Chow rings of Complex Algebraic Groups

Transcription

1 Chow rings of Complex Algebraic Groups Shizuo Kaji joint with Masaki Nakagawa Workshop on Schubert calculus 2008 at Kansai Seminar House Mar. 20, 2008

2 Outline Introduction Our problem (algebraic geometory) Cohomology of flag variety Borel presentation (algebraic topology) Schubert presentation (geometry) Divided difference operator (combinatorics) Computations and Main Theorems (man & computer power) Future Work

3 Notations G: simply connected simple complex Lie group B: Borel subgroup of G l: rank of G G/B: a projective variety called the flag variety associated to G H (G/B; Z): ordinary integral cohomology of G/B A (G): Chow ring of G

4 Main goal General Goal Determine A (G) for all simply connected simple complex Lie groups Classification Theorem tells that G is one of the following types: SL n, Spin(n), Sp(n), G 2, F 4, E 6, E 7, E 8 Grothendieck considered the problem in the 1950 s. He gave a formula to compute it from H (G/B; Z). Consequently, A (G) was determined to be trivial for G = SL n, Sp(n). A (G) Z/p was determined by Kac(1985) for all G. A (G) for G = Spin(n), G 2, F 4 were determined by R.Marlin(1974). His method seems to be hopeless for other exceptional types. (Note: Nakagawa also checked the result of Marlin by the same method we use here). Our Goal Today Determine A (G) for G = E 6, E 7, E 8.

5 What is Chow ring A (X ): the Chow ring of a non-singular variety X A (X ) = i 0 Ai (X ) A i (X ) is a group of the rational equivalence classes of algebraic cycles of codimension i. (an algebraic cycle is a linear sum of possibly singular subvarieties) intersection product A i (X ) A j (X ) A i+j (X )

6 Basic Facts Theorem (Grothendieck(1958)) the cycle map cl : A (G/B) H 2 (G/B; Z) is an isomorphism of rings: A (G/B) H 2 (G/B; Z) = H (G/B; Z). the pullback of the projection p : G G/B induces a surjection p : A (G/B) A (G), where the kernel is an ideal generated by A 1 (G/B). Corollary A (G) = H (G/B; Z)/(H 2 (G/B; Z)) Note: Since H (G/B; Q) is generated by degree 2 elements, A (G) Q = Q for all G. For G = SL n, Sp n, H (G/B; Z) is also generated by degree 2 elements, and so A (G) = Z.

7 Strategy A (G) = H (G/B; Z)/(H 2 (G/B; Z)) A presentation for H (G/B; Z) was given by Borel. It is called Borel presentation, which is a quotient of a polynomial ring divided by some ideal. Ring structure is clear, but generators have little geometric meaning. H (G/B; Z) has another module basis consisting of by Schubert classes. Schubert classes come from subvarieties called Schubert varieties. Ring structure is complicated, so it is difficult to use Grothendieck s Theorem. Hence, what we will do are: easy Compute A (G) purely algebraically from Borel presentation. difficult Find Schubert varieties representing the generators. Main tool We use the divided difference operator given by Demazure and Berstein-Gelfand-Gelfand.

8 Borel presentation K(= G R ): maximal compact subgroup of G T (= T R ): maximal compact torus of K (=K B = (S 1 ) l ) BT : classifying space of T (=(CP ) l ) W : Weyl group of K (=N(T )/T ) {ω i } 1 i l : fundamental weights and H (BT ; Z) = Symh Z = Z[ω 1,..., ω l ] Inclusion K G induces a diffeomorphism K/T = G/B. the classifying map K/T ι BT of the T -bundle T K K/T induces the characteristic map ι : H (BT ; Z) H (K/T ; Z) Theorem (Borel(1953)) ι induces H (BT ; Q)/I W H (K/T ; Q), where I W = (H + (BT ; Q) W ) an ideal generated by the W -invariants of positive degrees.

9 Borel presentation Toda(1975) extended Borel s work to give H (K/T ; Z) by a quotient ring of a polynomial ring. Based on Toda s method, H (G/B; Z) = H (K/T ; Z) were explictly determined for all G. Theorem ι : H (BT ; Z) Z[γ di ] (ideal) H (K/T ; Z), γ di = 2d i.

10 Schubert presentation The Bruhat decomposition of G gives a cell decomposition G = w W G/B = w W BwB BwB/B. l(w): length of w W, w 0 W : the longest element X w = closure of Bw 0 wb/b( = C l(w0w) ): Schubert variety σ w = {the cohomology class corresponding to X w } H 2l(w) (G/B; Z): Schubert class corresponding to w {σ w } w W forms an additive basis for H (G/B; Z).

11 Comparison of the two presentations Hence we have two descriptions for H (K/T ; Z) = H (G/B; Z) = A (G/B) Borel presentation Schubert presentation elements polynomials Schubert classes geometry no algebraic cycles ring structure easy hard Demazure and BGG s divided difference operator bridges those two presentations.

12 Divided difference operator We can switch between Borel presentation and Schubert presentation. Theorem (B-G-G(1973), Demazure(1973)) For w W, they defined w : H (BT ; Z) H 2l(w) (BT ; Z). (characteristic map) c : H 2k (BT ; Z) H 2k (K/T ; Z) defined by c(f ) = w (f )σ w (Note: w (f ) Z) (Giambelli formula) ( σ w = c ( w 1w0 How to calculate? l(w)=k α + α W α (ω β ) = δ αβ α (fg) = α (f )g + s α (f ) α (g) )), + : the set of positive roots.

13 Borel presentation for H (E 6 /T ; Z) Theorem (Toda-Watanabe(1974)) H (E 6 /T ; Z) = Z[t 1, t 2,..., t 6, t 0, γ 3, γ 4 ] (ρ 1, ρ 2, ρ 3, ρ 4, ρ 5, ρ 6, ρ 8, ρ 9, ρ 12 ) ( t i = 2, γ i = 2i) ρ 1 =c 1 3t 0 ρ 2 = c 2 4t 2 0 ρ 3 =c 3 2γ 3 ρ 4 = c 4 + 2t 4 0 3γ 4 ρ 5 =c 5 3t 0 γ 4 + 2t 2 0 γ 3 ρ 6 = γ c 6 3t 2 0 γ 4 + t 6 0 ρ 8 =3γ 4 2 6tγ 3 γ 4 9t 2 0 c t 4 0 γ 4 6t 5 0 γ 3 t 8 0 ρ 9 =2c 6 γ 3 3t 3 0 c 6 ρ 12 =3c 2 6 2γ t 0 γ 3 γ t 2 0 c 6γ 4 + 5t 3 0 c 6γ 3 15t 4 0 γ t 6 0 c t 8 0 γ 4 6t 9 0 γ 3 2t 12 0

14 Correspondence Using the characteristic map, we can translate the generators {t 1, t 2,..., t 6, t, γ 3, γ 4 } in Borel presentation into Schubert classes. Borel Schubert Borel Schubert t 1 σ 1 + σ 2 t 6 σ 6 t 2 σ 1 + σ 2 σ 3 t σ 2 t 3 σ 2 + σ 3 σ 4 γ 3 σ σ 542 t 4 σ 4 σ 5 γ 4 σ σ σ 6542 t 5 σ 5 σ 6 Furthermore, we wish to take a single Schubert class for each generator. In this E 6 case, for example, we can take the following classes: σ 342 = γ 3 + 2t 3 σ 1342 = γ 4 2tγ 3 + 2t 4

15 A(E 6 ) By Grothendieck s Theorem, where A (G) = A (G/B)/(A 1 (G/B)) = H (G/B; Z)/(H 2 (G/B; Z)) = H (K/T ; Z)/(H 2 (K/T ; Z)), H (K/T ; Z) = Z[t 1,..., t l, t 0, γ d1,...]/(ρ j1,...) H 2 (K/T ; Z) = Z{t 1,..., t l, t 0 } Therefore to obtain A (G) from H (K/T ; Z), we simply put t i = 0, (0 i l) in Borel presentation. H (E 6 /T ; Z)/(t 1,..., t 6, t 0 ) = Z[γ 3, γ 4 ]/(2γ 3, 3γ 4, γ 2 3, γ 3 4) = Z[σ 542, σ 6542 ]/(2σ 542, 3σ 6542, σ 2 542, σ )

16 Main Theorems p : G G/B: projection Theorem (K-Nakagawa) A(E 6 ) = Z[X 3, X 4 ]/(2X 3, 3X 4, X 2 3, X 3 4 ) X 3 = p (X w0s 5s 4s 2 ) = B(w 0 s 5 s 4 s 2 )B G X 4 = p (X w0s 6s 5s 4s 2 ) = B(w 0 s 6 s 5 s 4 s 2 )B G

17 A(E 7 ) Theorem (K-Nakagawa) A(E 7 ) = Z[X 3, X 4, X 5, X 9 ] /(2X 3, 3X 4, 2X 5, X 2 3, 2X 9, X 2 5, X 3 4, X 2 9 ) X 3 = p (X w0s 5s 4s 2 ) = B(w 0 s 5 s 4 s 2 )B G X 4 = p (X w0s 6s 5s 4s 2 ) = B(w 0 s 6 s 5 s 4 s 2 )B G X 5 = p (X w0s 7s 6s 5s 4s 2 ) = B(w 0 s 7 s 6 s 5 s 4 s 2 )B G X 9 = p (X w0s 6s 5s 4s 3s 7s 6s 5s 4s 2 ) = B(w 0 s 6 s 5 s 4 s 3 s 7 s 6 s 5 s 4 s 2 )B G

18 A(E 8 ) Proposition (K-Nakagawa) A(E 8 ) = Z[X 3, X 4, X 5, X 6, X 9, X 10, X 15 ] / 2X 3, 3X 4, 2X 5, 5X 6, 2X 9, X5 2 3X 10, X4 3, 2X 15, X9 2, 3X 10 2, X 3 8, X X X 6 5 X i = p (γ i ) (i = 3, 4, 5, 6, 9, 10, 15) Note: here X i may not be the pull-back of a single Schubert variety but a linear combination of them.

19 Case of G = Spin(n), G 2, F 4 Theorem (Marlin, Nakagawa) A(F 4 ) = Z[X 3, X 4 ]/(2X 3, 3X 4, X 2 3, X 3 4 ), where X 3 = B(w 0 s 1 s 2 s 3 )B, X 4 = B(w 0 s 1 s 2 s 3 s 4 )B. where X 3 = B(w 0 s 1 s 2 s 1 )B. A(G 2 ) = Z[X 3 ]/(2X 3, X 2 3 ), A(Spin(2n + 1)) = Z[X 3, X 5,..., X 2[ n+1 2 ] 1]/(2X i, X p i i ), where X i = B(w 0 s n i+1 s n 1 s n )B (1 i n) and p i = 2 [log 2 n i ]+1. A(Spin(2n)) = Z[X 3, X 5,..., X 2[ n 2 ] 1 ]/(2X i, X p i i ), where X 1 = B(w 0 s n )B, X i = B(w 0 s n i... s n 2 s n )B (2 i n 1)) and p i = 2 [log 2 n 1 i ]+1.

20 Future Work Determine which Schubert classes belong to the decomposable ideal. (equivalently, find indecomposable Schubert classes) Find a presentation of a given Schubert class σ w as a polynomial in a fixed set of ring generators. (Schubert polynomial of type G 2, F 4, E l (l = 6, 7, 8)) Replace B with any parabolic subgroup P in the above problems. (Note: there is a ring monomorphism H (G/P; Z) H (G/B; Z) described in terms of Schubert presentation)

21 Finding a set of ring generators How to determine which Schubert classes can be chosen as generators? This question can be formulated as follows. Definition R: graded commutative ring with R 0 = Z R : non-invertible elements of R decomposable ideal: (R R ) x R is indecomposable when x 0 R/(R R ) In our setting when R = H (G/B; Z): There is at most one ring generator in each degree H >2 (G/B; Z). If we find an indecomposable σ w H 2d (G/B; Z), then we take it as a generator γ d. Related question Which Schubert classes are indecomposable?

22 Example (finding indecomposables) H (F 4 /B; Z) has generators only in degrees 2, 6, and 8. H 2 (F 4 /B; Z) is spanned by σ w, where W = [1], [2], [3], [4], the length one elements in the Weyl group. Of course they are indecomposable. Out of 16(= dim H 6 (F 4 /B; Z)), the indecomposables are: W = [1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 2], [3, 2, 1], [3, 2, 3] Out of 25(= dim H 8 (F 4 /B; Z)), the indecomposables are: W =[1, 2, 3, 4], [1, 2, 3, 2], [1, 2, 4, 3], [1, 3, 2, 3], [1, 3, 2, 4], [1, 4, 3, 2] [2, 1, 3, 4], [2, 1, 4, 3], [2, 3, 2, 1], [2, 3, 2, 4], [2, 4, 3, 2] [3, 2, 1, 3], [3, 2, 1, 4], [3, 2, 3, 4] [4, 3, 2, 1], [4, 3, 2, 3] Note: there are more than one way to express an element of Weyl group by the products of the simple reflections.

23 Thank you for listening