J. Symbolic Computation (2002) 33, 507–517 doi:10.1006/jsco.2001.0506 Available online at http://www.idealibrary.com on
A Mathematica Program for the Degrees of Certain Schubert Varieties XU AN ZHAO† AND HAIBAO DUAN‡ †
Department of Mathematics,Beijing Normal University, Beijing 100875, People’s Republic of China ‡ Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China
This paper explains a Mathematica program that computes the degrees of Schubert varieties on the flag manifold G/T for G = U (n), SO(n) and Sp(n). c 2002 Elsevier Science Ltd
1. Introduction A fundamental invariant of a complex projective variety X ⊂ CP m is its degree, defined by Z deg X = κ(X)k , k = dimC X, X
where κ(X) is the restriction of the standard Kaehler form on CP m to X. The importance of this invariant is seen from its various interpretations (Griffith and Harris, 1978, P. 171). (1) If F is a polynomial system on CP m whose zero locus is X, then deg X is the number of solutions to the system F = 0; L = 0, where L is a general linear system on CP m of rank m − k; (2) deg X is equal to the number of intersection points of X with a general linear subspace of complementary dimension; (3) the number k! deg X gives the volume of X. Let G be a compact, connected Lie group with a fixed maximal torus T ⊂ G and Weyl group W . The length function on W is denoted by l : W → Z. It is well known that the flag variety G/T admits a canonical decomposition into cells, indexed by elements in W , [ G/T = Xw , dim Xw = 2l(w), w∈W
with each cell Xw a projective variety, known as a Schubert variety on G/T (cf. Chevalley, 1958; Borel, 1969; Bernstein et al., 1973). This note concerns itself with the following. † E-mail:
[email protected] [email protected] Dedicated to Prof. Boju Jiang on his 65th birthday.
‡ E-mail:
0747–7171/02/040507 + 11
$35.00/0
c 2002 Elsevier Science Ltd
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Problem 1. Find the correspondence deg : W → Z, w → deg Xw , for a given G. It was announced in Duan (2001) that, using the so called “enumerative formula” on a flag manifold G/T developed in Duan (2001), it is possible to compose Mathematica programs to solve Problem 1. This paper gives the mathematical background and a program detail for the case G = U (n), the unitary group of order n. This answers, in particular, a question raised by Monk (1959). We indicate by the way that similar algorithms apply equally well to the cases G = SO(n) the special orthogonal group of order n; and Sp(n) the symplectic group of order n. By doing this we wish to explain also to our audience that the computation of some seemingly sophisticated geometrical invariants can be boiled down to a pure algebraic or combinatorial computation. The paper is arranged as follows. Section 1 recalls basic cohomology properties of the flag manifolds we are considering. These will be useful, in Section 2, to introduce the divided difference operators that yield a set of Schubert polynomials for G/T (Bernstein et al., 1973; Demazure, 1973; Monk, 1959). In Section 4 the degree of a Schubert variety on G/T is expressed in terms of integration of certain variation of Schubert polynomials, and the enumerative formulas obtained in Duan (2001), playing the role of transforming the integration on G/T by derivative, will be reviewed in Section 5. Section 6 discusses the algorithm and finally the program is described in Section 7. All homology and cohomology of a manifold M , denoted by H∗ (M ) and H ∗ (M ), will have integer coefficient. The Kronecker pairing, between cohomology and homology, is denoted by h, i : H r (M ) × Hr (M ) → Z. Further, if M is a closed oriented n-manifold, [M ] ∈ Hn (M ) stands for the orientation class. It is understood that a flag manifold G/T has a canonical orientation inherited from the Kaehlerian form on G/T .
2. Preliminaries We recall some concepts and facts connected with the flag variety G/T . L(G) denotes the Lie algebra of G and T ⊂ G, a fixed maximal torus. Let ∆+ ⊂ L(T ) be the set of positive roots and π ⊂ ∆+ , the system of simple roots. There are canonical correspondences σ : ∆+ → W ;
c : ∆+ → H 2 (G/T )
defined respectively by σ(γ) = the reflection on L(T ) in the hyperplane through the origin orthogonal to γ ∈ ∆+ ; and c(γ) = the first Chern class of the complex line bundle (over G/T ) associated with γ ∈ ∆+ (Bernstein et al., 1973).
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The natural action of W on G/T yields a faithful representation W → Aut(H 2 (G/T ))(Duan and Zhao, 2000), and W will be identified with its image group in Aut(H 2 (G/T )). Let Σn be the permutation group on {1, 2, . . . , n} and Sn , the full permutation group on {±1, ±2, . . . , ±n}. The kernel of the standard homomorphism det : Sn → {±1} is denoted by
Sn+ .
It is well known that the Weyl groups for the Lie groups G = U (n), SO(2n), SO(2n + 1)
and
Sp(n)
are respectively Σn , Sn+ , Sn and Sn . Let ei (z1 , . . . , zn ) be the ith elementary symmetric function in the variables z1 , . . . , zn . If x ∈ H r (G/T ) we write |x| = r. We describe now the integral cohomologies, and make the above set-up more concrete for the four types of flag manifold An = U (n)/T : Bn = SO(2n)/T ;
Cn = SO(2n + 1)/T ;
Dn = Sp(n)/T.
Borel (1967) and Duan and Zhao (2000) might be appropriate references to results listed below. Theorem A-1. The integral cohomology of An can be given by H ∗ (An ) = Z[x1 , . . . , xn ]/hei (x1 , . . . , xn ), 1 ≤ i ≤ ni, |xi | = 2; so that the action of Σn on H 2 (An ) = span{x1 , . . . , xn } is xi → xw(i) , The set of positive roots ∆
+
w ∈ Σn .
can be ordered as {γij | 1 ≤ i < j ≤ n}; so that
(1) π = {γi,i+1 | 1 ≤ i ≤ n − 1}; and (2) the correspondences σ : ∆+ → Σn ; c : ∆+ → H 2 (An ) are given respectively by σ(γij ) = (i, j);
and
c(γij ) = xj − xi . 2
Theorem B-1. The integral cohomology of Bn can be given by H ∗ (Bn ) = Z[x1 , . . . , xn ; α1 , . . . , αn−1 ]/hei (x1 , . . . , xn ) − 2αi , 14 ei (x21 , . . . , x2n ), 1 ≤ i ≤ n − 1i, |xi | = 2; so that the action of Sn+ on H 2 (Bn ) = span{x1 , . . . , xn } is xw(i) if w(i) > 0 xi → , w ∈ Sn+ . −x|w(i)| if w(i) < 0 The set of positive roots ∆+ can be ordered as {γij ; βij | 1 ≤ i < j ≤ n}; so that (1) π = {γi,i+1 , βn−1,n | 1 ≤ i ≤ n − 1}; and (2) the correspondences σ : ∆+ → Sn+ ; c : ∆+ → H 2 (Bn ) are given respectively by σ(γij ) = (i, j);
σ(βij ) = (−i, −j);
and c(γij ) = xj − xi ;
c(βij ) = xj + xi . 2
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Theorem C-1. The integral cohomology of Cn can be given by H ∗ (Cn ) = Z[x1 , . . . , xn ; α1 , . . . , αn ]/ h ei (x1 , . . . , xn ) − 2αi , 14 1ei (x21 , . . . , x2n ), 1 ≤ i ≤ ni, |xi | = 2; so that the action of Sn on H 2 (Cn ) = span{x1 , . . . , xn } is xw(i) if w(i) > 0 xi → , w ∈ Sn . −x|w(i)| if w(i) < 0 The set of positive roots ∆+ can be ordered as {γij ; βij | 1 ≤ i < j ≤ n} ∪ {αi | 1 ≤ i ≤ n} so that (1) π = {γi,i+1 , αn | 1 ≤ i ≤ n − 1}; and (2) the correspondences σ : ∆+ → Sn ; c : ∆+ → H 2 (Cn ) are given respectively by σ(γij ) = (i, j);
σ(βij ) = (−i, −j);
σ(αi ) = (−i)
c(βij ) = xj + xi ;
c(αi ) = xi . 2
and c(γij ) = xj − xi ;
Theorem D-1. The integral cohomology of Dn can be given by H ∗ (Dn ) = Z[x1 , . . . , xn ]/hei (x21 , . . . , x2n ), 1 ≤ i ≤ ni,
|xi | = 2;
2
so that the action of Sn on H (Dn ) = span{x1 , . . . , xn } is xw(i) if w(i) > 0 xi → , w ∈ Sn . −x|w(i)| if w(i) < 0 The set of positive roots ∆+ can be ordered as {γij ; βij | 1 ≤ i < j ≤ n} ∪ {αi | 1 ≤ i ≤ n}; so that (1) π = {γi,i+1 , αn | 1 ≤ i ≤ n − 1}; and (2) the correspondences σ : ∆+ → Sn ; c : ∆+ → H 2 (Dn ) are given respectively by σ(γij ) = (i, j);
σ(βij ) = (−i, −j);
σ(αi ) = (−i)
c(γij ) = xj − xi ;
c(βij ) = xj + xi ;
c(αi ) = 2xi . 2
and
3. The Schubert Polynomials Let sw ∈ H2l(w) (G/T ), w ∈ W , be the image of the fundamental class [Xw ] ∈ H2l(w) (Xw ) under the map induced by the inclusion XS w ⊂ G/T . Since only even dimensional cells are involved in the decomposition G/T = w∈W Xw , the set {sw | w ∈ W } forms a free basis for H∗ (G/T ). Let gw ∈ H 2l(w) (G/T ) be the Kronecker dual of sw . The class sw (resp. gw ) is known as the Schubert cycle (resp. Schubert cocycle) associated with w ∈ W (Bernstein et al., 1973). Theorem 2. The Poincare duality on G/T , at the level of cohomology, is gw → gww0 , where w0 ∈ W is the element on which l attains its maximum 21 dim M (Bernstein et al., 1973). 2
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A common feature of the ring H ∗ (M ), M = G/T , is that it can be identified with a polynomial algebra Z[x1 , . . . , xn ] graded by |xi | = 2 , n = rankG, subject to certain homogeneous relations p1 , . . . , pn i.e. H ∗ (M ) = Z[x1 , . . . , xn ]/hp1 , . . . , pn i. For the four families of spaces under consideration, this has been indicated through Theorem A-1 to Theorem D-1. For simplicity, abbreviate the polynomial ring Z[x1 , . . . , xn ] by Z[M ], the ideal generated by the pi by rM , and write Z[M ]r for the subset of all polynomials of homogeneous degree r. Let pM : Z[M ] → H ∗ (M ) be the quotient map. A polynomial fw ∈ Z[M ]2l(w) , w ∈ W , with pM (fw ) = gw is known as a Schubert polynomial associated with w. The problem For an M = G/T , find a set of Schubert polynomials {fw ∈ Z[M ] | w ∈ W } was studied and solved at the same time by Bernstein et al. (1973) and Demazure (1973) (a survey for type An can be found in Macdonald and Harris, 1978). We review their result. In Bernstein et al. (1973) and Demazure (1973), the authors introduced, for each root γ ∈ ∆+ , the divided difference operator Aγ : Z[M ] → Z[M ] corresponding to γ (of degree −2) by f − σ(γ)f . Aγ (f ) = c(γ) They proved the following. Theorem 3. If w ∈ W has a reduced decomposition w = σ(γ1 ) ◦ σ(γ2 ) ◦ · · · ◦ σ(γl ) with γi ∈ π and l = l(w), then the composed operator Aw = Aγ1 ◦ · · · ◦ Aγl depends only on w and not on a specific choice of the decomposition w = σ(γ1 ) ◦ σ(γ2 ) ◦ · · · ◦ σ(γl ). Further, fix a f0 ∈ Z[M ]dim M with hpM (f0 ), [M ]i =1. Then a set of Schubert polynomials for M is given by {fw = Aw f0 | w ∈ W }. 2 Theorem 3 was used by Billey and Haiman (1995) to determine a set of Schubert polynomials for M = An , Bn , Cn and Dn . A Maple program to manipulate a set of Schubert polynomials for M = An was written by Veigneau (1997). For other methods to compute Schubert polynomials, see Remark 2 at the end of Section 6. 4. The Integration Along G//T Suppose that the integral cohomology of M = G/T is given as Z[x1 , . . . , xn ]/hp1 , . . . , pn i. Consider the additive correspondence Z : Z[x1 , . . . , xn ] → Z
Z
M
As indicated by the notation, the operator M ” in De Rham theory.
h = hpM (h), [M ]i.
by M
R M
can be interpreted as “integration along
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Problem 2. Find the correspondence
R M
: Z[x1 , . . . , xn ] → Z.
Assume that a set {fw ∈ Z[M ] | w ∈ W } of Schubert polynomials for M has been specified. Then, for an f ∈ Z[x1 , . . . , xn ], pM (f ) ∈ H ∗ (M ) can be expressed uniquely as a linear combination of the Schubert cocycles X pM (f ) = aw (f )gw , aw (f ) ∈ Z. w∈W
The Poincare duality on G/T (Theorem 2) implies the following. Theorem 4. aw (f ) =
R M
f · fww0 , w ∈ W . 2
We observe, by virtue of Theorem 4, that the effective computability of to a number of enumerative problems on the flag variety M = G/T .
R M
is related
Observation 1. The Kaehler form on M = G/T is known to be P κ(M ) = 12 c(γ). γ∈∆+
From this we get, by Theorem A-1, B-1, C-1 and D-1, that P (n − i)xi , if 1≤i≤n P if 1≤i≤n(n − i)xi , κ(M ) = P if (n − i + 21 )xi , 1≤i≤n P (n − i + 1)xi , if
M = An M = Bn M = Cn M = Cn .
1≤i≤n
For a w ∈ W , deg Xw is the coefficient of gw in f = κ(M )l(w) . In other word, the correspondence deg : W → Z is given by Z w → aw (κ(M )l(w) ) = κ(M )l(w) · fww0 . 2 M
Observation 2. The total Chern class of the cotangent bundle of M = G/T is Y c(M ) = (1 + c(γ)). γ∈∆+
From this we get, by Theorem A-1, B-1, C-1 and D-1, that Q (1 + xj − xi ), 1≤i
