R ESEARCH I NSTITUTEFOR M ATHEMATICAL S CIENCESKYOTOUNIVERSITY,Kyoto,Japan ByAnatolN.KIRILLOVandHiroshiNARUSEApril2015 ConstructionofdoubleGrothendieckpolynomialsofclassicaltypesusingId-Coxeteralgebras RIMS-1823

Construction of double Grothendieck polynomials of classical types using Id-Coxeter algebras

Dedicated to Professor Ken-ichi SHINODA

By

Anatol N. KIRILLOV and Hiroshi NARUSE

April 2015

R _ESEARCH I NSTITUTE FOR M ATHEMATICAL S _CIENCES

KYOTO UNIVERSITY, Kyoto, Japan

of classical types using Id-Coxeter algebras

Anatol N. Kirillov* and Hiroshi Naruse**

RIMS Kyoto* , IPMU* and University of Yamanashi**

Dedicated to Professor Ken-ichi SHINODA

Abstract. We construct double Grothendieck polynomials of classical types which are equiva- lent to the polynomials defined in [15] and compare with [14].

1. Introduction

Let G be a semisimple Lie group, B ⊂G be a Borel subgroup of G, T ⊂B be a maximal torus inB, F :=G/B andW :=NG(T)/T be the corresponding flag variety and the Weyl group. Let ℓ be the rank of G.

According to the famous Borel theorem, the cohomology ring H^∗(G/B,Q) is isomor- phic to the quotient Q[x₁, . . . , xℓ]/Jℓ, where xi :=c₁(Li)∈H²(G/B,Q), i = 1, . . . , ℓ, and c₁(L_i) denotes the first Chern class of the standard line bundle L_i over the flag variety in question, Jℓ stands for the ideal generated by the fundamental invariants of positive degree associated with the Weyl groupW .

To our best knowledge the first systematic and complete treatment of the Schubert Calculus has been done by I.N. Bernstein, I.M. Gelfand and S.I. Gelfand [2] and independently, by M. Demazure [5] in the beginning of 70’s of the last century. A Schubert polynomial S_w(Xℓ), ℓ=rk(G), corresponding to an element w of the Weyl group W, by definition is a polynomial which expresses the Poincar´e dual class of the homology class of the Schubert variety Xw :=BwB/B ⊂G/B in terms of the Borel generators xi,1 ≤ i ≤ ℓ, in the cohomology ring of the flag variety F. Therefore by the very definition, a Schubert polynomial S_w(X) is defined only modulo the ideal J_ℓ.

Mathematics Subject Classification: 05E05.

Key words and phrases: Grothendieck polynomial, Schubert calculus, Id-Coxeter algebra .

The second author is partially supported by the Grant-in-Aid for Scientific Research (C) 25400041, Japan Society for the Promotion of Science.

Hence it is an interesting problem: does there exist “natural representative” of a Shubert polynomial S_w(Xℓ) in the ring Q[x₁, . . . , xℓ] with “nice” combinatorial, algebraic and geometric properties ?

For the typeAn−1flag varieties A. Lascoux and M.-P. Sch¨utsenberger constructed a family of double Schubert polynomialsS_w(X_n−1, Y_n−1) ,w∈Sn with several “nice”

properties, when we setS_w(X_n−1) :=S_w(X_n−1,0) , such as

1. S_w(Xn−1) is a representative of the Schubert class Xw corresponding to w ∈ Sn, that isS_w(Xn−1)≡S_w(Xn−1)( mod Jn−1),

2. (Compatibility conditions)

∂_i^(x)S_w(Xn−1, Yn−1) =

(S_ws

i(Xn−1, Yn−1) if l(wsi) =ℓ(w)−1,

0 otherwise

∂_i^(y)S_w(X_n−1, Y_n−1) = (S_s

iw(Xn−1, Yn−1) if l(siw) =ℓ(w)−1,

0 otherwise

3. S_w(X_n−1, Y_n−1) has nonnegative integer coefficients, 4. S_w(Xn−1, Yn−1) is stable,

5. S_w(X_n−1, Y_n−1) satisfies the vanishing conditions, that is S_w(−v(Y_n−1), Y_n−1) = 0, unless w ≤v with respect to the Bruhat order≤ on the symmetric group Sn,

6. the structural constants for the multiplication of Schubert polynomials S_w(Xn−1), w ∈ Sn, coincide with the triple intersections numbers of Schubert va- rieties.

A new approach to the theory of type A Schubert polynomials which is based on the study of the type A nil–Coxeter algebras, has been initiated by S. Fomin and R. Stanley. The basic idea of that approach is to consider and study the generating function of all Schubert polynomials simultaneously, namely, to treat the following generating function

S(X_n−1) = X

w∈Sn

S_w(X_n−1)uw,

where uw denotes the standard linear basis in the nil–Coxeter algebraN Cn.

An unexpected and deep result discovered in [9] is that in the algebra N C_n[x₁, . . . , x_n−1] the polynomialS_n(X_n−1) is completely factorizable in the product of linear factors. The basic tool to prove the factorizability property is the usage of the Yang–Baxter relation among the elements hi(x) = 1 +xui in the algebra N Cn[x, y],

namely

(1 +xui)(1 + (x+y)u_i+1)(1 +yui) = (1 +yu_i+1)(1 + (x+y)ui)(1 +xu_i+1). (1) The main consequence of the Yang–Baxter relation (1) is that the polynomialsAk(x) = h_n−1(x)h_n−2(x). . . hk(x), commute, namely

[Ak(x), Ak(y)] = 0.

Now one can prove, [9], [8] that S(Xn−1) = X

w∈Sn

S_w(Xn−1)uw =A₁(x₁)A₂(x₂). . . An−1(xn−1).

This approach can be applied to a construction of type A double Schubert poly- nomials, Grothendieck and double Grothendieck polynomials, which originally had been introduced by A. Lascoux and M.-P. Sch¨utzenberger.

Construction of “good” representatives for the Schubert polynomials correspond- ing to the flag varieties of classical types B, C, D was initiated by S. Billey and M.Haiman [3] and independently by S. Fomin and A. N. Kirillov, [7]. In [7] the authors extended an algebro-combinatorial approach to a definition and study of the type A Schubert and Grothendieck polynomials to the case of those of types B and C. But it also works for typeD as well. The key tool in a construction of the aforementioned polynomials is a unitary exponential solution to the quantum Yang–Baxter equations ([22]) with values in the NiCoxeter algebras of types B, C, D correspondingly. The exponential solution to the quantum Yang–Baxter equation associated with nilCox- eter algebraN C(R), R:=A_n−1, Bn, Cn, Dn,allows to construct a family of elements R_i(x)∈N C(R)[x], i= 1, . . . , rk(R) such that

Ri(x)Ri(y) =Ri(y)Ri(x), i= 1, . . . , rk(R).

The elements Ri(x₁), . . . , Ri(xℓ), i = 1, . . . , ℓ := rk(R), are building blocks in the construction of the generating function for all Schubert polynomials corresponding to the flag variety associated with the root system R.

Now in order to ensure the coherency conditions one needs to specify the action of simple transpositions of the corresponding Weyl group on the ring of polynomials Q[x1, . . . , xℓ]. In [7] and [15] the authors have chosen the standardaction of the Weyl group on the cohomology ring of the corresponding flag varietyG/B. Namely,

s0(x1) =−x1, s0(xi) =xi, if i≥1,(types B, C), sˆ1 =−s₁, sˆ1(xi) =xi if i≥2,(type D).

Based on these (= standard !) choice of the action of the simple transpositions, the divided difference operators (resp. isobaricones) are defined uniquely. It is easy to see [7] that for root systems of typesB, C, D it is impossible to find “good” representatives for the Schubert classes which satisfy the properties 2,3,6 listed above. Nevertheless

in [7] the authors introduce the so called Schubert polynomials of the second kind with nice combinatorial properties including those 3,4,5,6, and therefore suitable for computation of the triple intersection numbers for Schubert polynomials of classical type, the main Problem of the Schubert Calculus, see [7] for details.

As for a construction of certain representatives the double Schubert and β- Grothendieck polynomials of classical types B, C, D, the author of [15] has used the following observation: if the a family of polynomials Sw(X), w ∈ W has “wanted properties” with respect to variables X, the the polynomials

Sw(X, Y) := X

u,v∈W,uv=w ℓ(uv)=ℓ(u)+ℓ(v)

S_u⁻¹(Y)Sv(X)

also will have “wanted properties” with respect to the set of variablesX andY. Based on this observation and using the Schubert and Grothendieck polynomials introduced in [6], the author of [15] has introduced a family of polynomials depending on two sets of variableXandY having nice combinatorial properties including among others, that 3,4,5,6 listed above. One example of such polynomials is the triple β-Grothendieck polynomialG^W

w (X, Y, Z),w∈W, whereW stands for the Weyl group of classical type B, C, or D. Indeed, let W be of type B, C, D, one can start with the W-type Schu- bert/Grothendieck expression of thesecond kindS^W(Z, X) :=p

H^W(Z)G^A(X), have been introduced for the Schubert polynomials in [3] for Schubert polynomials of types W, [7] for W = B, C types Schubert polynomials, [16] for Schubert/Grothendieck case. According toan observationmentioned above, a “good” candidate for the double Schubert/Grothendieck expression of type W is

S^W(Y, T, Z, X) :=S^W(−Y,−T)⁻¹S^W(Z, X) =G^A(−Y)⁻¹ q

H^W(T)H^W(Z)G^A(X).

To deduce this equality we have used the following facts:

H^W(T), H^W(Z)

= 0, H^W(−T)⁻¹ =H^W(T).

Finally, one can restrict the generating functionS^W(Y, T, Z, X) on the diagonalT =Z and come to the following expression for the generating function of a double Schu- bert/Grothendieck polynomials of typeW

S^W(Y, Z, X) =G^A(−Y)⁻¹H^W(Z)G^A(X).

Another algero-geometric interpretation of the generating function S^W(Y, Z, X) has been obtained in [12].

Advantage of the algebro-combinatorial approach is, for example, a possibility to define, among others, a plactic versions of polynomials G^W

w (X, Y), S^W

w (X, Y) and their generalizations, see [16] for the case of root systems of type A

In [3] the authors used non-standard action of Weyl group on the ring of super- symmetric functions of infinite number of variables Γ = (Z[x₁, x₂, . . .])^SS and define another family of Schubert polynomials.

In [12] the second authoret al. studied the double Schubert polynomials of type B, C, D using localization map of equivariant cohomology. ForK-theory there is anal- ogous map and the image has the so called Goresky-Kottwitz-MacPherson property [10]. As mentioned for the case of Grassmannians in [14], the Schubert classes can be characterized by recurrence relations.(c.f. §6.)

2. Definitions and Notations

In this paper W =W(X) is a Weyl group of type X =A, B, C, D. I^X is the set of simple reflections in W(X). We index the simple reflections by the same notation as in [12] §3.2. In particular , for typeB andC, s₀ corresponds to the left most node of the Dynkin diagram with the relation (s0s1)⁴ = 1 and (s0si)² = 1 for i ≥ 2. For type D, s_ˆ1 := s₀s₁s₀ and we consider W(D) as the subgroup of W(B) generated by s_ˆ1, s₁, . . ..

Following [7], we prepare some notations. Letβ be an indeterminate. We define operations ⊕ and ⊖ as follows.

x⊕y :=x+y+βxy, x⊖y := (x−y)/(1 +βy).

We also use the convention that

¯

x:=⊖x=− x 1 +βx.

Then we have x⊕x¯= 0. For a Weyl group W with the set S of Coxeter generators, we define Id-Coxeter algebra as follows.

Definition 1. (Id-Coxeter algebra)

Id-Coxeter algebra Idβ(W) for W is a Z[β] algebra with generators ui for each si ∈S and relations as follows.

u²_i =βui, u_iu_ju_i· · ·

| {z }

mi,jtermes

=u_ju_iu_j· · ·

| {z }

mi,jtermes

if m_i,j is the order of s_is_j.

For eachsi ∈I^X,we define divided-difference operator π_i^(a) and ψ_i^(a) with respect to the variablesa= (a₁, a₂, ...) as follows. Assume thatR⊃Z[β] is a ring with a group action ofW(X). We define the action of W(X) onR[a,a] :=¯ R[a₁, a₂, ...,a¯₁,¯a₂, ...] as follows.

Definition 2. The action of si ∈I^X on the variables a₁, a₂, . . . ,a¯₁,¯a₂, . . .

• If i≥1, si(ai) =a_i+1, si(a_i+1) =ai, si(¯ai) = ¯a_i+1, si(¯a_i+1) = ¯ai, and si(ak) =ak, si(¯ak) = ¯ak for k 6=i, i+ 1.

• s₀(a₁) = ¯a₁, s₀(¯a₁) =a₁, and s₀(ak) =ak, s₀(¯ak) = ¯ak for k >1.

• s_ˆ1(a₁) = ¯a₂, s_ˆ1(a₂) = ¯a₁, s_ˆ1(¯a₁) = a₂, s_ˆ1(¯a₂) = a₁, and s_ˆ1(ak) = ak, s_ˆ1(¯ak) = ¯ak

for k >2.

We write the induced action on R[a,¯a] by s^(a)_i . Divided difference operators π_i^(a) and ψ_i^(a) are defined as follows. For f ∈R[a,a] =¯ R[a₁, a₂, ...,a¯₁,¯a₂, ...],

π^(a)_i (f) := f −(1 +βαi(a))s^(a)_i (f)

αi(a) and ψ_i^(a) :=π^(a)_i +β,

where αi(a) is the element in Z[β][a,¯a] corresponding to the root αi, i.e. αi(a) = ai⊕¯ai+1 for i= 1,2, ..., α^B₀(a) = ¯a1, α₀^C(a) = ¯a1⊕¯a1 and αˆ1(a) = ¯a1⊕¯a2.

( Formally we can think as αi(a) = e^βαi −1

β . c.f. [6])

Proposition 1. We have the following relations of operators.

π_i² =−βπi, ψ_i² =βψi for all si ∈I^X. πiπjπi· · ·

| {z }

mi,jtermes

=πjπiπj· · ·

| {z }

mi,jtermes

, ψiψjψi· · ·

| {z }

mi,jtermes

=ψjψiψj· · ·

| {z }

mi,jtermes

if mi,j is the order of sisj.

We can check the relations by direct calculations.

The explicit form of ψ_i^(a) is as follows, ψ_i^(a)(F) = ^s_a^{(a)i F−F}

i+1⊖ai for i≥1 , ψ_0,B^(a)(F) = ^s(a)0 _a^F−F

1 , ψ_0,C^(a)(F) = ^s(a)0_a ^F−F

1⊕a1 and ψ_ˆ^(a)

1 (F) = ^s

(a) ˆ1 F−F a1⊕a2 .

Similarly we can define divided difference operators π^(b)_i and ψ_i^(b) corresponding to the variables b₁, b₂, ....

3. Basic Properties

Let h_i(x) := 1 +xu_i. Then it follows that h_i(x)h_i(y) =h_i(x⊕y).

Lemma 1. (Yang-Baxter relation)

h_i(x)h_j(y) = h_j(y)h_i(x) m_i,j = 2

hi(x)hj(x⊕y)hi(y) = hj(y)hi(x⊕y)hj(x) mi,j = 3 h_i(x)h_j(x⊕y)h_i(x⊕y⊕y)h_j(y) = h_j(y)h_i(x⊕y⊕y)h_j(x⊕y)h_i(x) m_i,j = 4

These can be proved by direct calculations.

Definition 3.

A⁽ⁿ⁾_i (x) :=h_n−1(x)h_n−2(x)· · ·h_i(x) (i= 1,2, ..., n−1) F_n^B(x) :=A⁽ⁿ⁾₁ (x)h0(x)A⁽ⁿ⁾₁ (¯x)⁻¹

=h_n−1(x)h_n−2(x)· · ·h₁(x)h₀(x)h₁(x)· · ·h_n−2(x)h_n−1(x) F_n^C(x) :=A⁽ⁿ⁾₁ (x)h0(x)² A⁽ⁿ⁾₁ (¯x)⁻¹

=h_n−1(x)h_n−2(x)· · ·h₁(x)h₀(x)²h₁(x)· · ·h_n−2(x)h_n−1(x) F_n^D(x) :=A⁽ⁿ⁾₂ (x)h_1ˆ(x)h₁(x)A⁽ⁿ⁾₂ (¯x)⁻¹

=h_n−1(x)· · ·h₂(x)h₁(x)h_ˆ1(x)h₂(x)· · ·h_n−1(x) Lemma 2.

(1) A⁽ⁿ⁾_i (x)A⁽ⁿ⁾_i (y) =A⁽ⁿ⁾_i (y)A⁽ⁿ⁾_i (x)

(2) F_n^X(x)F_n^X(y) =F_n^X(y)F_n^X(x) for X =B, C, D (3) F_n^X(x)F_n^X(¯x) = 1

Note that from (1) we have A⁽ⁿ⁾_i (x)A⁽ⁿ⁾_i (y)⁻¹ =A⁽ⁿ⁾_i (y)⁻¹A⁽ⁿ⁾_i (x) and A⁽ⁿ⁾_i (x)⁻¹A⁽ⁿ⁾_i (y)⁻¹ =A⁽ⁿ⁾_i (y)⁻¹A⁽ⁿ⁾_i (x)⁻¹.

Proof.

(1) For the case i = n−1 is trivial. By reverse induction on i, we can assume i < n−1 andA⁽ⁿ⁾_i+1(x)A_i+1⁽ⁿ⁾(y) =A⁽ⁿ⁾_i+1(y)A⁽ⁿ⁾_i+1(x). Then

A⁽ⁿ⁾_i (x)A⁽ⁿ⁾_i (y) = A_i+1⁽ⁿ⁾(x)hi(x)A⁽ⁿ⁾_i+1(y)hi(y)

= A⁽ⁿ⁾_i+1(x)A⁽ⁿ⁾_i+1(y)h_i+1(¯y)hi(x)h_i+1(y)hi(y⊖x)hi(x)

= A⁽ⁿ⁾_i+1(y)A⁽ⁿ⁾_i+1(x)hi+1(¯y)hi+1(y⊖x)hi(y)hi+1(x)hi(x)

= A⁽ⁿ⁾_i+1(y)A⁽ⁿ⁾_i+1(x)h_i+1(¯x)hi(y)h_i+1(x)hi(x)

= A⁽ⁿ⁾_i+1(y)A⁽ⁿ⁾_i+2(x)hi(y)h_i+1(x)hi(x)

= A⁽ⁿ⁾_i+1(y)hi(y)A⁽ⁿ⁾_i+2(x)hi+1(x)hi(x)

= A⁽ⁿ⁾_i (y)A⁽ⁿ⁾_i (x)

(2) Using Lemma 3.1 and (1) we can show the equalities as follows. For X =B, F_n^B(x)F_n^B(y)

= A⁽ⁿ⁾₁ (x)h₀(x)A⁽ⁿ⁾₁ (¯x)⁻¹A⁽ⁿ⁾₁ (y)h₀(y)A⁽ⁿ⁾₁ (¯y)⁻¹

= A⁽ⁿ⁾₁ (x)h0(x)A⁽ⁿ⁾₁ (y)A₁⁽ⁿ⁾(¯x)⁻¹h0(y)A⁽ⁿ⁾₁ (¯y)⁻¹

= A⁽ⁿ⁾₁ (x)A⁽ⁿ⁾₁ (y)h₁(¯y)h₀(x)h₁(y)h₁(x)h₀(y)h₁(¯x)A⁽ⁿ⁾₁ (¯x)⁻¹A⁽ⁿ⁾₁ (¯y)⁻¹

= A⁽ⁿ⁾₁ (y)A⁽ⁿ⁾₁ (x)h₁(¯y)h₀(x)h₁(y)h₁(x)h₀(y)h₁(¯x)A⁽ⁿ⁾₁ (¯y)⁻¹A⁽ⁿ⁾₁ (¯x)⁻¹

= A⁽ⁿ⁾₁ (y)A⁽ⁿ⁾₂ (x)h1(x⊕y)h¯ 0(x)h1(x⊕y)h0(y)h1(¯x⊕y)A⁽ⁿ⁾₂ (¯y)⁻¹A⁽ⁿ⁾₁ (¯x)⁻¹

= A⁽ⁿ⁾₁ (y)A⁽ⁿ⁾₂ (x)h₀(y)h₁(x⊕y)h₀(x)A⁽ⁿ⁾₂ (¯y)⁻¹A⁽ⁿ⁾₁ (¯x)⁻¹

= A⁽ⁿ⁾₁ (y)h₀(y)A⁽ⁿ⁾₁ (x)A₁⁽ⁿ⁾(¯y)⁻¹h₀(x)A⁽ⁿ⁾₁ (¯x)⁻¹

= A⁽ⁿ⁾₁ (y)h₀(y)A⁽ⁿ⁾₁ (¯y)⁻¹A⁽ⁿ⁾₁ (x)h₀(x)A⁽ⁿ⁾₁ (¯x)⁻¹

= F_n^B(y)F_n^B(x)

Similar arguments with appropriate modifications will giveX =C, D cases.

The essential equalities to be used are

h₁(x⊕y)h¯ ₀(x⊕x)h₁(x⊕y)h₀(y⊕y)h₁(¯x⊕y) =h₀(y⊕y)h₁(x⊕y)h₀(x⊕x) and h₂(x⊕y)h¯ ₁(x)h_ˆ1(x)h₂(x⊕y)h₁(y)h_ˆ1(y)h₂(¯x⊕y) =h₁(y)h_ˆ1(y)h₂(x⊕y)h₁(x)h_ˆ1(x).

(3) This esentially follows by the relation hi(x)hi(¯x) = 1.

4. β-super symmetric functions

Definition 4. β-super symmetric function is a symmetric function which sat- isfies the following property.

f(t,¯t, x₃, ..., x_n) =f(0,0, x₃, ..., x_n) for every t.

Remark

The β-supersymmetric property is translated to usual supersymmetricity by the change of variables xi to e^βxi −1

β .

Let SSβ(x₁, . . . , xn) := {f ∈ Z[β][x₁, ..., xn] | f : β-supersymmetric} and set SSβ(x) := lim

←nSSβ(x₁, . . . , xn).

SSβ(x) is the ring of β-supersymmetric functions and we denote it as Γ_β^′(x). If β = 0 this becomes the the ring of supersymmetric functions Γ^′.

4.1. K-theoretic Schur functions GPλ(x), GQλ(x) In [14] β- supersymmetric functions GPλ(x), GQλ(x) are defined. Let b₁, b₂, ... be indetermi- nates, and set [x|b]^k = (x⊕b₁)· · ·(x⊕bk) and [[x|b]]^k = (x⊕x)(x⊕b₁)· · ·(x⊕b_k−1).

Let SP_n be the set of strict partitions of length at most n. i.e. λ = (λ₁ > λ₂ >

· · ·> λr >0) such that r ≤n.

Definition 5. (Ikeda-Naruse [14]) For a strict partition λ∈SPn, GPλ(x₁, . . . , xn|b) := 1

(n−r)!

X

w∈Sn

w



 Y

1≤i≤r



[xi|b]^λi Y

i<j≤n

xi⊕xj

xi⊖xj









GQλ(x₁, . . . , xn|b) := 1 (n−r)!

X

w∈Sn

w



 Y

1≤i≤r



[[xi|b]]^λi Y

i<j≤n

xi⊕xj

xi⊖xj









where w∈Sn acts x1, . . . , xn as permutation of indices.

We also define

GPλ(x1, . . . , xn) :=GPλ(x1, . . . , xn|0), GQλ(x1, . . . , xn) :=GQλ(x1, . . . , xn|0), GPλ(x) := lim

←nGPλ(x1, ..., xn) and GQλ(x) := lim

←nGQλ(x1, ..., xn|b).

GPλ(x|b) := lim

←nGPλ(x₁, ..., x_2n|b) and GQλ(x|b) := lim

←nGQλ(x₁, ..., xn|b).

Examples.

GP1(x1, . . . , xn) =x1⊕x2⊕ · · · ⊕xn.

GQ₁(x₁, . . . , xn) = (x₁⊕x₁)⊕(x₂⊕x₂)⊕ · · · ⊕(xn⊕xn).

Lemma 3.

(1) GPλ(x₁, . . . , xn) and GQλ(x₁, . . . , xn) are β-supersymmetric functions.

(2) {GP_λ(x₁, . . . , x_n)}λ∈SPn forms a basis of SS_β(x₁, . . . , x_n) over Z[β].

(3) Let SS_β^C(x₁, . . . , xn) be the Z[β]-subspace of SSβ(x₁, . . . , xn) spanned by GQ_λ(x₁, . . . , x_n)(λ ∈SP_n). Then {GQ_λ(x₁, . . . , x_n)}λ∈SPn forms a basis of SS_β^C(x₁, . . . , x_n) over Z[β].

Proof.

(1) follows from the definition.

(2) and (3) follows from the fact for corresponding properties for usual Schur P, Q-functions.

Remark 1. We remark that the definition ofβ-supersymmetry and the polyno- mialsGP_λ, GQ_λ can be generalized in more general setting such as algebraic cobordism [20]. We are planning to study the details elsewhere. (cf.[21])

Lemma 4. ([14])

GPλ(x|b)andGQλ(x|b)are characterized by (left) divided difference relations and initial conditions. i.e.

π_i^(b)GXλ(x|b) =

(GX_λ⁽ⁱ⁾(x|b) if siλ < λ

−β GXλ(x|b) if siλ≥λ and

GX_∅(x|b) = 1

where GBλ(x|b) =GPλ(x|0, b), GCλ(x|b) =GQλ(x|b), GDλ(x|b) =GPλ(x|b).

See [14] Theorem 6.1 and Theorem 7.1.

4.2. Stable symmetric functions Fw^X(x₁, ..., x_n) Definition 6. For X =B, C, D, we define

F_n^X(x₁, x₂, . . . , xn) :=

Yn

i=1

F_n^X(xi) and F_∞^X(x) := lim

←nF_n^X(x₁, x₂, . . . , xn).

We also defineFw^X(x₁, ..., xn) and Fw^X(x) by the following expression.

F_n^X(x1, x2, . . . , xn) = X

w∈W_n^X

Fw^X(x1, ..., xn)uw , F_∞^X(x) = X

w∈W^X

Fw^X(x)uw

Lemma 5. For each w ∈W_n^X, Fw^X(x1, x2, . . . , xn) is a β-supersymmetric func- tion.

Proof.

This follows from Lemma 3.3 (2) and (3).

Lemma 6. (0) For X =B, C, D, Fw^X−1(x1, x2, . . . , xn) =Fw^X(x1, x2, . . . , xn).

(1) For X =B or D, Fw^X(x₁, x₂, . . . , xn) can be expanded in GPλ(x₁, x₂, . . . , xn) with coefficients in Z[β].

(2) For a (maximal) Grassmannian element w∈W_n^X, Fw^B(x₁, x₂, . . . , xn) =GP_λB(w)(x₁, x₂, . . . , xn)

Fw^C(x₁, x₂, . . . , xn) =GQ_λC(w)(x₁, x₂, . . . , xn) Fw^D(x₁, x₂, . . . , xn) =GP_λD(w)(x₁, x₂, . . . , xn)

where λB(w), λC(w), λD(w) are strict partitions corresponding to w. cf.[14]

Proof.

(0) This follows from the symmetry of F_n^X. (1) This follows from Lemma 4.3 (2).

(2) This follows from Proposition 6.

Remark 2. We state conjecture that the coefficients in the expansion of (1) is positive. This will be a consequence of K-theory analogue of “transition equation”

for type B,C,D.(cf. [12]) Example

Fs^B0(x₁, . . . , xn) =GP₁(x₁, . . . , xn) Fs^C0(x₁, . . . , xn) =GQ₁(x₁, . . . , xn) Fs^D_ˆ1(x1, . . . , xn) =GP1(x1, . . . , xn)

Proposition 2. (Compatible sequence formula) cf. ([3],[7])

For w∈W_n^X, we have

Fw^B(x₁, ..., xn) = X

˜ a∈R(w)˜

X

˜b∈C^B(˜a)

β^{ℓ(˜a)−ℓ(w)}2^{|˜b|−γ(˜a,˜b)−oB(˜a)}x˜b

Fw^C(x₁, ..., xn) = X

˜a∈R(w)˜

X

˜b∈C^C(˜a)

β^{ℓ(˜a)−ℓ(w)}2^{|˜b|−γ(˜a,˜b)}x˜b

Fw^D(x₁, ..., xn) = X

˜a∈R(w)˜

X

˜b∈C^D(˜a)

β^{ℓ(˜a)−ℓ(w)}2^{|˜b|−γ(˜a,˜b)−oD(˜a)}x˜b,

where we used the following notations.

R(w) is the set of sequence of indices ˜˜ a= (˜a₁,˜a₂, . . . ,˜a_ℓ) such thatu_sa˜1 · · ·u_s˜aℓ = uw.

ℓ(˜a) is the lengthℓ of the sequence ˜a.

C^B(˜a) = C^C(˜a) is the set of compatible sequences ˜b with respect to ˜a , i.e.

˜b= (1≤˜b₁ ≤˜b₁ ≤ · · · ≤˜b_ℓ(˜a) ≤n) such that ˜a_i−1 ≤˜ai ≥˜a_i+1 =⇒ ˜b_i−1 <˜b_i+1. C^D(˜a) is the set of compatible sequence for the flattened word ˜˜aof ˜awith further properties that if ˜ai = ˜ai+1 = 1 or ˜ai = ˜ai+1 = ˆ1 then ˜bi < ˜bi+1. Note that the flattened word ˜˜a is obtained from ˜a by replacing ˆ1 with 1. cf.[3].

o^B(˜a) is the number of appearance of 0’s in ˜a.

o^D(˜a) is the total number of appearance of 1 and ˆ1 in ˜a.

|˜b| is the number of distinct ˜bi’s.

γ(˜a,˜b) := #|{i|˜ai = ˜a_i+1 and ˜bi = ˜b_i+1}|. x_˜b :=x_˜b1x_˜b2· · ·x_˜b

ℓ for ˜b= (˜b₁. . . ,˜b_ℓ).

Proof.

This follows essentially from the expansion of the defining generating function.

Example.

type D, n= 2 case w= [¯1,¯2,3] =s1sˆ1

˜b= (1,1) is a compatible sequence for ˜a= (1,ˆ1),(ˆ1,1).

˜b= (1,2) is a compatible sequence for ˜a= (1,ˆ1),(ˆ1,1).

˜b= (2,2) is a compatible sequence for ˜a= (1,ˆ1),(ˆ1,1).

˜b= (1,1,2) is a compatible sequence for ˜a = (1,ˆ1,1),(ˆ1,1,ˆ1),(1,ˆ1,ˆ1),(ˆ1,1,1).

˜b= (1,2,2) is a compatible sequence for ˜a = (1,ˆ1,1),(ˆ1,1,ˆ1),(1,1,ˆ1),(ˆ1,ˆ1,1).

˜b= (1,1,2,2) is a compatible sequence for

˜

a= (1,ˆ1,1,ˆ1),(ˆ1,1,ˆ1,1),(ˆ1,1,1,ˆ1),(1,ˆ1,ˆ1,1).

There are no other compatible sequences and the sum of the terms becomes Fs^D1sˆ1(x₁, x₂) =x²₁+ 2x₁x₂+x²₂+ 2βx₁²x₂+ 2βx₁x²₂+β²x²₁x²₂ = (x₁⊕x₂)².

5. Main results

First we recall the type A Grothendieck polynomials [7].

We set G_An−1(a₁, ..., a_n−1) :=A⁽ⁿ⁾₁ (a₁)A⁽ⁿ⁾₂ (a₂)· · ·A⁽ⁿ⁾_n−1(a_n−1).

Then for w∈Sn, we define G^wAn−1(a) as the coefficient of uw.

GAn−1(a₁, ..., a_n−1) = X

w∈Sn

Gw^An−1(a)uw. Furthermore, we can considerGA(a) := lim

←nGAn−1(a1, ..., an−1) and get strongly stable polynomials Gw^A(a) by

GA(a) = X

w∈S∞

Gw^A(a)uw.

Strongly stable means that ifw∈Sn then Gw^A(a) =G^wAn−1(a) (which does not depend on n).

5.1. The first definition

Definition 7. We define for X =B, C or D,

G^X_n(a, b;x) :=GAn−1(¯b₁, ...,¯b_n−1)⁻¹F_n^X(x)GAn−1(a₁, ..., a_n−1) and defineGn,w^X (a, b;x) as the coefficient of u_w.

G^X_n(a, b;x) = X

w∈W_n^X

Gn,w^X (a, b;x)uw.

In this case Gn,w^X (a, b;x)∈SSβ(x1, . . . , xn)[a1, . . . , an−1, b1, . . . , bn−1].

Furthermore, we can define Gw^X(a, b;x) by GA(¯b)⁻¹F_∞(x)GA(a) = X

w∈W^X

Gw^X(a, b;x)uw.

Then Gw^X(a, b;x) has strong stability (cf. Proposition 5), and when we set β = 0 this is the double Schubert polynomial defined in [12]. It is clear that if w ∈ W_n^X then Gn,w^X (a, b;x) =Gw^X(a, b;x1, . . . , xn,0,0, . . .).

We will writew·v=z (called Demazure product) if uwuv =βℓ(w)+ℓ(v)−ℓ(z)uz. Proposition 3. For X =B, C, D and w∈W^X, we have

Gw^X(a, b;x) = X

(v1,u,v2)∈R(w)

G_v^A−1

1 (b)Fu^X(x)Gv^A2(a)

where R(w) ={(v₁, u, v₂)∈S_∞×W^X ×S_∞ |v₁·u·v₂ =w}.

Definition 8. The action of Weyl groupW_n^X onSS_β⁽ⁿ⁾(x)⊗Z[β][a,a]¯⊗Z[β][b,¯b]

is derived from the action as follows. For f(x)∈SSβ(x),

s^(a)₀ f(x) =f(a₁, x), s^(b)₀ f(x) =f(b₁, x),

s^(a)_ˆ

1 f(x) =f(a₁, a₂, x), s^(b)_ˆ

1 f(x) =f(b₁, b₂, x).

These actions can be clarified by the change of variables explained in the second definition below (cf.§5.2 Remark 3).

Proposition 4. We have

π_i^(a)G^X_n(a, b;x) =G^X_n(a, b;x)(ui−β) and π_i^(b)G^X_n(a, b;x) = (ui−β)G^X_n(a, b;x).

N.B. These mean that

π^(a)_i Gw^X(a, b;x) =

(Gws^Xi(a, b;x) if l(wsi) =ℓ(w)−1,

−βGw^X(a, b;x) otherwise and

π_i^(b)Gw^X(a, b;x) =

(Gs^Xiw(a, b;x) if l(s_iw) =ℓ(w)−1,

−βGw^X(a, b;x) otherwise . Proof.

We will prove ψ^(a)_i G^X_n(a, b;x) = G^X_n(a, b;x)u_i. Recall the explicit formula of ψ_i after the Prop. 2.2.

G_An−1(¯b)⁻¹ is invariant for the action of s^(a)_i , i ∈ I^X. For i > 0, ψ_i^(a)F_n^X(x) = F_n^X(x) and ψ_i^(a)G_An−1(a) = G_An−1(a)u_i(cf. [6]), therefore ψ_i^(a)F_n^X(x)G_An−1(a) = F_n^X(x)GAn−1(a)ui .

ψ_0,B^(a)(F_n^B(x)G_An−1(a)) = ^F

B

n(x)F^B(a1)G_An−1(¯a1,a2,...,an−1)−Fn^B(x)G_An−1(a)

a1 =

F_n^B(x)GAn−1(a)u₀

ψ_0,C^(a)(F_n^C(x)GAn−1(a)) = ^F

C

n(x)H^C(a1)G_An−1(¯a1,a2,...,an−1)−Fn^C(x)G_An−1(a)

a1⊕a1 =

F_n^C(x)GAn−1(a)u₀

ψ_ˆ1^(a)(F_n^D(x)GAn−1(a)) = ^F

D

n(x)F^D(a1,a2)G_An−1(¯a2,¯a1,...,an−1)−F_n^D(x)G_An−1(a)

a1⊕a2 =

F_n^D(x)GAn−1(a)u_ˆ1

Similar arguments hold for the action ofψ_i^(b).

Proposition 5. (strong stability)

Gw^X(a, b;x) has strong stability i.e. if in : W_n^X →W_n+1^X is the natural inclusion, then

Gi^Xn(w)(a, b;x) =Gw^X(a, b;x).

Proposition 6. (Grassmannian elements) For a Grassmannian element w ∈ W^X, we have the following equality.

Gw^B(a, b;x) =GP_λB(w)(x|0, b) Gw^C(a, b;x) =GQ_λC(w)(x|b) Gw^D(a, b;x) =GP_λD(w)(x|b)

where λX(w) is the strict partition corresponding to w∈W^X (cf. [14]).

5.2. The second definition As [7], we can use “change of variables” for xi, i= 1,2. . . ..

F(xi) = q

F(¯ai)F(¯bi)

to define the double Grothendieck polynomialGw^Xn(a, b) with two sets of variablesa, b.

Remark 3. As s^(a)₀ (p

F(¯a1,¯a2, . . .)) = p

F(a1,¯a2, . . .) and by the supersym- metric property of F, this is = p

F(a₁, a₁,a¯₁,¯a₂, . . .) = F(a₁)p

F(¯a₁,¯a₂, . . .). This explains the action s^(a)₀ (F(x)) = F(a₁, x) and s^(b)₀ (F(x)) = F(b₁, x). The action of s^(a)_ˆ1 and s^(b)_ˆ1 as well.

Definition 9. Let X =B, C, D. For w∈W_n^X, we define G^X_n(a) and G^X_n(a, b) as follows.

G^X_n(a) :=

q

F_n^X(¯a₁, ...,¯an)GA_n−1(a) and G^X_n(a, b) :=G^X_n(¯b)⁻¹G^X_n(a).

By expanding these in terms ofuw, we can define Gn,w^X (a) and Gn,w^X (a, b) by G^X_n(a) = X

w∈Wn^X

Gn,w^X (a)uw and G^X_n(a, b) = X

w∈Wn^X

Gn,w^X (a, b)uw.

Remark 4. This double Grothendieck polynomial Gn,w^X (a, b) is essentially the same as defined in [15]. This has weak stability. i.e. Gn,w^X = Gn+1,w^X |^an+1=bn+1=0 for w∈W_n^X. But it doesn’t have strong stability.

Note that forw ∈W_n^X, then

Gn,w^X (a)∈Q[β][[a₁, ..., an,¯a₁, ...,a¯n]] and

Gn,w^X (a, b)∈Q[β][[a₁, ..., an,¯a₁, ...,a¯n, b₁, ..., bn,¯b₁, ...,¯bn]].

Examples G2,s^B 0(a, b) =

√1+(¯a1⊕¯a2⊕¯b1⊕¯b2)β−1

β = ^¯a1⊕¯a2₂^{⊕¯b1⊕¯b2} −β^{(¯a1⊕¯a2⊕}₈^{¯b1⊕b¯2)2} +· · · G2,s^C 0(a, b) = ¯a1⊕¯a2⊕¯b1⊕¯b2, G3,s^C 0(a, b) = ¯a1⊕¯a2⊕a¯3⊕¯b1⊕¯b2⊕¯b3. G3,s^D ˆ1(a, b) =

√1+(¯a1⊕¯a2⊕¯a3⊕¯b1⊕¯b2⊕¯b3)β−1 β

Proposition 7. The following holds for X =B, C, D and i ∈IXn. π^(a)_i G^X_n(a, b) =G^X_n(a, b)(u_i−β)

π^(b)_i G^X_n(a, b) = (ui−β)G^X_n(a, b) Proof.

These are Prop. 5.3 with change of variables.

6. Identification with Schubert class

Let R^(b)β := Z[β][b₁,¯b₁, b₂,¯b₂, . . .]. K-theory Schubert classes are determined by the localization (Prop. 2.10 in [17]). And they are determined uniquely by either

“right hand” recurrence ((2.12) in [17]) or “left hand” recurrence (Remark 2.3 in [17]

).

(right recurrence) G^e = 1, and π_i^(a)G^w =G^wsi if wsi < w and π^(a)_i G^w = −βG^w if ws_i> w.

(left recurrence) G^e = 1, and π_i^(b)G^w = G^siw if siw < w and π_i^(b)G^w = −βG^w if siw > w.

Therefore we can identify the polynomials Gw^X(a, b;x) defined above as Schubert classes. In particular we have

Theorem 1. Assume Gu^X(a, b;x)Gv^X(a, b;x) = X

w∈W^X

c^w,X_u,v (β)Gw^X(a, b;x), c^w,X_u,v (β) ∈ R^(b)β . Then c^w,X_u,v (β)|^β=−1 is the generalized Littlewood-Richardson coef- ficient for equivariant K-theory of type X. (b_i is considered as 1−e^ti.)

Remark 5. c^w_u,v(0) is the generalized Littlewood-Richardson coefficient for equivariant cohomology if we replace bi to −ti. (cf. [12].)

Example

Gs^C0(a, b;x)Gs^C0(a, b;x) =Gs^C1s0(a, b;x) +βGs^C0s1s0(a, b;x)

7. Adjoint polynomials

The Grothendieck polynomial represents the K-theory Schubert class of the structure sheaf O^Xw of the Schubert variety X^w = B−wB/B ⊂ X = G/B. We can also define the adjoint polynomials Hn,w^X , for each w ∈ W_n^X, corresponding to the ideal sheaf O^Xw(−∂X^w) of boundary ∂X^w in X^w. cf. [11, 18]. The pairing h·,·i:KT(X)⊗R(T)KT(X)→R(T) is given by

hv₁, v₂i=χ(X, v₁⊗v₂) where χ(X,F) =X

p≥0

(−1)^pch H^p(X,F).

We define the relative adjoint polynomialH^Xw,v for w≤vbyHw,v^X :=ψ^(a)_w−1v(Gv^X).

The adjoint polynomial for w ∈ W_n^X is H^Xn,w := H^X_w,w⁽ⁿ⁾

0

, where w₀⁽ⁿ⁾ is the longest element in W_n^X (cf. [18]). These polynomials are no more stable but have similar properties as Grothendieck polynomials.

Proposition 8. For w ∈W_n^X H^Bn,e = Q

1≤i≤n−1(1 +βai)ⁿ⁻ⁱQ

1≤i≤n−1(1 +βbi)ⁿ⁻ⁱQ

1≤i≤n(1 +βxi)²ⁿ⁻¹ H^Cn,e = Q

1≤i≤n−1(1 +βai)ⁿ⁻ⁱQ

1≤i≤n−1(1 +βbi)ⁿ⁻ⁱQ

1≤i≤n(1 +βxi)²ⁿ H^Dn,e = Q

1≤i≤n−1(1 +βai)ⁿ⁻ⁱQ

1≤i≤n−1(1 +βbi)ⁿ⁻ⁱQ

1≤i≤n(1 +βxi)²ⁿ⁻² and

H^Xn,w = (−1)^ℓ(w)H^XeⁿGn,w^X

where Gn,w^X =Gn,w^X (a, b;x).

We can derive these formula using generating functions. Let us defineH_n^X(a, b;x) as

H_n^X(a, b;x) := X

w∈W_n^X

(−1)^ℓ(w)H^Xn,w(a, b;x)uw. Then we get the following formula.

Proposition 9.

H_n^X(a, b;x) =H^Xn,eG^X_n(¯a,¯b; ¯x).

Actually we can show the following property.

Proposition 10. For si ∈I_n^X we have

π_i^(a)H_n^X(a, b;x) =H_n^X(a, b;x)(−ui)