INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES
WARSZAWA 1996
TWISTED ACTION OF THE SYMMETRIC GROUP ON THE COHOMOLOGY OF A FLAG MANIFOLD
A L A I N L A S C O U X L.I.T.P., Universit´ e Paris 7
2, Place Jussieu, 75251 Paris Cedex 05, France E-mail: al@litp.ibp.fr
B E R N A R D L E C L E R C L.I.T.P., Universit´ e Paris 7
2, Place Jussieu, 75251 Paris Cedex 05, France E-mail: bl@litp.ibp.fr
J E A N - Y V E S T H I B O N
Institut Gaspard Monge, Universit´ e de Marne-la-Vall´ ee 2, rue de la Butte-Verte, 93166 Noisy-le-Grand Cedex, France
E-mail: jyt@litp.ibp.fr
Abstract. Classes dual to Schubert cycles constitute a basis on the cohomology ring of the flag manifold F , self-adjoint up to indexation with respect to the intersection form. Here, we study the bilinear form
(X, Y ) := hX · Y, c(F )i
where X, Y are cocycles, c(F ) is the total Chern class of F and h , i is the intersection form.
This form is related to a twisted action of the symmetric group of the cohomology ring, and to the degenerate affine Hecke algebra. We give a distinguished basis for this form, which is a deformation of the usual basis of Schubert polynomials, and apply it to the computation of the Schubert cycle expansions of Chern classes of flag manifolds.
1. Introduction and preliminaries. Let V be a complex vector space of dimen- sion n, and F = F (V ) be the variety of complete flags in V . It is well known that the cohomology ring H
∗(F , C) is the quotient of the polynomial ring C[X] = C[x
1, x
2, . . . , x
n] by the ideal I
+of symmetric polynomials without constant term.
Let σ
i, i = 1, . . . , n − 1 be the simple transposition exchanging x
iand x
i+1. Denote 1991 Mathematics Subject Classification: 14M15, 05E15, 20G20.
Supported by PRC Math-Info and EEC grant n
0ERBCHRXCT930400.
The paper is in final form and no version of it will be published elsewhere.
[111]
by ∂
ithe linear operator on C[x
1, . . . , x
n] defined by
∂
if := f − σ
if
x
i− x
i+1(1)
(Newton’s divided difference). The operators ∂
1, . . . , ∂
n−1induce operators on H
∗(F ).
According to [1] and [4], the basis of Schubert cycles can be obtained from the class of a point P =
n!1Q
i<j
(x
i− x
j) by successive applications of divided difference oper- ators. Taking as representative of P the polynomial X := x
n−11x
n−22· · · x
01, one obtains polynomials X
µ, µ ∈ S
n, called Schubert polynomials, which represent the Schubert subvarieties in the cohomology ring [11]. A detailed account of the algebraic theory of Schubert polynomials can be found in Macdonald’s treatise [14].
Divided differences satisfy the braid relations
∂
i∂
i+1∂
i= ∂
i+1∂
i∂
i+1∂
i∂
j= ∂
j∂
ifor |i − j| > 1, (2) but the squares ∂
i2are null. These relations allow to define operators ∂
µfor any per- mutation µ ∈ S
n: if µ = σ
i1σ
i2· · · σ
imis a reduced decomposition of µ, one sets
∂
µ= ∂
i1∂
i2· · · ∂
im. The result does not depend on the choice of a particular reduced decomposition of µ.
To recover an action of the symmetric group, one can take any q ∈ C and define D
i:= σ
i+ q∂
i, 1 ≤ i ≤ n − 1. (3) These operators still satisfy the braid relations
D
iD
i+1D
i= D
i+1D
iD
i+1D
iD
j= D
jD
i(|i − j| > 1) (4) together with
D
2i= 1, (5)
so that they generate a representation of the symmetric group S
non the polynomial ring C[x
1, . . . , x
n], as well as on the cohomology ring H
∗(F , C). These operators have been considered by Cherednik and Bernstein (cf. [2], [3]). Similar operators, acting on the equivariant K-theory of flag manifolds, have been used by Lusztig [13]. More general operators satisfying braid relations have been given in [12].
As ∂
idecreases degrees by 1, all q 6= 0 will give equivalent representations of S
n, and by homogeneity, the general case can be recovered from the case q = 1. For simplicity, we set q = 1, and write
s
i:= σ
i+ ∂
i. (6)
We denote as above by s
µthe product of operators s
icorresponding to a permutation µ.
Remark that the operator algebra generated by the s
iand the variables x
j(interpreted as operators f 7→ x
jf ) is isomorphic to the degenerate affine Hecke algebra considered in [2].
Schubert calculus for other classical groups can be found in the work of Fulton [7]
and of Pragacz and Ratajski [15].
This paper is organized as follows. We first define certain elements (Yang-Baxter
operators) of the degenerate affine Hecke algebra. Then we use them to define a bilinear
form on the cohomology of a flag manifold. We exhibit a distinguished basis, called affine
Schubert polynomials, and compute its adjoint basis. We then apply this formalism to the computation of the Schubert expansions of Chern classes.
Acknowledgements. The preparation of this paper has been facilitated by the use of the program system SYMMETRICA [9] and of the Maple package SP [16].
2. Yang-Baxter operators. We shall define inductively operators
µand ∇
µas- sociated with any permutation µ in S
n. Set
12...n= 1, ∇
12...n= 1, and, if µ = σ
iα with
`(µ) = `(α) + 1, and β = α
−1,
µ
=
s
i+
β 1i+1−βi
α
∇
µ=
s
i−
β 1i+1−βi
∇
α(7)
Using the braid relations (4), one can check that this definition is consistent, i.e.
does not depend on the chosen factorization (see [2, 3] and [6]). This follows in fact from a classical solution of the Yang-Baxter equation. In [17], C. N. Yang observed that the operators defined by Y
i(u) = u
−1+ σ
i, where u is a scalar parameter and σ
ithe transposition (i, i + 1) satisfy the “Quantum Yang-Baxter Equation with spectral parameter”:
Y
i(u − v)Y
i+1(u − w)Y
i(v − w) = Y
i+1(v − w)Y
i(u − w)Y
i+1(u − v) (8) It follows that given a n-tuple of parameters u = (u
1, . . . , u
n), one can define for any per- mutation µ ∈ S
nan operator R
µ(u) by the following prescription: Y
µ(u) = Y
i(u
β(i+1)− u
β(i))R
α(u), where, as above, R
12...n= 1, µ = σ
iα, `(µ) = `(α) + 1 and β = α
−1. Then, our operators (7) are respectively R
µ(u) and R
µ(−u), where u = (1, 2, . . . , n) and σ
iis interpreted as s
i.
For the maximal element ω = (n, n−1, . . . , 1) of S
n, one has the following factorization property (given in [6] for the case of the Hecke algebra):
Proposition 2.1. Define θ = Y
1≤i<j≤n
(1 + x
i− x
j) and θ
∗= Y
1≤i<j≤n
(1 − x
i+ x
j).
Then, for any polynomial f , (i) ∇
ωf = θ
∗∂
ωf
(ii)
ωf = ∂
ω(θf ).
P r o o f. Recall that the classes of the Schubert polynomials X
µ, µ ∈ S
n, form a basis of H
∗(F ) = C[X]/I
+. Given µ and i such that `(µσ
i) > `(µ), the polynomial X
µis symmetrical in x
iand x
i+1. As such, it is sent to 0 by the operator ∇
σi= σ
i+ ∂
i− 1.
Now, for any permutation µ 6= ω, there exists an i such that `(µσ
i) > `(µ). If we choose a reduced decomposition of ω ending by σ
i, ω = νσ
i, say, we see that X
µis sent to 0 by ∂
ω= ∂
ν∂
σiand by ∇
ω= ∇
µ∇
σi.
Thus, ∇
ωas well as ∂
ωannihilate all Schubert polynomials X
µfor µ 6= ω. Finally, X
ω= x
n−11. . . x
0nis sent to 1 by ∂
ω. To conclude, it remains to prove that
∇
ω(X
ω) = Y
1≤i<j≤n
(1 − x
i+ x
j).
This formula can be proved by induction on n using the factorization
ω
n= σ
1σ
2· · · σ
n−1ω
n−1,
which gives
∇
ωn= s
1− 1
· · ·
s
n−1− 1 n − 1
∇
ωn−1.
3. Quadratic form. Recall that the intersection form of the cohomology ring H
∗(F , C) is induced by the form on C[X]
hf, gi = ∂
ω(f g)|
0= ∂
ω(f g|
`(ω)) (9) where f |
kdenotes the homogeneous component of degree k of f (cf. [1], [5]). With respect to this form, the Schubert polynomials satisfy
hX
µ, X
νi = 1 if ν = ωµ
0 otherwise. (10)
The tangent bundle T F of the flag manifold has a composition sequence {L
iL
−1j}
i<jwhere L
1, L
2, . . . , L
nare the tautological line bundles on F . The total Chern class of L
ibeing c(L
i) = 1 + x
i, the total Chern class of the tangent bundle of F is
c(F ) = Y
i<j
(1 + x
i− x
j) (11)
(see e.g. [8], our convention is L
i= ξ
i∗in the notation of [8]). Consider now the following quadratic form on C[X]:
Definition 3.1.
(f, g) :=
ω(f g)|
0.
Thus, in the cohomology ring, we see from Proposition 2.1 that
(f, g) = hf, g c(F )i = hf c(F ), gi. (12) Lemma 3.2. The operators
iare self-adjoint with respect to the quadratic form ( , ).
P r o o f. For any i,
ω i= 2
ω, since
2i= 2
iand since one can find a reduced decomposition of ω ending with σ
i. Now,
(
if, g) =
ω((
if )g) |
0= 1
2
ω i((
if )g) |
0= 1
2
ω((
if )(
ig)) |
0since
if is a scalar for
i, being symmetrical in x
i, x
i+1. The last expression being symmetrical in f, g, this proves that (
if, g) = (f,
ig).
4. Affine Schubert polynomials. Let H
nbe the linear subspace of C[x
1, . . . , x
n] generated by the monomials x
I= x
i11x
i22· · · x
innsuch that i
k≤ n − k. Let Π be the projector from C[X] onto H
nassociating to a polynomial P the unique representative in H
nof its class P ∈ C[X]/I
+.
Definition 4.1. Let µ ∈ S
n. The affine Schubert polynomial of index µ is defined by Z
µ= Π
µ−1ωZ
ωwhere Z
ω:= X
ω= x
n−11x
n−22· · · x
0n.
Example 4.2. For n = 3,
Z
321= x
21x
2Z
312= x
21Z
231= x
1x
2Z
213= x
1− 1/2x
1x
2− x
21Z
132= x
1+ x
2− x
1x
2− 1/2x
21Z
123= 1
In general, one has Z
ω= X
ω, Z
µ= X
µ+(terms of degree > `(µ)), and Z
id= 1, the last identity being due to the fact that
ω(Z
ω) is symmetrical with term of lowest degree X
id= 1.
Example 4.3. For n = 3,
Z
321= X
321Z
312= X
312Z
231= X
231Z
213= X
213− 1/2X
231− X
312Z
132= X
132− X
231− 1/2X
312Z
123= X
123Theorem 4.4. The polynomials Z
µ, µ ∈ S
n, form a basis of H
n. The quadratic form ( , ) is positive definite, and the adjoint basis of {Z
µ} is {Z
µ∨} where Z
µ∨= Π(∇
µ−1ωX
ω).
P r o o f. Z
µis a non-homogeneous polynomial with the Schubert polynomial X
µas its term of smallest degree. Since the classes of the Schubert polynomials form a basis of H
∗(F ), the same is true for the Z
µ.
The polynomials Z
ω∨Z
µθ (for µ 6= id) have no component of degree `(ω). Therefore, their images under ∂
ωare symmetric polynomials without constant term, which proves that for all µ 6= id, (Z
ω∨, Z
µ) = 0. On the other hand,
(Z
ω∨, Z
12...n) = (Z
ω∨, 1) = ∂
ω(X
ωθ) |
0= ∂
ω(X
ωθ)|
`(ω)= ∂
ωX
ω= X
12...n= 1.
For the general case of a Z
ν∨, one uses induction on the length of ν. Let ν and i be such that `(νσ
i) < `(ν). Then, for any µ and an appropriate constant k
(Z
µ, Z
ν∨) = (Z
µ, (s
i− k)Z
ν∨) = ((s
i− k)Z
µ, Z
ν∨) = k
0(Z
µ, Z
ν∨) + k
00(Z
µσi, Z
ν∨) (for some other scalars k
0, k
00). By induction, one can suppose (Z
µ, Z
ν∨) = 0 for µν
−16= ω.
One is thus reduced to study the case
µσ
iν
−1= ω, `(µσ
i) > `(µ).
In that case,
Z
µσi=
s
i+ 1
r
Z
µand Z
µσ∨i=
s
i− 1
r
Z
µ∨for a certain integer r. Then, we check
(Z
µσi, Z
νσ∨i
) =
s
i+ 1 r
Z
µ,
s
i− 1
r
Z
µ∨=
s
2i− 1 r
2Z
µ, Z
ν∨= 0,
and
(Z
µ, Z
νσ∨i
) =
s
i+ 1 r − 2
r
Z
µ, Z
ν∨= (Z
µσi, Z
ν∨) − 2
r (Z
µ, Z
ν∨) = 1 − 0.
Example 4.5. Again for n = 3, Z
321∨= x
21x
2Z
312∨= x
21− 2x
21x
2Z
231∨= x
1x
2− 2x
21x
2Z
213∨= x
1− 3/2x
1x
2− 3x
21+ 3x
21x
2Z
132∨= x
1+ x
2− 3x
1x
2− 3/2x
21+ 3x
21x
2Z
123∨= 1 − 4x
1− 2x
2+ 6x
1x
2+ 6x
21− 6x
21x
25. Change of basis. The operators ∂
iare self-adjoint with respect to h , i, but σ
iis adjoint to −σ
i. This implies that −s
iis adjoint to ¯ s
i:= σ
i− ∂
i.
Let us define
µ, ∇
µto be the images of
µand ∇
µunder the replacement s
i7→ ¯ s
i. We also define
Z
µ:= (−1)
`(ωµ)Π
µ−1ω
Z
ω, Z
∨µ:= (−1)
`(ωµ)Π
∇
µ−1ωZ
ω.
Then, (−1)
`(µ)Z
µis obtained from Z
µunder the transformation x
i7→ −x
i, since signs in the expansion of Z
µcorrespond to the degree.
Lemma 5.1. {Z
ωµ} is the adjoint basis of {Z
µ} with respect to h , i, i.e. one has hZ
ωµ, Z
µi = 1 and hZ
ωµ, Z
νi = 0 for ν 6= µ.
Similarly, {Z
∨ωµ} is the adjoint basis of {Z
µ∨} for h , i.
P r o o f. As in Section 4, the lemma is proved by induction on the length of µ, starting from the case
hZ
ωµ, Z
ωi = 0 if µ 6= ω.
Take i such that `(µσ
i) > `(µ). Then, hZ
ωµσi, Z
νi =
s
i+ 1 r
Z
ωµ, Z
ν=
Z
ωµ,
−¯ s
i+ 1 r
Z
ν.
Since (−¯ s
i+
1r)Z
νis a linear combination of Z
νand Z
νσi, the nullity of the scalar products hZ
ωµσi, Z
νi follows from those of hZ
ωµ, Z
νi for ν 6= µ and ν 6= µσ
i. In the special case ν = µσ
i, one has
hZ
ωµσi, Z
µi =
Z
ωµ,
−¯ s
i+ 1 r
−¯ s
i− 1 r
Z
µσi=
Z
ωµ,
1 − 1
r
2Z
µσiwhich is null.
Example 5.2.
hZ
23514, Z
41352i =
s
2+ 1
2
Z
25314,
−¯ s
2− 1 2
Z
43152=
s
2− 1
2
s
2+ 1 2
Z
25314, Z
43152=
1 − 1
4
Z
25314, Z
43152= 0.
Let {A
µ} and {B
ν} be two bases of H
n. We denote by M (A, B) the transition matrix from the basis {A
µ} to the basis {B
ν}, with the convention
A
µ= X
ν
M (A, B)
µνB
ν. (13)
For example, M (Z, X)
µν= hZ
µ, X
ωνi and M (X, Z)
µν= (X
µ, Z
ων∨). These matrices have a symmetry property, thanks to the following property of the scalar product:
hωP, ωQi = (−1)
`(ω)hP, Qi. (14)
Indeed, taking into account the two identities
ω(X
µ) = (−1)
`(µ)X
ωµω, ω(Z
µ) = (−1)
`(µ)Z
ωµω(15) we see that the four matrices
M (Z, X), M (X, Z), M (Z
∨, X) and M (X, Z
∨) possess the symmetry
M
µν= M
ωµω,ωνω. (16)
Furthermore, we have the following relation between these matrices and their inverses:
Theorem 5.3. The inverse of M (Z, X) is a matrix with nonnegative entries, given by
M (X, Z)
µν= |M (Z, X)
νω,µω|.
Similarly,
M (X, Z
∨)
µν= |M (Z
∨, X)
νω,µω|, where | · | denotes the absolute value.
P r o o f. The first matrix corresponds to the expansions Z
µ= X
ν
hZ
µ, X
ωνiX
ν. The inverse formulas are, according to Lemma 5.1,
X
ν= X
µ
hZ
ωµ, X
νiZ
µ.
But now, hZ
ωµ, X
νi = |hZ
ωµ, X
νi|, whence the first part of the theorem follows. The proof of the second part is similar.
Thus, the inverse of the matrix M (Z, X) is obtained from M (Z, X) by reflection through the antidiagonal (µ, ν) −→ (ων, ωµ) and suppression of the signs.
Corollary 5.4. For any pair of permutations, (X
µ, Z
η∨) = |hX
ωµω, Z
ωηωi|
and
(X
µ, Z
η) = |hX
ωµω, Z
ωηω∨i|.
Indeed,
X
µ= X
η
(X
µ, Z
η∨)Z
ωηbut we have just seen that the coefficients of the expansion of X
µin the basis Z
ωηare
the hZ
η, X
µi.
6. Schubert expansions of Chern classes. Let |I| be a composition of n, i.e.
I ∈ N
rwith |I| := i
1+ . . . + i
r= n, and let J
1, . . . , J
rbe the associated decomposition of the interval [1, n], that is
J
1= [1, i
1], J
2= [i
1+ 1, i
1+ i
2], . . . , J
r= [i
1+ · · · + i
r−1+ 1, n].
Let F
Ibe the variety of flags
V
0= {0} ⊂ V
1⊂ V
2⊂ . . . ⊂ V
r= V
such that dim V
k= i
1+ · · · + i
k. Let also S
Idenote the Young subgroup S(J
1) × S(J
2) × · · · × S(J
r) ⊂ S
nassociated to the composition I. The Chern class θ
Iof the tangent bundle of F
Iis the maximal S
I-invariant factor of θ:
θ
I= θ/(θ
J1θ
J2· · · θ
Jr) where
θ
Jk= Y
i<j i,j∈Jk
(1 + x
i− x
j).
A basis of the cohomology ring H
∗(F
I) is the set of Schubert polynomials X
µwith µ minimal in its right coset µS
I. In other words, one restricts the Schubert basis (X
µ) to those µ such that µ
1< . . . < µ
i1, µ
i1+1< . . . < µ
i1+i2, . . ., µ
i1+...+ir−1+1< . . . < µ
n. Since µ
i< µ
i+1iff X
µis symmetrical in x
iand x
i+1, the Schubert basis of H
∗(F
I) consists of those Schubert polynomials which are invariant under S
I.
Define the Chern coefficient c
µ[F
I] of the variety F
Ias the coefficient of (the class) of X
µin the expansion of θ
Ion the Schubert basis of H
∗(F
I). In other words,
c
µ[F
I] = hθ
I, X
ωµi. (17)
As in the case of the full flag variety F , these scalar products can be computed with the help of the scalar product ( , ).
Let ω
Ibe the maximal element of S
I, and ζ
I:= ω
Iω. We have seen that
ω
= ∂
ωθ = ∂
ζI∂
ωIθ
Iθ
J1θ
J2· · · θ
Jr. The operator ∂
ωfactorizes
∂
ω= ∂
ζi∂
ωIso that
c
µ[F
I] = ∂
ωθ
IX
ωµ= ∂
ζi∂
ωiθ
IX
ωµ= ∂
ζiθ
I∂
ωiX
ωµ(18)
= ∂
ζIθ
IX
ωµωI= ∂
ζiθ
IX
ωµωI(∂
ωIθ
J1θ
J2· · · θ
Jr/I!) (19)
= 1
I! ∂
ω(θ X
ωµωI) (20)
= 1
I! (1, X
ωµωI)
Equality (18) follows from the fact that θ
Iis invariant under S
Iand thus commutes
with ∂
ωI. Now, θ is of degree
n2, and ∂
ωdecreases degrees by `(ω) =
n2. Thus ∂
ω(θ)
is a scalar which is checked to be n!. More generally, by direct product, one has for the maximal element of the Young subgroup S
I∂
ωIθ
J1· · · θ
Jr= I! := i
1! · · · i
r! and equality (19) follows from this identity.
Since θ
Ias well as X
ωµωIare invariant under S
I, they commute with ∂
ωI, which is step (20).
Summarizing, we have the following expression for the components of the Chern class of F
Ion the Schubert basis.
Theorem 6.1. Let I = (i
1, . . . , i
r) be a composition of n, S
Iand F
Ithe correspond- ing Young subgroup and flag variety. Let µ be a permutation which is minimum in its coset µS
I. Then, the Chern coefficient c
µ[F
I] is given by
c
µ[F
I] = (1, X
ωµωI)/I!.
In particular, for the full flag variety (case I = (1, 1, . . . , 1)), one has
c
µ[F ] = (1, X
ωµ) =
ω(X
ωµ) (21) and these numbers constitute the first column of the matrix M (X, Z
∨). Equivalently, they are equal to the absolute values of the entries of the last row of M (Z
∨, X).
In the case of a Grassmann manifold G(p, p + q) = F
(p,q), the basis of H
∗(F
(p,q)) consists of those X
µfor which µ
1< . . . < µ
pand µ
p+1< . . . < µ
p+q(Grassmannian permutations). In fact, for such a permutation, X
µis equal to the Schur function indexed by the partition (µ
1− 1, µ
2− 2, . . . , µ
p− p) on the set of variables {x
1, . . . , x
p}. Thus, the Chern coefficient c
µ[F
(p,q)] is
c
µ[F
(p,q)] =
ωX
(n+1−µp,...,n+1−µ1,n+1−µn,...,n+1−µp+1). (22) For example, up to a factor (2!)
2, the Chern coefficients of F
(2,2)are 4, 16, 28, 28, 48, 24.
They are given by the absolute values of the six entries of the bottom row of the matrix M (Z
∨, X) corresponding to columns indexed by permutations ωµ where µ is Grassman- nian.
7. Tables for n = 4.
7.1. Affine Schubert polynomials.
Z
4321= x
31x
22x
3Z
4312= x
31x
22Z
4231= x
31x
2x
3Z
4213= x
31x
2− 1/2 x
31x
2x
3− x
31x
22Z
4132= x
31x
3+ x
31x
2− x
31x
2x
3− 1/2 x
31x
22Z
4123= x
31Z
3421= x
21x
22x
3Z
3412= x
21x
22Z
3241= x
21x
2x
3− 1/2 x
21x
22x
3− x
31x
2x
3Z
3214= x
21x
2− 2/3 x
21x
2x
3− 3/2 x
21x
22+ 1/3 x
21x
22x
3− 2 x
31x
2+ 2/3 x
31x
2x
3+ x
31x
22Z
3142= x
21x
3+ x
21x
2− x
21x
2x
3− 2/3 x
21x
22− x
31x
3− x
31x
2Z
3124= x
21− 1/2 x
21x
3− 1/2 x
21x
2+ 1/2 x
21x
2x
3− 2 x
31+ 1/2 x
31x
3+ 1/2 x
31x
2Z
2431= x
1x
22x
3+ x
21x
2x
3− x
21x
22x
3− 1/2 x
31x
2x
3Z
2413= x
1x
22− 1/2 x
1x
22x
3+ x
21x
2− 1/2 x
21x
2x
3− 2 x
21x
22+ 1/2 x
21x
22x
3− 1/2 x
31x
2+ 1/4 x
31x
2x
3+ 1/2 x
31x
22Z
2341= x
1x
2x
3Z
2314= x
1x
2− 2/3 x
1x
2x
3− x
1x
22− x
21x
2Z
2143= x
1x
3+ x
1x
2− 3/2 x
1x
2x
3− 2/3 x
1x
22+ 1/3 x
1x
22x
3+ x
21− 2 x
21x
3− 8/3 x
21x
2+ 7/3 x
21x
2x
3+ 4/3 x
21x
22− 1/3 x
21x
22x
3− 3/2 x
31+ 2 x
31x
3+ 7/3 x
31x
2− 2/3 x
31x
2x
3− 1/3 x
31x
22Z
2134= x
1− 1/2 x
1x
2+ 1/2 x
1x
22− x
21+ 1/2 x
21x
2+ x
31Z
1432= x
22x
3+ x
1x
2x
3+ x
1x
22− 2 x
1x
22x
3+ x
21x
3+ x
21x
2− 2 x
21x
2x
3− 3/2 x
21x
22+ x
21x
22x
3− 2/3 x
31x
3− 2/3 x
31x
2+ 2/3 x
31x
2x
3+ 1/3 x
31x
22Z
1423= x
22+ x
1x
2− x
1x
22+ x
21− x
21x
2− 2/3 x
31Z
1342= x
2x
3+ x
1x
3+ x
1x
2− 2 x
1x
2x
3− 1/2 x
21x
3− 1/2 x
21x
2+ 1/2 x
21x
2x
3+ 1/2 x
31x
3+ 1/2 x
31x
2Z
1324= x
2−1/2 x
2x
3− x
22+ x
1−1/2 x
1x
3−5/2 x
1x
2+ x
1x
2x
3+2 x
1x
22−3/2 x
21+ 1/4 x
21x
3+ 9/4 x
21x
2− 1/4 x
21x
2x
3− 1/2 x
21x
22+ x
31− 1/4 x
31x
3− 1/4 x
31x
2Z
1243= x
3+ x
2− x
2x
3− 1/2 x
22+ x
1− x
1x
3− 3/2 x
1x
2+ x
1x
2x
3+ 1/2 x
1x
22− 1/2 x
21+ 1/2 x
21x
2Z
1234= 1
7.2. Adjoint polynomials.
Z
4321∨= x
31x
22x
3Z
4312∨= x
31x
22− 2 x
31x
22x
3Z
4231∨= x
31x
2x
3− 2 x
31x
22x
3Z
4213∨= x
31x
2− 3/2 x
31x
2x
3− 3 x
31x
22+ 3 x
31x
22x
3Z
4132∨= x
31x
3+ x
31x
2− 3 x
31x
2x
3− 3/2 x
31x
22+ 3 x
31x
22x
3Z
4123∨= x
31− 2 x
31x
3− 4 x
31x
2+ 6 x
31x
2x
3+ 6 x
31x
22− 6 x
31x
22x
3Z
3421∨= x
21x
22x
3− 2 x
31x
22x
3Z
3412∨= x
21x
22− 2 x
21x
22x
3− 2 x
31x
22+ 4 x
31x
22x
3Z
3241∨= x
21x
2x
3− 3/2 x
21x
22x
3− 3 x
31x
2x
3+ 3 x
31x
22x
3Z
3214∨= x
21x
2− 4/3 x
21x
2x
3− 5/2 x
21x
22+ 2 x
21x
22x
3− 4 x
31x
2+ 4 x
31x
2x
3+ 6 x
31x
22− 4 x
31x
22x
3Z
3142∨= x
21x
3+ x
21x
2− 3 x
21x
2x
3− 4/3 x
21x
22+ 8/3 x
21x
22x
3− 3 x
31x
3− 3 x
31x
2+ 8 x
31x
2x
3+ 8/3 x
31x
22− 16/3 x
31x
22x
3Z
3124∨= x
21− 3/2 x
21x
3− 7/2 x
21x
2+ 9/2 x
21x
2x
3+ 5 x
21x
22− 4 x
21x
22x
3− 4 x
31+ 9/2 x
31x
3+ 25/2 x
31x
2− 12 x
31x
2x
3− 12 x
31x
22+ 8 x
31x
22x
3Z
2431∨= x
1x
22x
3+ x
21x
2x
3− 3 x
21x
22x
3− 3/2 x
31x
2x
3+ 3 x
31x
22x
3Z
2413∨= x
1x
22− 3/2 x
1x
22x
3+ x
21x
2− 3/2 x
21x
2x
3− 4 x
21x
22+ 9/2 x
21x
22x
3− 3/2 x
31x
2+ 9/4 x
31x
2x
3+ 9/2 x
31x
22− 9/2 x
31x
22x
3Z
2341∨= x
1x
2x
3− 2 x
1x
22x
3− 4 x
21x
2x
3+ 6 x
21x
22x
3+ 6 x
31x
2x
3− 6 x
31x
22x
3Z
2314∨= x
1x
2− 4/3 x
1x
2x
3− 3 x
1x
22+ 8/3 x
1x
22x
3− 5 x
21x
2+ 16/3 x
21x
2x
3+ 10 x
21x
22− 8 x
21x
22x
3+ 8 x
31x
2− 8 x
31x
2x
3− 12 x
31x
22+ 8 x
31x
22x
3Z
2143∨= x
1x
3+ x
1x
2−5/2 x
1x
2x
3−4/3 x
1x
22+2 x
1x
22x
3+ x
21−4 x
21x
3−16/3 x
21x
2+ 9 x
21x
2x
3+ 16/3 x
21x
22− 6 x
21x
22x
3− 5/2 x
31+ 7 x
31x
3+ 9 x
31x
2− 12 x
31x
2x
3− 6 x
31x
22+ 6 x
31x
22x
3Z
2134∨= x
1− 2 x
1x
3− 7/2 x
1x
2+ 5 x
1x
2x
3+ 9/2 x
1x
22− 4 x
1x
22x
3− 5 x
21+ 8 x
21x
3+ 31/2 x
21x
2− 18 x
21x
2x
3− 15 x
21x
22+ 12 x
21x
22x
3+ 11 x
31− 14 x
31x
3− 26 x
31x
2+ 24 x
31x
2x
3+ 18 x
31x
22− 12 x
31x
22x
3Z
1432∨= x
22x
3+ x
1x
2x
3+ x
1x
22− 4 x
1x
22x
3+ x
21x
3+ x
21x
2− 4 x
21x
2x
3− 5/2 x
21x
22+ 6 x
21x
22x
3− 4/3 x
31x
3− 4/3 x
31x
2+ 4 x
31x
2x
3+ 2 x
31x
22− 4 x
31x
22x
3Z
1423∨= x
22− 2 x
22x
3+ x
1x
2− 2 x
1x
2x
3− 5 x
1x
22+ 8 x
1x
22x
3+ x
21− 2 x
21x
3− 5 x
21x
2+ 8 x
21x
2x
3+ 10 x
21x
22− 12 x
21x
22x
3− 4/3 x
31+ 8/3 x
31x
3+ 16/3 x
31x
2− 8 x
31x
2x
3− 8 x
31x
22+ 8 x
31x
22x
3Z
1342∨= x
2x
3− 2 x
22x
3+ x
1x
3+ x
1x
2− 6 x
1x
2x
3− 2 x
1x
22+ 8 x
1x
22x
3− 7/2 x
21x
3− 7/2 x
21x
2+ 25/2 x
21x
2x
3+ 5 x
21x
22− 12 x
21x
22x
3+ 9/2 x
31x
3+ 9/2 x
31x
2− 12 x
31x
2x
3− 4 x
31x
22+ 8 x
31x
22x
3Z
1324∨= x
2− 3/2 x
2x
3− 3 x
22+ 3 x
22x
3+ x
1− 3/2 x
1x
3− 15/2 x
1x
2+ 9 x
1x
2x
3+ 13 x
1x
22−12 x
1x
22x
3−9/2 x
21+21/4 x
21x
3+73/4 x
21x
2−75/4 x
21x
2x
3−22 x
21x
22+ 18 x
21x
22x
3+ 6 x
31− 27/4 x
31x
3− 75/4 x
31x
2+ 18 x
31x
2x
3+ 18 x
31x
22− 12 x
31x
22x
3Z
1243∨= x
3+ x
2− 3 x
2x
3− 3/2 x
22+ 3 x
22x
3+ x
1− 5 x
1x
3− 13/2 x
1x
2+ 14 x
1x
2x
3+ 15/2 x
1x
22−12 x
1x
22x
3−7/2 x
21+11 x
21x
3+31/2 x
21x
2−26 x
21x
2x
3−15 x
21x
22+ 18 x
21x
22x
3+ 5 x
31− 14 x
31x
3− 18 x
31x
2+ 24 x
31x
2x
3+ 12 x
31x
22− 12 x
31x
22x
3Z
1234∨= 1 −2 x
3−4 x
2+6 x
2x
3+6 x
22−6 x
22x
3−6 x
1+10 x
1x
3+22 x
1x
2−28 x
1x
2x
3−
26 x
1x
22+ 24 x
1x
22x
3+ 16 x
21− 22 x
21x
3− 48 x
21x
2+ 52 x
21x
2x
3+ 44 x
21x
22− 36 x
21x
22x
3− 22 x
31+ 28 x
31x
3+ 52 x
31x
2− 48 x
31x
2x
3− 36 x
31x
22+ 24 x
31x
22x
37.3. Transition matrices with Schubert polynomials. The following matrices give the decompositions of the polynomials Z
µand Z
µ∨in the basis of Schubert polynomials. Rows and columns are indexed by permutations in reverse lexicographic order:
[4321, 4312, 4231, 4213, 4132, 4123, 3421, 3412, 3241, 3214, 3142, 3124, 2431, 2413, 2341, 2314, 2143, 2134, 1432, 1423, 1342, 1324, 1243, 1234]
The bar over a number is to be interpreted as a minus sign.
7.3.1. M (Z, X). The entry in row µ and column ν of the following matrix is equal to the coefficient of X
νin Z
µ. This number is also the coefficient of Z
ωµ∨in X
ων.
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1
121 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0
121 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0
120 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1
232 0 0
13 32 231 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0
231 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0
122 0 0
120
121 0 0 0 0 0 0 0 0 0 0 0 0
0 0
120 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0
12 14 120 0
122 0 0 0 0
121 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1
231 0 0 0 0 0 0 0 0
0
13 23 132
32 13 432 0 2 0
13 23 320 1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 1 0
120
120 1 0 0 0 0 0 0
0
13 230
230 1
320 0 0 0 2 0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0
230 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0
0 0 0 0
120 0 0
120
120 0 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0
141 0
12 140
14 120 2 1 1 0 0 0 1
121 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
121 0 0 0 0
121 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
7.3.2. M (Z
∨, X). The entry in row µ and column ν of the following matrix is equal to the coefficient of X
νin Z
µ∨. This number is also the coefficient of Z
ωµin X
ων.
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 3
321 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3
323 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
6 6 6 2 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 2 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 0 3 0 0 0
320 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 6 4 4 0 0 2
52 431 0 0 0 0 0 0 0 0 0 0 0 0 0 0
16 3
8
3
8 0 3 0
83 433 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
8 12 12 8
924 4 5
922
321 0 0 0 0 0 0 0 0 0 0 0 0
3 0
320 0 0 3 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
9 2
9 2
9 4
3
2
