Computers work exceptionally on Lie groups
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1 Computers work exceptionally on Lie groups Shizuo Kaji Fukuoka University First Global COE seminar on Mathematical Research Using Computers at Kyoto University Oct. 24, 2008 S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 1 / 1
2 Outline Introduction Computation of topological invariants s of Lie groups Schubert calculus (geometry) Divided difference operator (combinatorics) Borel presentation (algebraic topology) Computer assisted part Open problems S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 2 / 1
3 Introduction About presentation Latex-Beamer This presentation slide is made with LaTeX-Beamer. It is an easy-to-use L A TEXpackage, free of charge It produces a PDF file, which is almost environment independent There are a lot of people who use it; you can ask, consult web pages, even request new features It has the capability of and hyperlinks Main Theorem this kind of gimmicks S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 3 / 1
4 Introduction About presentation Latex-Beamer Theorem A mathematician is an optimist Proof. I have discovered a truly remarkable proof which this margin is too small to contain. S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 4 / 1
5 Introduction Computer assisted mathematics Advantages and Disadvantages Advantages of using computers are: Theorem can be proven while you are sleeping Everyone can confirm the computation The development of computers and softwares might produce new theorems Disadvantages of using computers are: It requires non-essential, non-mathematical work to make a practical program (sometimes we may have to struggle with bugs in compilers, OS, CPU...) It looks less elegant than human proof consequently, results are often underestimated by those who don t use computers Where and in what form can we submit results? S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 5 / 1
6 Common process Computation of topological invariants General setting Problem Compute some invariant F (X) for a space X, where F is a functor from spaces to some algebras Look for a theory which enables a concrete calculation Compute easy cases by hand to get insight Translate the mathematical theory into computer algorism Search for parts (libraries) made by others Implement of the data structure that corresponds to mathematical objects (polynomial, DGA, Lie algebra, Hopf algebra,... ) Choose programming language, software, etc. In algebraic topology, we usually resort to symbolic computation rather than numerical one Run the program and pray! Optimize it from both mathematical and computer s point of view Confirm the result by different algorism, softwares, platforms,... Look at the result to find general theory behind it S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 6 / 1
7 Lie group basic Computation of topological invariants Lie group G: simple, simply-connected, compact Lie group G T : maximal torus (dim T = l is the rank of G) t g: their Lie algebras with an invariant inner product (, ) t Π = {α i } 1 i l : simple roots 2α j {ω i } 1 i l : fundamental weights ( ( (α, ω j,α j) i) = δ ij ) H (BT ; Z) = Z[w 1,..., w l ], where w i = 2 s i GL(t ): simple reflection corresponding to α i, s i Aut(Z[w 1,..., w l ]) ( s i (e) = e ( 2αi (α, e)α i,α i) i ) W = N(T )/T : Weyl group of G (= a finite group generated by {s i } 1 i l ) Classification: A n, B n, C n, D n, G 2, F 4, E 6, E 7, E 8 These objects are all concrete (symbolic). However, for example, w i s are possibly irrational vectors so we need to handle with care. (Stembridge s coxeter/weyl package in Maple is so convenient that this often encourage me to choose Maple) S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 7 / 1
8 Computation of topological invariants Lie group Computable invariants rational cohomology is the invariant ring H (BG; Q) = H (BT ; Q) W mod p cohomology H (BG; F p ) the invariant rings H (BT ; F p ) W and H (B(Z/p) n ; Z/p) Wp rational cohomology of flag variety is the coinvariant ring H (G/T ; Q) = H (BT ; Q)/(H + (BT ; Q) W ) stable homotopy group π S (G) free resolution of H (G; F p ) as A p -algebra Grothendieck s torsion index of G Gröbner basis of (H + (BT ; Z) W ) H (BT ; Z) H (ΩG), H (G/P ), K (ΩG), cat(g),... self-equivalence of generalized flag variety Aut(G/P ) Aut(H (G/P ; Q)), where P is a parabolic subgroup S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 8 / 1
9 An example of invariants: The Chow ring A (G) S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 9 / 1
10 Notation Notation G: simply connected simple complex Lie group (SL(n, C)) B: Borel subgroup of G (the subgroup of upper triangular matrices) G/B: a projective variety called the flag variety (the space of flags, 0 V 1 V 2 V n 1 V n = C n, dim C (V i ) = i). H (G/B; Z): ordinary integral cohomology of G/B A (G): Chow ring of G A (G) = L i 0 Ai (G) A i (G) 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 (G) A j (G) A i+j (G) S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 10 / 1
11 Chow ring of G Problem Goal Determine A (G) for all simply connected simple complex Lie groups Classification Theorem tells that G is one of the following: SL n, Spin n, Sp n, G 2, F 4, E 6, E 7, E 8 Grothendieck considered the problem in the 1950 s (Grothendieck) A (G) = H (G/B; Z)/(H 2 (G/B; Z)) Consequently, A (G) Q = Q for all G and A (G) = Z for G = SL n, Sp n A (G) for G = Spin n, G 2, F 4 were determined by R.Marlin(1974) Remaining cases are when G = E 6, E 7, E 8. S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 11 / 1
12 Grothendieck s Theorem Theory 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) 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)) A (G) = Z, G = SL n, Sp n S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 12 / 1
13 Schubert class Schubert calculus The Bruhat decomposition gives a cell decomposition G/B = BwB/B w W X w = closure of BwB/B( = C l(w) ): Schubert variety Z w = {the cohomology class corresponding to [X w0w]} H 2l(w) (G/B; Z): Schubert class {Z w } w W forms an additive basis for H (G/B; Z) (indexed by W ) In particular, H (G/B; Z) is torsion free S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 13 / 1
14 Schubert calculus Structure constant The intersection product of two Schubert classes Z w, Z w can be written in the linear sum of Schubert classes: Z w Z w = c v ww Z v The coefficients c v ww A goal in Schubert calculus Give a combinatorial formula for c v ww l(v)=l(w)+l(w ) Z are called the structure constants Littlewood-Richardson rule for Grassmaniann Chevalley formula for Z w Z w when l(w) = 1 Schubert polynomial S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 14 / 1
15 Divided difference operator Combinatorial machinery (translator) Definition (B-G-G(1973), Demazure(1973)) 1 For α i Π, i : H (BT ; Z) H 2 (BT ; Z) i (f) = f s i(f) α i, f H (BT ; Z) = Z[ω 1,..., ω l ] 2 For w W, w = s i1 s i2 s ik : a reduced decomposition, w = i1 i2 ik : H (BT ; Z) H 2k (BT ; Z) S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 15 / 1
16 Combinatorial machinery (translator) Theorem (B-G-G(1973), Demazure(1973)) the characteristic map c : H 2k (BT ; Z) H 2k (G/B; Z): c(f) = X w(f)z w (Note: w(f) Z) l(w)=k the following composition is induced by the inclusion of Z Q: H (G/B; Z) H (G/B; Q) = H (BT ; Q)/H + (BT ; Q) W c H (G/B; Q) (Giambelli formula) Q ««α α Z w = c + w 1w0 W Inductive formula: α(ω β ) = δ αβ α(fg) = α(f)g + s α(f) α(g) w = i1 i2 ik S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 16 / 1
17 Combinatorial machinery (translator) Ring structure of H (G/B; Z) polynomial characteristic map Giambelli formula 1 Given elements Z w, Z w H (G/B; Z) Schubert classes (Weyl group) 2 by Giambelli formula, we have polynomials f, f H (BT ; Q) which correspond to Z w, Z w 3 by characteristic map, we have c(f f ) H (G/B; Z) which correspond to the intersection product Z w Z w 4 we obtain the ring structure of H (G/B; Z) 5 hence we obtain A(G) = H (G/B; Z)/H 2 (G/B; Z) S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 17 / 1
18 Optimization Computational complexity characteristic map: Giambelli formula: c(f) = l(w)=k w (f)z w ( α Z w = c ( α )) + w 1w0 W For G = E 8, W = = l(w 0 ) = + = 1 2 dim G/B = 120 Today s computer cannot handle polynomials of degree 120! The above strategy is not practical as it is S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 18 / 1
19 Borel presentation Optimization There are two descriptions for H (G/B; Z) = A (G/B) Borel presentation Schubert presentation quotient of a polynomial ring Z-basis indexed by Weyl group elements polynomials Schubert classes geometry no algebraic cycles ring structure easy hard (main theme of Schubert calculus) Borel presentation characteristic map Schubert presentation Giambelli formula S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 19 / 1
20 Optimization Results from algebraic topology By spectral sequence argument, Borel presentation for each H (G/B; Z) was computed by Borel, Toda-Watanabe, Bott-Samelson, and Nakagawa. They have the following form in general. H (G/B; Z) = Z[t i, γ j ]/(ideal), ( t i = 2, γ j > 2 Our strategy is: 1 Compute A (G) purely algebraically from Borel presentation A (G) = H (G/B; Z)/(H 2 (G/B; Z)) = Z[γ j ]/(ideal) 2 Find Schubert varieties representing the generators γ j S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 20 / 1
21 More optimization for type E Convenient presentation of H (BT ; Z) Let G be either E 6, E 7, or E 8. α 1 α 3 α 4 α 5 α 6 α 2 If we take away α 2, then the Dynkin diagram becomes type A. By this observation, We take another set of generators for H (BT ; Z) = Z[ω 1, ω 2,..., ω l ]: t l = ω l t i = s i+1 (t i+1 ) = t 1 = s 1 (t 2 ) = ω 1 + ω 2 t = ω 2 { ω i ω i+1 (4 i l 1) ω i 1 + ω i ω i+1 (i = 2, 3) S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 21 / 1
22 More optimization for type E Toda-Watanabe s magical basis Let c i = i-th elementary symmetric function in t 1,..., t l (1 i l) H (BT ; Z) = Z[ω 1, ω 2,..., ω l ] = Z[t 1, t 2,..., t l, t]/(c 1 3t). s i (i 2) act on {t i } 1 i l as permutations and trivially on t. For example, for f Z[t, c 2,..., c l ], i f = 0 if i 2 this reduces the computation S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 22 / 1
23 Ingredients from algebraic topology Theorem (Nakagawa(2001)) H (E 7 /B; Z) = Z[t 1, t 2,..., t 7, t, γ 3, γ 4, γ 5, γ 9 ] /(ρ 1, ρ 2, ρ 3, ρ 4, ρ 5, ρ 6, ρ 8, ρ 9, ρ 10, ρ 12, ρ 14, ρ 18 ), ρ 1 = c 1 3t, ρ 2 = c 2 4t 2, ρ 3 = c 3 2γ 3, ρ 4 = c 4 + 2t 4 3γ 4, ρ 5 = c 5 3tγ 4 + 2t 2 γ 3 2γ 5, ρ 6 = γ c 6 2tγ 5 3t 2 γ 4 + t 6, ρ 8 = 3γ 4 2 2γ 3 γ 5 + t(2c 7 6γ 3 γ 4 ) 9t 2 c t 3 γ t 4 γ 4 6t 5 γ 3 t 8, S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 23 / 1
24 Ingredients from algebraic topology From the result in previous slide, an easy calculation by hand shows A (E 7 ) = A (E 7 /B)/(A 1 (E 7 /B)) = H (E 7 /B; Z)/(H 2 (E 7 /B; Z)) = Z[γ 3, γ 4, γ 5, γ 9 ]/(2γ 3, 3γ 4, 2γ 5, γ 2 3, 2γ 9, γ 2 5, γ 3 4, γ 2 9) By using a Maple script, we obtain γ 3 = Z Z 542 γ 4 = Z Z Z 6542 γ 5 = Z γ 9 = 2Z Z Note that we abbreviate s i1 s i2 s ik W as i 1 i k A (E 7 ) = Z[Z 542, Z 6542, Z 76542, Z ] / 2Z 542, 3Z 6542, 2Z 76542, Z542 2, 2Z , Z , Z3 6542, Z S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 24 / 1
25 Final results Theorem (K-Nakagawa) A(E 6 ) = Z[Z 542, Z 6542 ]/(2Z 542, 3Z 6542, Z 2 542, Z ), (Z 542 = B(w 0 s 6 s 5 s 4 s 2 )B G etc.) 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 = Z 542, X 4 = Z 6542, X 5 = Z 76542, X 9 = Z ) 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, 3X2 10, X8 3, X X X5 6 S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 25 / 1
26 Computation to theory We encounter spin-off problems during the process. Definition Z w is indecomposable Z w Z v l(v)<l(w) torsion index and decomposability t(g) = min{t t Z w Im(c), w W } combinatorics of Weyl group and decomposability Schubert polynomial for exceptional types (Schubert polynomials live in H (BT ; Z) Z[γ i ], where γ i s correspond to indecomposable Schubert classes) Cohomology ( Chow rings ) of generalized flag varieties G/P I hope further experimentation and visualization will lead to the solution. S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 26 / 1
27 Open problems Open problems Cohomology ring Invariant ring of Weyl group Z[w1,..., w l ] W Invariant ring of mod p Weyl group Fp[v 1,..., v m] Wp Action of Steenrod operations on H (BG; F p) Homotopy groups Handy free resolutions for algebras which arose as cohomology rings Homology of the λ-algebra (E2-term of Adams spectral sequence) How to handle, for example, algebras over Steenrod algebras? or generally, algebras over operads? How to deal with spectral sequences? S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 27 / 1
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