔖̃_{wt_{ij}}(x, y) − ∑_{i<j, l(w)=l(wt_{ij})+l(t_{ij})} q_{ij}𝔖̃_{wt_{ij}}(x, y) (mod J̃).

Remark 15 i) 𝔖̃_{[b,d]} = e_{d}(x_{1}, …, x_{b} | q_{1}, …, q_{b−1}) coincides with quantum elementary symmetric function.

ii) It is clear that (initial data)

𝔖̃_{sk}^{2} = 𝔖̃_{s_{k+1}s_{k}} + 𝔖̃_{s_{k−1}s_{k}} + q_{k},

i.e. ⟨𝔖̃_{k}𝔖̃_{k}𝔖̃_{w_{0}}⟩ = q_{k}.

iii) To our knowledge, in the classical case q = 0, the Pieri rule for Schubert polynomials was first stated in [LS1], (2.2). Our formulation of Theorem 12 is very close to that given in [BB]. The difference is: we use the paths in the extended Ehresman–Bruhat order (quantum case) instead of the paths in the ordinary Ehresman–Bruhat order (classical case). Very transparent proof of Monk’s formula one can find in the I. Macdonald book [M1], (4.15). It is the proof that was generalized in [FGP] to the case of quantum Schubert polynomials. Recently, F. Sotile [S] gave a proof of the Pieri rule based on geometrical approach.

Acknowledgement. We would like to acknowledge our special indebtedness to Dr. N.A. Liskova for the inestimable help in preparing the manuscript for publication.

Appendix A

Quantum double Schubert polynomials for S_{4}.

𝔖̃_{121321}(x, y) = Δ_{1}(y_{3} | x_{1})Δ_{2}(y_{2} | x_{1}, x_{2})Δ_{3}(y_{1} | x_{1}, x_{2}, x_{3}),
𝔖̃_{21321}(x, y) = q_{1}^{2}x_{1} + q_{1}q_{2}x_{1} − q_{2}x_{1}^{3} + 2q_{1}x_{1}^{2}x_{2} + x_{1}^{3}x_{2}^{2} + q_{1}x_{1}^{2}y_{1} − q_{2}x_{1}^{2}y_{1}
+ q_{1}x_{1}x_{2}y_{1} + x_{1}^{3}x_{2}y_{1} + x_{1}^{2}x_{2}^{2}y_{1} + q_{1}x_{1}y_{1}^{2} + x_{1}^{2}x_{2}y_{1}^{2} + q_{1}x_{1}^{2}y_{2}
− q_{2}x_{1}^{2}y_{2} + q_{1}x_{1}x_{2}y_{2} + x_{1}^{3}x_{2}y_{2} + x_{1}^{2}x_{2}^{2}y_{2} − q_{2}x_{1}y_{1}y_{2} + x_{1}^{3}y_{1}y_{2}
+ 2x_{1}^{2}x_{2}y_{1}y_{2} + x_{1}x_{2}^{2}y_{1}y_{2} + x_{1}^{2}y_{1}^{2}y_{2} + x_{1}x_{2}y_{1}^{2}y_{2} + q_{1}x_{1}y_{2}^{2}
+ x_{1}^{2}x_{2}y_{2}^{2} + x_{1}^{2}y_{1}y_{2}^{2} + x_{1}x_{2}y_{1}y_{2}^{2} + x_{1}y_{1}^{2}y_{2}^{2} + q_{1}^{2}y_{3} + q_{1}q_{2}y_{3}
− q_{2}x_{1}^{2}y_{3} + 2q_{1}x_{1}x_{2}y_{3} + x_{1}^{2}x_{2}^{2}y_{3} + q_{1}x_{1}y_{1}y_{3} − q_{2}x_{1}y_{1}y_{3}
+ q_{1}x_{2}y_{1}y_{3} + x_{1}^{2}x_{2}y_{1}y_{3} + x_{1}x_{2}^{2}y_{1}y_{3} + q_{1}y_{1}^{2}y_{3} + x_{1}x_{2}y_{1}^{2}y_{3}
+ q_{1}x_{1}y_{2}y_{3} − q_{2}x_{1}y_{2}y_{3} + q_{1}x_{2}y_{2}y_{3} + x_{1}^{2}x_{2}y_{2}y_{3} + x_{1}x_{2}^{2}y_{2}y_{3}
− q_{2}y_{1}y_{2}y_{3} + x_{1}^{2}y_{1}y_{2}y_{3} + 2x_{1}x_{2}y_{1}y_{2}y_{3} + x_{2}^{2}y_{1}y_{2}y_{3} + x_{1}y_{1}^{2}y_{2}y_{3}
+ x_{2}y_{1}^{2}y_{2}y_{3} + q_{1}y_{2}^{2}y_{3} + x_{1}x_{2}y_{2}^{2}y_{3} + x_{1}y_{1}y_{2}^{2}y_{3} + x_{2}y_{1}y_{2}^{2}y_{3}
+ y_{1}^{2}y_{2}^{2}y_{3},
𝔖̃_{12321}(x, y) = −q_{1}q_{2}x_{1} + q_{2}x_{1}^{3} − q_{1}^{2}x_{3} + q_{1}x_{1}^{2}x_{3} − q_{1}x_{1}x_{2}x_{3} + x_{1}^{3}x_{2}x_{3}
− q_{1}^{2}y_{1} − q_{1}q_{2}y_{1} + q_{1}x_{1}^{2}y_{1} + q_{2}x_{1}^{2}y_{1} − q_{1}x_{1}x_{2}y_{1} + x_{1}^{3}x_{2}y_{1}
− q_{1}x_{1}x_{3}y_{1} + x_{1}^{3}x_{3}y_{1} − q_{1}x_{2}x_{3}y_{1} + x_{1}^{2}x_{2}x_{3}y_{1} − q_{1}x_{1}y_{1}^{2} + x_{1}^{3}y_{1}^{2}
− q_{1}x_{2}y_{1}^{2} + x_{1}^{2}x_{2}y_{1}^{2} − q_{1}x_{3}y_{1}^{2} + x_{1}^{2}x_{3}y_{1}^{2} − q_{1}y_{1}^{3} + x_{1}^{2}y_{1}^{3} + q_{2}x_{1}^{2}y_{2}
+ q_{1}x_{1}x_{3}y_{2} + x_{1}^{2}x_{2}x_{3}y_{2} + q_{1}x_{1}y_{1}y_{2} + q_{2}x_{1}y_{1}y_{2} + x_{1}^{2}x_{2}y_{1}y_{2}
+ x_{1}^{2}x_{3}y_{1}y_{2} + x_{1}x_{2}x_{3}y_{1}y_{2} + x_{1}^{2}y_{1}^{2}y_{2} + x_{1}x_{2}y_{1}^{2}y_{2} + x_{1}x_{3}y_{1}^{2}y_{2}
+ x_{1}y_{1}^{3}y_{2} + q_{2}x_{1}^{2}y_{3} + q_{1}x_{1}x_{3}y_{3} + x_{1}^{2}x_{2}x_{3}y_{3} + q_{1}x_{1}y_{1}y_{3}
+ q_{2}x_{1}y_{1}y_{3} + x_{1}^{2}x_{2}y_{1}y_{3} + x_{1}^{2}x_{3}y_{1}y_{3} + x_{1}x_{2}x_{3}y_{1}y_{3} + x_{1}^{2}y_{1}^{2}y_{3}
+ x_{1}x_{2}y_{1}^{2}y_{3} + x_{1}x_{3}y_{1}^{2}y_{3} + x_{1}y_{1}^{3}y_{3} + q_{2}x_{1}y_{2}y_{3} + q_{1}x_{3}y_{2}y_{3}
+ x_{1}x_{2}x_{3}y_{2}y_{3} + q_{1}y_{1}y_{2}y_{3} + q_{2}y_{1}y_{2}y_{3} + x_{1}x_{2}y_{1}y_{2}y_{3}
+ x_{1}x_{3}y_{1}y_{2}y_{3} + x_{2}x_{3}y_{1}y_{2}y_{3} + x_{1}y_{1}^{2}y_{2}y_{3} + x_{2}y_{1}^{2}y_{2}y_{3}
+ x_{3}y_{1}^{2}y_{2}y_{3} + y_{1}^{3}y_{2}y_{3},
𝔖̃_{12132}(x, y) = q_{1}q_{2}x_{1} + q_{2}x_{1}^{2}x_{2} + q_{1}^{2}x_{3} + 2q_{1}x_{1}x_{2}x_{3} + x_{1}^{2}x_{2}^{2}x_{3} + q_{1}^{2}y_{1}
+ q_{1}q_{2}y_{1} + 2q_{1}x_{1}x_{2}y_{1} + q_{2}x_{1}x_{2}y_{1} + x_{1}^{2}x_{2}^{2}y_{1} + q_{1}x_{1}x_{3}y_{1}
+ q_{1}x_{2}x_{3}y_{1} + x_{1}^{2}x_{2}x_{3}y_{1} + x_{1}x_{2}^{2}x_{3}y_{1} + q_{1}x_{1}y_{1}^{2} + q_{1}x_{2}y_{1}^{2}
+ x_{1}^{2}x_{2}y_{1}^{2} + x_{1}x_{2}^{2}y_{1}^{2} + q_{1}x_{3}y_{1}^{2} + x_{1}x_{2}x_{3}y_{1}^{2} + q_{1}y_{1}^{3} + x_{1}x_{2}y_{1}^{3}

+ q_{2}x_{1}^{2}y_{2} + q_{2}x_{1}x_{2}y_{2} + q_{1}x_{1}x_{3}y_{2} + q_{1}x_{2}x_{3}y_{2} + x_{1}^{2}x_{2}x_{3}y_{2}
+ x_{1}x_{2}^{2}x_{3}y_{2} + q_{1}x_{1}y_{1}y_{2} + q_{2}x_{1}y_{1}y_{2} + q_{1}x_{2}y_{1}y_{2} + q_{2}x_{2}y_{1}y_{2}
+ x_{1}^{2}x_{2}y_{1}y_{2} + x_{1}x_{2}^{2}y_{1}y_{2} + x_{1}^{2}x_{3}y_{1}y_{2} + 2x_{1}x_{2}x_{3}y_{1}y_{2} + x_{2}^{2}x_{3}y_{1}y_{2}
+ x_{1}^{2}y_{1}^{2}y_{2} + 2x_{1}x_{2}y_{1}^{2}y_{2} + x_{2}^{2}y_{1}^{2}y_{2} + x_{1}x_{3}y_{1}^{2}y_{2} + x_{2}x_{3}y_{1}^{2}y_{2}
+ x_{1}y_{1}^{3}y_{2} + x_{2}y_{1}^{3}y_{2} + q_{2}x_{1}y_{2}^{2} + q_{1}x_{3}y_{2}^{2} + x_{1}x_{2}x_{3}y_{2}^{2} + q_{1}y_{1}y_{2}^{2}
+ q_{2}y_{1}y_{2}^{2} + x_{1}x_{2}y_{1}y_{2}^{2} + x_{1}x_{3}y_{1}y_{2}^{2} + x_{2}x_{3}y_{1}y_{2}^{2} + x_{1}y_{1}^{2}y_{2}^{2}
+ x_{2}y_{1}^{2}y_{2}^{2} + x_{3}y_{1}^{2}y_{2}^{2} + y_{1}^{3}y_{2}^{2},
𝔖̃_{1321}(x, y) = −q_{1}^{2} − q_{1}q_{2} + q_{1}x_{1}^{2} − q_{1}x_{1}x_{2} + x_{1}^{3}x_{2} − q_{1}x_{1}y_{1} + x_{1}^{3}y_{1}
− q_{1}x_{2}y_{1} + x_{1}^{2}x_{2}y_{1} − q_{1}y_{1}^{2} + x_{1}^{2}y_{1}^{2} + q_{1}x_{1}y_{2} + x_{1}^{2}x_{2}y_{2}
+ x_{1}^{2}y_{1}y_{2} + x_{1}x_{2}y_{1}y_{2} + x_{1}y_{1}^{2}y_{2} + q_{1}x_{1}y_{3} + x_{1}^{2}x_{2}y_{3} + x_{1}^{2}y_{1}y_{3}
+ x_{1}x_{2}y_{1}y_{3} + x_{1}y_{1}^{2}y_{3} + q_{1}y_{2}y_{3} + x_{1}x_{2}y_{2}y_{3} + x_{1}y_{1}y_{2}y_{3}
+ x_{2}y_{1}y_{2}y_{3} + y_{1}^{2}y_{2}y_{3},
𝔖̃_{2321}(x, y) = −q_{1}^{2} − q_{1}q_{2} + q_{1}x_{1}^{2} − q_{1}x_{1}x_{2} + x_{1}^{3}x_{2} − 2q_{1}x_{1}x_{3} + x_{1}^{3}x_{3}
− q_{1}x_{2}x_{3} − q_{1}x_{1}y_{1} + x_{1}^{3}y_{1} − q_{1}x_{2}y_{1} + x_{1}^{2}x_{2}y_{1} − q_{1}x_{3}y_{1}
+ x_{1}^{2}x_{3}y_{1} − q_{1}y_{1}^{2} + x_{1}^{2}y_{1}^{2} − q_{1}x_{1}y_{2} + x_{1}^{3}y_{2} − q_{1}x_{2}y_{2} + x_{1}^{2}x_{2}y_{2}
− q_{1}x_{3}y_{2} + x_{1}^{2}x_{3}y_{2} − q_{1}y_{1}y_{2} + 2x_{1}^{2}y_{1}y_{2} + x_{1}x_{2}y_{1}y_{2} + x_{1}x_{3}y_{1}y_{2}
+ x_{1}y_{1}^{2}y_{2} − q_{1}y_{2}^{2} + x_{1}^{2}y_{2}^{2} + x_{1}y_{1}y_{2}^{2} + q_{1}x_{1}y_{3} + x_{1}^{2}x_{2}y_{3} − q_{1}x_{3}y_{3}
+ x_{1}^{2}x_{3}y_{3} + x_{1}^{2}y_{1}y_{3} + x_{1}x_{2}y_{1}y_{3} + x_{1}x_{3}y_{1}y_{3} + x_{1}y_{1}^{2}y_{3}
+ x_{1}^{2}y_{2}y_{3} + x_{1}x_{2}y_{2}y_{3} + x_{1}x_{3}y_{2}y_{3} + 2x_{1}y_{1}y_{2}y_{3} + x_{2}y_{1}y_{2}y_{3}
+ x_{3}y_{1}y_{2}y_{3} + y_{1}^{2}y_{2}y_{3} + x_{1}y_{2}^{2}y_{3} + y_{1}y_{2}^{2}y_{3},
𝔖̃_{2132}(x, y) = q_{1}^{2} + q_{1}q_{2} − q_{2}x_{1}^{2} + 2q_{1}x_{1}x_{2} + x_{1}^{2}x_{2}^{2} + q_{1}x_{1}y_{1} − q_{2}x_{1}y_{1}
+ q_{1}x_{2}y_{1} + x_{1}^{2}x_{2}y_{1} + x_{1}x_{2}^{2}y_{1} + q_{1}y_{1}^{2} + x_{1}x_{2}y_{1}^{2} + q_{1}x_{1}y_{2}
− q_{2}x_{1}y_{2} + q_{1}x_{2}y_{2} + x_{1}^{2}x_{2}y_{2} + x_{1}x_{2}^{2}y_{2} − q_{2}y_{1}y_{2} + x_{1}^{2}y_{1}y_{2}
+ 2x_{1}x_{2}y_{1}y_{2} + x_{2}^{2}y_{1}y_{2} + x_{1}y_{1}^{2}y_{2} + x_{2}y_{1}^{2}y_{2} + q_{1}y_{2}^{2} + x_{1}x_{2}y_{2}^{2}
+ x_{1}y_{1}y_{2}^{2} + x_{2}y_{1}y_{2}^{2} + y_{1}^{2}y_{2}^{2},
𝔖̃_{1213}(x, y) = q_{2}x_{1}^{2} + q_{1}x_{1}x_{3} + x_{1}^{2}x_{2}x_{3} + q_{1}x_{1}y_{1} + q_{2}x_{1}y_{1} + x_{1}^{2}x_{2}y_{1}
+ x_{1}^{2}x_{3}y_{1} + x_{1}x_{2}x_{3}y_{1} + x_{1}^{2}y_{1}^{2} + x_{1}x_{2}y_{1}^{2} + x_{1}x_{3}y_{1}^{2} + x_{1}y_{1}^{3}
+ q_{2}x_{1}y_{2} + q_{1}x_{3}y_{2} + x_{1}x_{2}x_{3}y_{2} + q_{1}y_{1}y_{2} + q_{2}y_{1}y_{2} + x_{1}x_{2}y_{1}y_{2}
+ x_{1}x_{3}y_{1}y_{2} + x_{2}x_{3}y_{1}y_{2} + x_{1}y_{1}^{2}y_{2} + x_{2}y_{1}^{2}y_{2} + x_{3}y_{1}^{2}y_{2} + y_{1}^{3}y_{2},
𝔖̃_{1232}(x, y) = q_{2}x_{1}^{2} + q_{2}x_{1}x_{2} + q_{1}x_{1}x_{3} + q_{1}x_{2}x_{3} + x_{1}^{2}x_{2}x_{3} + x_{1}x_{2}^{2}x_{3}

+ q_{1}x_{1}y_{1} + q_{2}x_{1}y_{1} + q_{1}x_{2}y_{1} + q_{2}x_{2}y_{1} + x_{1}^{2}x_{2}y_{1} + x_{1}x_{2}^{2}y_{1}
+ x_{1}^{2}x_{3}y_{1} + 2x_{1}x_{2}x_{3}y_{1} + x_{2}^{2}x_{3}y_{1} + x_{1}^{2}y_{1}^{2} + 2x_{1}x_{2}y_{1}^{2} + x_{2}^{2}y_{1}^{2}
+ x_{1}x_{3}y_{1}^{2} + x_{2}x_{3}y_{1}^{2} + x_{1}y_{1}^{3} + x_{2}y_{1}^{3} + q_{2}x_{1}y_{2} + q_{1}x_{3}y_{2}
+ x_{1}x_{2}x_{3}y_{2} + q_{1}y_{1}y_{2} + q_{2}y_{1}y_{2} + x_{1}x_{2}y_{1}y_{2} + x_{1}x_{3}y_{1}y_{2}
+ x_{2}x_{3}y_{1}y_{2} + x_{1}y_{1}^{2}y_{2} + x_{2}y_{1}^{2}y_{2} + x_{3}y_{1}^{2}y_{2} + y_{1}^{3}y_{2} + q_{2}x_{1}y_{3}
+ q_{1}x_{3}y_{3} + x_{1}x_{2}x_{3}y_{3} + q_{1}y_{1}y_{3} + q_{2}y_{1}y_{3} + x_{1}x_{2}y_{1}y_{3} +
+ x_{1}x_{3}y_{1}y_{3} + x_{2}x_{3}y_{1}y_{3} + x_{1}y_{1}^{2}y_{3} + x_{2}y_{1}^{2}y_{3} + x_{3}y_{1}^{2}y_{3} + y_{1}^{3}y_{3},
𝔖̃_{121}(x, y) = q_{1}x_{1} + x_{1}^{2}x_{2} + x_{1}^{2}y_{1} + x_{1}x_{2}y_{1} + x_{1}y_{1}^{2} + q_{1}y_{2} + x_{1}x_{2}y_{2}
+ x_{1}y_{1}y_{2} + x_{2}y_{1}y_{2} + y_{1}^{2}y_{2},
𝔖̃_{132}(x, y) = q_{1}x_{1} − q_{2}x_{1} + q_{1}x_{2} + x_{1}^{2}x_{2} + x_{1}x_{2}^{2} − q_{2}y_{1} + x_{1}^{2}y_{1} + 2x_{1}x_{2}y_{1}
+ x_{2}^{2}y_{1} + x_{1}y_{1}^{2} + x_{2}y_{1}^{2} + q_{1}y_{2} + x_{1}x_{2}y_{2} + x_{1}y_{1}y_{2} + x_{2}y_{1}y_{2}
+ y_{1}^{2}y_{2} + q_{1}y_{3} + x_{1}x_{2}y_{3} + x_{1}y_{1}y_{3} + x_{2}y_{1}y_{3} + y_{1}^{2}y_{3},
𝔖̃_{232}(x, y) = q_{1}x_{1} + q_{1}x_{2} + q_{2}x_{2} + x_{1}^{2}x_{2} + x_{1}x_{2}^{2} − q_{1}x_{3} + x_{1}^{2}x_{3} + x_{1}x_{2}x_{3}
+ x_{2}^{2}x_{3} + x_{1}^{2}y_{1} + 2x_{1}x_{2}y_{1} + x_{2}^{2}y_{1} + x_{1}x_{3}y_{1} + x_{2}x_{3}y_{1} + x_{1}y_{1}^{2}
+ x_{2}y_{1}^{2} + x_{1}^{2}y_{2} + 2x_{1}x_{2}y_{2} + x_{2}^{2}y_{2} + x_{1}x_{3}y_{2} + x_{2}x_{3}y_{2} + 2x_{1}y_{1}y_{2}
+ 2x_{2}y_{1}y_{2} + x_{3}y_{1}y_{2} + y_{1}^{2}y_{2} + x_{1}y_{2}^{2} + x_{2}y_{2}^{2} + y_{1}y_{2}^{2} + q_{1}y_{3} + q_{2}y_{3}
+ x_{1}x_{2}y_{3} + x_{1}x_{3}y_{3} + x_{2}x_{3}y_{3} + x_{1}y_{1}y_{3} + x_{2}y_{1}y_{3} + x_{3}y_{1}y_{3}
+ y_{1}^{2}y_{3} + x_{1}y_{2}y_{3} + x_{2}y_{2}y_{3} + x_{3}y_{2}y_{3} + y_{1}y_{2}y_{3} + y_{2}^{2}y_{3},
𝔖̃_{123}(x, y) = q_{2}x_{1} + q_{1}x_{3} + x_{1}x_{2}x_{3} + q_{1}y_{1} + q_{2}y_{1} + x_{1}x_{2}y_{1} + x_{1}x_{3}y_{1}
+ x_{2}x_{3}y_{1} + x_{1}y_{1}^{2} + x_{2}y_{1}^{2} + x_{3}y_{1}^{2} + y_{1}^{3},
𝔖̃_{213}(x, y) = q_{1}x_{1} + x_{1}^{2}x_{2} − q_{1}x_{3} + x_{1}^{2}x_{3} + x_{1}^{2}y_{1} + x_{1}x_{2}y_{1} + x_{1}x_{3}y_{1} + x_{1}y_{1}^{2}
+ x_{1}^{2}y_{2} + x_{1}x_{2}y_{2} + x_{1}x_{3}y_{2} + 2x_{1}y_{1}y_{2} + x_{2}y_{1}y_{2} + x_{3}y_{1}y_{2} + y_{1}^{2}y_{2}
+ x_{1}y_{2}^{2} + y_{1}y_{2}^{2},
𝔖̃_{321}(x, y) = −2q_{1}x_{1} + x_{1}^{3} − q_{1}x_{2} − q_{1}y_{1} + x_{1}^{2}y_{1} − q_{1}y_{2} + x_{1}^{2}y_{2} + x_{1}y_{1}y_{2}
− q_{1}y_{3} + x_{1}^{2}y_{3} + x_{1}y_{1}y_{3} + x_{1}y_{2}y_{3} + y_{1}y_{2}y_{3},
𝔖̃_{23}(x, y) = q_{1} + q_{2} + x_{1}x_{2} + x_{1}x_{3} + x_{2}x_{3} + x_{1}y_{1} + x_{2}y_{1} + x_{3}y_{1} + y_{1}^{2}
+ x_{1}y_{2} + x_{2}y_{2} + x_{3}y_{2} + y_{1}y_{2} + y_{2}^{2},
𝔖̃_{32}(x, y) = −q_{1} − q_{2} + x_{1}^{2} + x_{1}x_{2} + x_{2}^{2} + x_{1}y_{1} + x_{2}y_{1} + x_{1}y_{2} + x_{2}y_{2}
+ y_{1}y_{2} + x_{1}y_{3} + x_{2}y_{3} + y_{1}y_{3} + y_{2}y_{3},
𝔖̃_{13}(x, y) = x_{1}^{2} + x_{1}x_{2} + x_{1}x_{3} + 2x_{1}y_{1} + x_{2}y_{1} + x_{3}y_{1} + y_{1}^{2} + x_{1}y_{2} + y_{1}y_{2}

+ x_{1}y_{3} + y_{1}y_{3},
𝔖̃_{12}(x, y) = q_{1} + x_{1}x_{2} + x_{1}y_{1} + x_{2}y_{1} + y_{1}^{2},
𝔖̃_{21}(x, y) = −q_{1} + x_{1}^{2} + x_{1}y_{1} + x_{1}y_{2} + y_{1}y_{2},
𝔖̃_{3}(x, y) = x_{1} + x_{2} + x_{3} + y_{1} + y_{2} + y_{3},
𝔖̃_{2}(x, y) = x_{1} + x_{2} + y_{1} + y_{2},
𝔖̃_{1}(x, y) = x_{1} + y_{1},
𝔖̃_{id}(x, y) = 1.

Appendix B

Quantum residue generating function Ψ(t) for S_{3}.

We define the functions g_{v}(t_{1}) and h_{v}(t_{2}) by the formulas

\frac{t_{1}}{t_{1} − x_{1}} = ∑_{v∈S_{3}} g_{v}(t_{1})S_{v}^{q},

\frac{t_{2}}{t_{2} − x_{2}} = ∑_{v∈S_{3}} h_{v}(t_{2})S_{v}^{q}

in the quantum cohomology ring QH^{∗}(F l_{3}). Then, we have

⟨\frac{t_{1}}{t_{1} − x_{1}} \frac{t_{2}}{t_{2} − x_{2}} ⟩ = ∑_{v∈S_{3}} g_{v}(t_{1})h_{vw_{0}}(t_{2}).

Since ⟨x_{1}^{5}⟩ = q_{1}, x_{2} = q_{1}^{−1}x_{1}^{3} − 2x_{1} and

QH^{∗}(F l_{3}) ≃ Z[q_{1}, q_{2}][x_{1}]/(f_{1}(x_{1})),

the correlation function ⟨P (x_{1}, x_{2})⟩ is expressed as

⟨P (x_{1}, x_{2})⟩ = q_{1}Res_{f_{1}}P (x_{1}, q_{1}^{−1}x_{1}^{3} − 2x_{1}) = q_{1} ∑_{f_{1}(μ)=0} \frac{1}{f′_{1}(μ)}P (μ, q_{1}^{−1}μ^{3} − 2μ).

Similarly, it also holds that

⟨P (x_{1}, x_{2})⟩ = (q_{2} − q_{2}) ∑_{f_{2}(μ)=0} \frac{1}{f′_{2}(μ)}P ((q_{1} − q_{2})^{−1}(μ^{3} − (2q_{1} + q_{2})μ), μ).

The functions g_{v} and h_{v} are given as follows:

f_{1}(t_{1})g_{121}(t_{1}) = q_{1}t_{1},
f_{1}(t_{1})g_{12}(t_{1}) = q_{1}t_{1}^{2},
f_{1}(t_{1})g_{21}(t_{1}) = t_{1}^{2}(t_{1}^{2} − q_{1}),
f_{1}(t_{1})g_{2}(t_{1}) = q_{1}t_{1}(t_{1}^{2} − q_{1}),
f_{1}(t_{1})g_{1}(t_{1}) = t_{1}(t_{1}^{4} − q_{1}t_{1}^{2} + q_{1}^{2}),
f_{1}(t_{1})g_{id}(t_{1}) = t_{1}^{2}(t_{1}^{4} − q_{1}t_{1}^{2} + q_{1}^{2}),
f_{2}(t_{2})h_{121}(t_{2}) = (q_{2} − q_{1})t_{2},
f_{2}(t_{2})h_{12}(t_{2}) = t_{2}^{2}(2q_{1} + q_{2} − t_{2}^{2}),
f_{2}(t_{2})h_{21}(t_{2}) = t_{2}^{2}(q_{1} + 2q_{2} − t_{2}^{2}),
f_{2}(t_{2})h_{2}(t_{2}) = \frac{(q_{1} + q_{2})t_{2}^{5} − q_{1}(q_{1} + q_{2})t_{2}^{3} + q_{1}q_{2}(2q_{2} − q_{1})t_{2}}{(q_{1} − q_{2})} ,
f_{2}(t_{2})h_{1}(t_{2}) = −t_{2}^{5} + (2q_{1} + q_{2})t_{2}^{3} + q_{1}(q_{2} − q_{1})t_{2},
f_{2}(t_{2})h_{id}(t_{2}) = \frac{(q_{1} + q_{2})t_{2}^{6} − 2q_{1}t_{2}^{2} + (q_{1}^{3} + 2q_{1}^{2}q_{2} − q_{2}^{3})t_{2}^{2}}{(q_{1} − q_{2})} .

The characteristic polynomials f_{1} and f_{2} are given by

f_{1}(t) = (t^{2} − q_{1})^{3} − q_{1}^{2}q_{2},

f_{2}(t) = t^{6} − 3(q_{1} + q_{2})t^{4} + 3(q_{1}^{2} + q_{1}q_{2} + q_{2}^{2})t^{2} − q_{1}^{3} + q_{1}^{2}q_{2} + q_{1}q_{2}^{2} − q_{2}^{3}. Hence, we have

D_{1} = \frac{1}{720t_{1}} f_{1}(t_{1})\frac{∂^{6}}{∂t_{1}^{6}} + (\frac{q_{1}^{3} + q_{1}^{2}q_{2}}{120t_{1}^{2}} + \frac{1}{40}q_{1}^{2} − \frac{3}{40}q_{1}t_{1}^{2} + \frac{1}{24}t_{1}^{4}) \frac{∂^{5}}{∂t_{1}^{5}}
+ (−\frac{q_{1}^{3} + q_{1}^{2}q_{2}}{24t_{1}^{3}} − \frac{3}{8}q_{1}t_{1} + \frac{5}{12}t_{1}^{3}) \frac{∂^{4}}{∂t_{1}^{4}} + (\frac{q_{1}^{3} + q_{1}^{2}q_{2}}{6t_{1}^{4}} − \frac{1}{2}q_{1} + \frac{5}{3}t_{1}^{2}) \frac{∂^{3}}{∂t_{1}^{3}}
+ (−\frac{q_{1}^{3} + q_{1}^{2}q_{2}}{2t_{1}^{5}} + \frac{5}{2}t_{1}) \frac{∂^{2}}{∂t_{1}^{2}} + (\frac{q_{1}^{3} + q_{1}^{2}q_{2}}{t_{1}^{6}} + 1) \frac{∂}{∂t_{1}} − \frac{q_{1}^{3} + q_{1}^{2}q_{2}}{t_{1}^{7}} ,
D_{2} = \frac{f_{2}(t_{2})}{720t_{2}} \frac{∂^{6}}{∂t_{2}^{6}}
+ (\frac{q_{1}^{3} − q_{1}^{2}q_{2} − q_{1}q_{2}^{2} + q_{2}^{3}}{120t_{2}^{2}} + \frac{q_{1}^{2} + q_{1}q_{2} + q_{2}^{2}}{40} + \frac{3}{40}(q_{1} + q_{2})t_{2}^{2} + \frac{t_{2}^{4}}{24}) \frac{∂^{5}}{∂t_{2}^{5}}

+ (−\frac{q_{1}^{3} − q_{1}^{2}q_{2} − q_{1}q_{2}^{2} + q_{2}^{3}}{24t_{2}^{3}} − \frac{3}{8}(q_{1} + q_{2})t_{2} + \frac{5}{12}t_{2}^{3}) \frac{∂^{4}}{∂t_{2}^{4}}
+ (\frac{q_{1}^{3} − q_{1}^{2}q_{2} − q_{1}q_{2}^{2} + q_{2}^{3}}{6t_{2}^{4}} − \frac{1}{2}(q_{1} + q_{2}) + \frac{5}{3}t_{2}^{2}) \frac{∂^{3}}{∂t_{2}^{3}}
+ (−\frac{q_{1}^{3} − q_{1}^{2}q_{2} − q_{1}q_{2}^{2} + q_{2}^{3}}{2t_{2}^{5}} + \frac{5}{2}t_{2}) \frac{∂^{2}}{∂t_{2}^{2}}
+ (\frac{q_{1}^{3} − q_{1}^{2}q_{2} − q_{1}q_{2}^{2} + q_{2}^{3}}{t_{2}^{6}} + 1) \frac{∂}{∂t_{2}} − \frac{q_{1}^{3} − q_{1}^{2}q_{2} − q_{1}q_{2}^{2} + q_{2}^{3}}{t_{2}^{7}} .

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