Chow rings of Complex Algebraic Groups Shizuo Kaji joint with Masaki Nakagawa Workshop on Schubert calculus 2008 at Kansai Seminar House Mar. 20, 2008
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
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
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.
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 )
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.
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.
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.
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.
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).
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.
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.
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 3 2 + 2c 6 3t 2 0 γ 4 + t 6 0 ρ 8 =3γ 4 2 6tγ 3 γ 4 9t 2 0 c 6 + 15t 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 4 3 + 6t 0 γ 3 γ 4 2 + 3t 2 0 c 6γ 4 + 5t 3 0 c 6γ 3 15t 4 0 γ 4 2 10t 6 0 c 6 + 19t 8 0 γ 4 6t 9 0 γ 3 2t 12 0
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 342 + 2σ 542 t 4 σ 4 σ 5 γ 4 σ 1342 + 2σ 3542 + 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
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 3 6542)
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
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
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, X15 2 + X 10 3 + 2X 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.
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.
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)
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?
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.
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