数理解析研究所講究録1086 巻 1999 年 17-25

17Quantum double Schubert polynomials, quantum
Schubert polynomials and Vafa-Intriligator formula(joint work with Anatol N. Kirillov)
京大理学研究科前野俊昭 (Toshiaki Maeno)
Introduction.
of typeis isomorphic to theThe cohomology ring of flag variety
by the ideal generated
quotient ring of the polynomial ringby symmetric polynomials. It is well-known that the dual classes of theSchubert cycles are represented by so-called Schubert polynomials in thecohomology ring. On the other hand, physicists introduced another ringstructure on the cohomology group, which is called quantum cohomology
ring. The quantum cohomology ring of the flag variety also has a structure. In theof the quotient ring of the polynomial ring
quantum cohomology ring of flag variety, the Schubert polynomials do not
represent the Schubert classes any more. Fomin, Gelfand and Postnikov[FGP] first introduced the quantum Schubert polynomials, which correspond to the Schubert classes in the quantum cohomology ring. They gave
a commuting family of operators acting on the polynomial ring and thequantization of polynomials. In this note, we define the quantum Schubert polynomials based on the orthogonality of the Schubert classes withrespect to the intersection pairing. We also introduce the quantum double
Schubert polynomials and see that they satisfy an analogue of the Cauchy
formula. As applications of quantum Cauchy formula, we consider theVafa-Intriligator type formula and the quantization map. For details, see
Kirillov and Maeno [KM1].
$Fl_{n}$$A_{n-1}$$\mathrm{Z}[x_{1}, \ldots , x_{n}]$$\mathrm{Z}[x_{1}, \ldots , x_{n}]([\mathrm{C}],[\mathrm{G}\mathrm{K}])$

181

Cohomology ring of flag variety and Schubert polynomials.

First of all, we make a brief review of the Schubert polynomials. Moredetails are found in Macdonald [M]. Let $0=E_{0}\subset E_{1}\subset\cdots\subset E_{n}=$be the universal flag of subbundles over $F=Fl_{n}$ . Then thecohomology ringis isomorphic to the quotient ring
, whereand is the ideal generated by symmetric polynomials,
and the natural identification is given by$\mathrm{C}^{n}\otimes O_{F}$$H^{*}(F, \mathrm{Z})$$P_{n}=\mathrm{Z}[x_{1}, \ldots , x_{n}]$$P_{n}/I_{n}$$I_{n}$$x_{i}\vdash+c_{1}(E_{i}/E_{i-1}),$$i=1,$$\ldots,$$n$.

Let us consider the natural sequence of quotient bundles$E_{n}=L_{n}arrow\cdotsarrow L_{1}arrow 0$,

where $L_{i}=E_{n}/E_{n-i}$ on $F$. Now we fix a flag
$0=V_{0}\subset V_{1}\subset\cdots\subset V_{n}=\mathrm{C}^{n}$Then we have induced morphisms$f_{pq}$:$V_{p}\otimes \mathcal{O}_{F}arrow L_{q}$for $1\leq p,$ $q\leq n$ .

Definition 1.1. For a permutation $w\in S_{n}$ , the Schubert cycleis thelocus where rank $f_{pq}\leq r_{w}(q,p)$ for any $1\leq p,$ $q\leq n-1$ , where $r_{w}(q,p)=$$\#\{i|i\leq q, w(i)\leq p\}$ .
It is easy to see that the dual class
where $l(w)$ is thelength of and that the Schubert cycles give an orthonormal basis of ofthe cohomology group with respect to the intersection pairing, namely$\Omega_{w}$$[\Omega_{w}]\in H^{2l(w)}(F, \mathrm{Z})$$w$$\langle\Omega_{u}, \Omega_{v}\rangle=\delta_{v},w0u$’

whereis the permutation of maximal length.
Next we consider the Schubert polynomials which represent the Schubert classes in the cohomology ring. We introduce the divided difference
operators, $n-1$ acting on the polynomial ringas follows:$w_{0}\in S_{n}$$\partial_{i}i=1,$$\ldots$$P_{n}$$( \partial_{i}f)(_{X})=\frac{f(x_{1},\ldots,x_{n})-f(x1\cdots,xi+1,Xi,\ldots Xn)}{x_{i}-X_{i1}+},$.

19

We choose a permutation $w\in S_{n}$ and its reduced decomposition$w=s_{a_{1}}\cdots s_{a_{p}}$where$s_{i}$,

is a simple transposition $(i, i+1)$ and $p=l(w)$ . Then the operator$\partial_{wa_{1}a_{p}}=\partial\cdots\partial$

does not depend on the choice of reduced decomposition ofand Sch\"utzenberger [LS1]).
Definition 1.2(For each permutation $w\in S_{n}$ , the Schubert polynomial$w$.

$\mathrm{L}\mathrm{a}\mathrm{S}\mathrm{C}\mathrm{o}\mathrm{u}\mathrm{x}$$\mathfrak{S}_{w}(x)$is defined to$be$$\mathfrak{S}_{w}(x)=\partial_{w}-1(W_{0}X)\delta$

where$\delta$,

is a multi-index $(n-1, n-2, \ldots , 0)$ .

. Moreover, the intersection pairin
lt is known thatis naturally identified with the induced pairing from theing on
residue$f,$ $g\in P_{n}$ ,$\mathfrak{S}_{w}(x)=\Omega_{w}$$H^{*}(F, \mathrm{Z})$$H^{*}(F, \mathrm{Z})$$\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}-$$\langle f, g\rangle_{I_{n}}=\mathrm{R}\mathrm{e}\mathrm{s}I_{n}(fg),$. Hence, the Schubert polyis the Grothendieck residue
wherenomials are orthonormal with respect to the pairing , . Conversely, theorthogonality characterizes the Schubert polynomials.
$([\mathrm{G}\mathrm{H}])$${\rm Res}_{I_{n}}$$\langle$$\rangle$

Proposition 1.3. The Schubert polynomials are uniquely characterized by
the following properties.

.
1.
,
2.
and multi-indices $I=$
where $c(w)=(c_{1}(w), \ldots , c_{n}(w))$ is the code of
run over $I\subset\delta(i.e0\leq i_{k}\leq n-k)$ and I degree-lexicographicallysmaller than $c(w)$ .
$\langle \mathfrak{S}_{u}, \mathfrak{S}_{v}\rangle=\delta u,w_{0}v$$\mathfrak{S}_{w}(x)=X(w)+c\Sigma\alpha Ix^{I}$$w$$(i_{1}, \ldots , i_{n})$Remarks. 1) (Definition of the code)
For a permutation $w\in S_{n}$ , we define$([\mathrm{M}], \mathrm{p}.9)$.

$c_{i}(w)=\#\{j|i<j, w(i)>w(j)\}$ .

20

The sequence $c(w)=(c_{1}(w), \ldots , c_{n}(w))$ is called the code of .
2) The Schubert polynomials are obtained as Gram-Schmidt’s orthogonalization of the monomials
ordered degree-lexicographically.
$w$$(x^{I})_{I\subset\delta}$

2

Quantum Schubert polynomials and quantum double Schubert plynomials

Quantum cohomology ring is a deformed ring of the ordinary cohomology
ring and its structure constants are given by the Gromov-Witten invariants. For general definition and properties of quantum cohomology ring,see [KM2], [MS] and [RT]. Let us remind the structure of quantum cohomology ring of flag variety.

Theorem 2.1(Givental and Kim [GK], Ciocan-Fontanine [C]).
The quantum cohomology ring of flag variety $F$ is generated by $x_{i}=$$c_{1}(E_{i}/E_{i-1}),$ $i=1,$, , as a-algebra and$\ldots$$n$$\mathrm{Z}[q_{1}, \ldots , q_{n-1}]$$QH^{*}(F)\cong \mathrm{z}[x_{1}, \ldots, x_{n}, q_{1}, \ldots, qn-1]/\tilde{I}_{n}$,where the ideal is generated by the quantum elementary symmetric functions $e_{i}(\sim x):=e_{i}(x|q)$ , with generating function$\tilde{I}_{n}$$\triangle_{n}(t|_{X)}$$:=$$\sum_{i=1}^{n}e_{i}(x|q)t^{i}=\det$

We define a pairing onby$\overline{P}_{n}=\mathrm{Z}[x_{1}, \ldots , x_{n}, q_{1}, \ldots , q_{n-1}]$$\mathrm{Z}[q_{1}, \ldots, qn-1]$$\langle f, g\rangle_{Q}={\rm Res}_{\tilde{I}n}(fg),$$f,$ $g\in\overline{P}_{n}$.

with values in21

This pairing induces a nondegenerate pairing on $QH^{*}(F)$ .
Definition 2.2([KM1], Definition 5).
as Gram-Schmidt’s orWe define the quantum Schubert polynomials
with respect to thethogonalization of the classical Schubert polynomials$\tilde{\mathfrak{S}}_{w}$$\mathfrak{S}_{w}$pairing1)
2)
whereorder.

$\langle$,:$\rangle_{Q}$$\langle\tilde{\mathfrak{S}}_{u},\tilde{\mathfrak{S}}_{v}\rangle Q=\delta v,w_{0}u$’$\tilde{\mathfrak{S}}_{w}(x)=x(w)+c\Sigma_{I}<C(w)aI(q)x^{I}$$a_{I}(q)\in \mathrm{Z}[q_{1}, \ldots , q_{n-1}]$,

and $I<c(w)$ means the degree-lexicographic

definition is the analogue of the characterization ofRemarks. )Schubert polynomials in Proposition 1.3.
2) The polynomials obtained by the Gram-Schmidt type orthogonalization. However,are, a priori, defined over the fleld of fractions offrom Theorem 3.1.
it turns out that they are defined over
Next we introduce quantum double Schubert polynomials.
$1$$\mathrm{T}\mathrm{h}\mathrm{i}\mathrm{s}$$\mathrm{Z}[q_{1}, \ldots , q_{n-1}]$$\mathrm{Z}[q_{1}, \ldots , q_{n-1}]$

Definition 2.3([KM1], Definition 4).
Let $x=$$(x_{1}, \ldots , x_{n}),$$y=(y_{1}, \ldots , y_{n})$be two sets of variables, put$\tilde{\mathfrak{S}}_{w_{0}}(x, y):=\tilde{\mathfrak{S}}_{w0}((q)x, y)=\prod_{i=1}^{n-}\triangle i(yn-1i|X_{1}, \ldots, X_{i})$.

For each permutation $w\in S_{n}$ , the quantum double Schubert polynomial isdefined to be,$\tilde{\mathfrak{S}}_{w}(x, y)=\partial_{www}(y)(0^{\tilde{\mathfrak{S}}}0)x,$$y$

acts on the variables.
where divided difference operator
Remarks. 1) If we put $q_{1}=\cdots=qn-1=0$ in the quantum double
Schubert polynomial, we get the classical one, namely$y$$\partial_{ww_{0}}^{(y)}$$\tilde{\mathfrak{S}}_{w}(x, y)|_{q=0^{=\mathfrak{S}}w}(x, y)$where the classical double Schubert polynomial:,$\mathfrak{S}_{w}(x, y)$$([\mathrm{L}\mathrm{S}2])$$\mathfrak{S}_{w_{0}}(x, y)=i+i\prod_{\leq n}(_{X+}iy_{j})$,defined as follows

22$\mathfrak{S}_{w}(x, y)=\partial_{w}^{()}y\mathfrak{S}w0(w_{0}x, y)$.

2) The quantum double Schubert polynomials are stable under the naturalembedding $i:S_{n}arrow S_{m}$ for $m>n$ :$\mathfrak{S}_{w}(x, y)=\mathfrak{S}_{i}(w)(x, y)$3.

Quantum Cauchy formula and its applications.

Theorem 3.1 ([KM1], Theorem 3). The following analogue of the Cauchy
formula holds for the quantum double Schubert polynomial:$\tilde{\mathfrak{S}}_{W_{0}}(x, y)=\sum\tilde{\mathfrak{S}}_{w}(xw\in Sn)\mathfrak{S}ww0(y)$For multi-index $I=$
elementary polynomials$(i_{1}, \ldots , i_{n})\subset\delta$$e_{I}(x)$and.

, the elementary and the quantum$\tilde{e}_{I}(x)$are defined to be$e_{I}(x)= \prod_{k=1}^{n-}1e_{i_{k}}(_{Xx_{n-k}}1, \ldots,)$,$\tilde{e}_{I}(x)=\prod_{k=1}^{n-}1\tilde{e}_{i_{k}}(X_{1}, \ldots, x_{n-k})$.

The theorem above is equivalent to that$\langle\tilde{e}_{I}(X),\tilde{\mathfrak{S}}w(_{X})\rangle Q=\langle eI(x), \mathfrak{S}_{w}(x)\rangle$,

for any permutation $w\in S_{n}$ . This means that
represents in the quantum cohomology ring the same class that
represents in the classicalcohomology ring. We can prove this fact geometrically. The proof can befound in [KM1], Section 7.
$\tilde{e}_{I}(x)$$e_{I}(x)$Corollary 3.2. The quantum Schubert polynomial
by the formula$\tilde{\mathfrak{S}}_{w}(X)=(\partial_{ww0}^{(y)}\mathfrak{S}_{w_{0}}(x, y))|_{y=0}$$\tilde{\mathfrak{S}}_{w}(x)$can be calculated.

In particular, the quantum Schubert polynomials are stable under the natural embedding$m>n$ .
$S_{n}arrow S_{m},$23

is obtained as

Corollary 3.3. The quantum Schubert polynomiala product of quantum elementary symmetric polynomials$\tilde{\mathfrak{S}}_{w_{0}}(x)$.

$\tilde{\mathrm{e}}_{w_{0}}(x)=\tilde{e}_{1}(x_{1})\tilde{e}2(X_{1}, x2)\cdots\tilde{e}_{n-1}(X1, \ldots, x_{n-1})$

For a polynomial $f(x)\in P_{n}$ , we denote by $[f(x)]$ the corresponding class
in $QH^{*}(F)$ . The correlation function (of genus zero) is given by$\langle f(x)\rangle=\int_{F}[f(x)]$.

This correlation function is a generating function of Gromov-Witten invariants of genus zero. By using genus invariants instead of genus zeroinvariants, we can define the correlation function of genus , which is de. For Gromov-Witten invariants of higher genus, seenoted byorder to calculate the higher genus correlation function, weand [RT].
$g$

$g$

$[\mathrm{I}<\mathrm{M}2]$$\langle f(x)\rangle_{g}$$\ln$introduce a polynomial$C^{(q,q)};(X, y)= \sum_{\in ws_{n}}\tilde{\mathfrak{S}}((wX)q)\tilde{\mathfrak{S}}_{w_{0}w}((q’))y$.

From Theorem 3.1, we have$c^{(q,q’)}(x, y)=\langle\tilde{\mathfrak{S}}((w0x, z)q),(q’0\tilde{\mathfrak{S}}(wy), z)\rangle^{(z})$,

is the classical pairing with respect to the variables . The
where ,
higher genus correlation functiton can be calculated by the following VafaIntriligator type formula.
$\langle$$z$$\rangle^{(z)}$Theorem 3.4.([KM1], Theorem 10).
Let $\Phi(x)=C^{(q,q)}(X, X)$ . Then we have$\langle f(x)\rangle_{g}={\rm Res}(\overline{I}nf(_{X)\Phi}(X)^{g})$.

As another application of the quantum Cauchy formula, we consider the. Let $f(x)\in P_{\infty}$ , then we havequantization map on the polynomial ringthe interpolation formula$P_{\infty}$$f(x)= \sum_{w\in S^{(n}})\mathfrak{S}_{w}(X, y)\partial(y)fw(y)$,24whereequal to$S^{(n)}$$n$

is the set of permutations whose codes have length less than or. We define the quantization of a polynomial $f(x)\in P_{\infty}$ by$y))\partial_{w}^{(y)_{f}}(y)$$\tilde{f}(x)=\sum_{w\in s(n}\tilde{\mathfrak{S}}_{w}(x,$,and obtain a linear map. If we put $x_{n+1}=x_{n+2}=\cdots=0$ ,and $q_{n}=q_{n+1}=\cdots=0$ , we have a quantization map. Thisquantization map preserves the pairings:$P_{\infty}arrow\overline{P}_{\infty}$$P_{n}arrow\overline{P}_{n}$$\langle\tilde{f},\tilde{g}\rangle_{Q}=\langle f, g\rangle_{I}n$.

for $f,$ $g\in P_{n}$ . Moreover, the quantization map maps the ideal
into
the ideal.
Quantization map does not commute with the product, namelyin general. The difference
the structure constants of thequantum cohomology ring. If is a linear form, the following quantum
Pieri formula gives an explicit result.
$I_{n}\subset P_{n}$$\tilde{I}_{n}\subset\overline{P}_{n}$$\overline{fg}\neq\tilde{f}\cdot\tilde{g}$$\overline{fg}-\tilde{f}\cdot\tilde{g}\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{c}- \mathrm{t}\mathrm{s}$$f$Theorem 3.5. Let
be a transposition of the integers
quantum cohomology ring $QH^{*}(F)$ ,$t_{ij}$$i$and . In the$j$$(x_{1}+ \cdots x_{k})\cdot\tilde{\mathfrak{S}}w(X)=\sum_{(*)}\tilde{\mathfrak{S}}w\iota_{ij}(_{X})+\sum q(**)iq_{i1}+\cdots qi+s-1\tilde{\mathfrak{S}}_{w}t_{i}j(x)$,where $(*)$ ranges the integers
such that $1\leq i\leq k<j\leq n$ and $l(wt_{ij})=$$l(w)+1$ , and $(**)$ ranges the integers
such that $1\leq i\leq k<j\leq n$ and$i,$ $j$$i,$ $j$$2s=\iota(w)-\iota(wt_{ij})+1\geq l(t_{ij})$.

Remark. The quantum Pieri formula was first proved in [FGP]. For thequantum Pieri formula forand equivariant quantum Pieri formula, see also [KM1], Section 9.
$\tilde{\mathfrak{S}}_{w_{0}}(x, y)$$*\vee\not\equiv\vee \mathrm{X}\ovalbox{\tt\small REJECT}$[C]

Ciocan-Fontanine 1., Quantum cohomology of flag varieties, Intern.
Math.Research Notes, 1995,, p.263-277;$\mathrm{n}.6$

25[FGP] Fomin S., Gelfand S., and Postnikov A., Quantum Schubert polynomials, Preprint 1996;
B., Quantum cohomology of flag manifolds and[GK] Givental \‘A.,Toda lattices, Comm.Math.Phys., 1995, v.168, p.609-641;$\mathrm{K}\mathrm{i}^{-}\mathrm{m}$[GH] Griffiths Ph., Harris J., Principles of algebraic geometry, John Wiley&Sons, New York, 1978;[KM1] Kirillov-A. N. and Maeno T., Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula,
Preprint 1996;
quantum coho[KM2] Kontsevich M., Manin Yu., Gromov-Witten
mology, and enumerative geometry, Comm.Math.Phys., 1994, v.164,
p.525-562;$C\iota_{as}seS_{f}$[LS1] Lascoux A., Sch\"utzenberger M.-P., Polyn\^omes de Schubert,, t.294, 1982, p.447-450;
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$\mathrm{C}.\mathrm{R}$.

$\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{i}_{\mathrm{S}}-$[LS2] Lascoux A., Sch\"utzenberger M.-P., Symmetry and fiag manifolds,Lect. Notes in Math., 1983, v.996, p.118-144;[M][MS]

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