computers work exceptionally on lie groups
TRANSCRIPT
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Computers work exceptionally on Lie groups
Shizuo Kaji
Fukuoka University
First Global COE seminaron
Mathematical Research Using Computersat Kyoto University
Oct. 24, 2008
S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 1 / 1
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Outline
Introduction
Computation of topological invariants
An example: Chow rings of Lie groupsI Schubert calculus (geometry)I Divided difference operator (combinatorics)I Borel presentation (algebraic topology)I Computer assisted part
Open problems
S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 2 / 1
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Introduction About presentation
Latex-Beamer
This presentation slide is made with LaTeX-Beamer.
It is an easy-to-use LATEXpackage, 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, evenrequest new features
It has the capability of
this kind of gimmicksand hyperlinks Main Theorem
S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 3 / 1
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Introduction About presentation
Latex-Beamer
TheoremA 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
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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 usecomputers
⇒ 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
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Computation of topological invariants General setting
Common process
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 algorismI Search for “parts” (libraries) made by othersI Implement of the data structure that corresponds to mathematical objects
(polynomial, DGA, Lie algebra, Hopf algebra,. . . )I Choose programming language, software, etc.I 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
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Computation of topological invariants Lie group
Lie group basic
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 = αi1≤i≤l: simple roots
ωi1≤i≤l: fundamental weights ( (2αj
(αj ,αj) , ωi) = δij )
H∗(BT ; Z) = Z[w1, . . . , wl], where |wi| = 2
si ∈ GL(t∗): simple reflection corresponding to αi, si ∈ Aut(Z[w1, . . . , wl])( si(e) = e− ( 2αi
(αi,αi), e)αi )
W = N(T )/T : Weyl group of G (= a finite group generated by si1≤i≤l)
Classification:An, Bn, Cn, Dn, G2, F4, E6, E7, E8
These objects are all concrete (symbolic). However, for example, wi’s are possiblyirrational vectors so we need to handle with care.(Stembridge’s coxeter/weyl package in Maple is so convenient that this oftenencourage me to choose Maple)
S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 7 / 1
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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; Fp)↔ the invariant rings H∗(BT ; Fp)
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 πS∗ (G) ↔ free resolution of H∗(G; Fp) as Ap-algebra
Grothendieck’s torsion index of G ↔ Grobner 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
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An example: Chow ring
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
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An example: Chow ring 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 ⊆ V1 ⊆ V2 ⊆ · · · ⊆ Vn−1 ⊆ Vn = Cn, dimC(Vi) = i).
H∗(G/B; Z): ordinary integral cohomology of G/B
A∗(G): Chow ring of GI A∗(G) =
Li≥0 Ai(G)
I Ai(G) is a group of the rational equivalence classes of algebraic cycles ofcodimension i.(an algebraic cycle is a linear sum of possibly singular subvarieties)
I intersection product Ai(G)⊗Aj(G) → Ai+j(G)
S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 10 / 1
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An example: Chow ring Problem
Chow ring of G
GoalDetermine A∗(G) for all simply connected simple complex Lie groups
Classification Theorem tells that G is one of the following:SLn,Spinn,Spn,G2,F4,E6,E7,E8
Grothendieck considered the problem in the 1950’sI (Grothendieck) A∗(G) ∼= H∗(G/B; Z)/(H2(G/B; Z))I Consequently, A∗(G)⊗Q = Q for all G and A∗(G) = Z for G = SLn, Spn
A∗(G) for G = Spinn,G2,F4 were determined by R.Marlin(1974)
Remaining cases are when G = E6,E7,E8.
S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 11 / 1
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An example: Chow ring Theory
Grothendieck’s Theorem
Theorem (Grothendieck(1958))
the cycle map cl : A∗(G/B) → H2∗(G/B; Z) is an isomorphism of rings:
A∗(G/B)'−→ H2∗(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 A1(G/B).
Corollary
A∗(G) ∼= H∗(G/B; Z)/(H2(G/B; Z))A∗(G) = Z, G = SLn, Spn
S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 12 / 1
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An example: Chow ring Schubert calculus
Schubert class
The Bruhat decomposition gives a cell decomposition
G/B =∐
w∈W
BwB/B
Xw = closure of BwB/B(∼= Cl(w)): Schubert variety
Zw = the cohomology class corresponding to [Xw0w] ∈ H2l(w)(G/B; Z):Schubert class
Zww∈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
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An example: Chow ring Schubert calculus
Structure constant
The intersection product of two Schubert classes Zw, Z ′w can be written in the
linear sum of Schubert classes:
Zw · Z ′w =
∑l(v)=l(w)+l(w′)
cvww′Zv
The coefficients cvww′ ∈ Z are called the structure constants
A goal in Schubert calculus
Give a combinatorial formula for cvww′
Littlewood-Richardson rule for Grassmaniann
Chevalley formula for Zw · Zw′ when l(w) = 1
Schubert polynomial
S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 14 / 1
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An example: Chow ring Combinatorial machinery (translator)
Divided difference operator
Definition (B-G-G(1973), Demazure(1973))
1 For αi ∈ Π, ∆i : H∗(BT ; Z) → H∗−2(BT ; Z)
∆i(f) =f − si(f)
αi, f ∈ H∗(BT ; Z) = Z[ω1, . . . , ωl]
2 For w ∈ W , w = si1si2 · · · sik: 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
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An example: Chow ring Combinatorial machinery (translator)
Theorem (B-G-G(1973), Demazure(1973))
the characteristic map c : H2k(BT ; Z) → H2k(G/B; Z):
c(f) =X
l(w)=k
∆w(f)Zw (Note:∆w(f) ∈ Z)
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)
Zw = c
„∆w−1w0
„ Qα∈∆+ α
|W |
««Inductive formula:
∆α(ωβ) = δαβ
∆α(fg) = ∆α(f)g + sα(f)∆α(g)
∆w = ∆i1 ∆i2 · · · ∆ik
S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 16 / 1
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An example: Chow ring Combinatorial machinery (translator)
Ring structure of H∗(G/B; Z)
polynomialcharacteristic map⇔Giambelli formula
Schubert classes (Weyl group)
1 Given elements Zw, Z ′w ∈ H∗(G/B; Z)
2 by Giambelli formula, we have polynomials f, f ′ ∈ H∗(BT ; Q) whichcorrespond to Zw, Z ′
w
3 by characteristic map, we have c(f · f ′) ∈ H∗(G/B; Z) which correspond tothe intersection product Zw · Z ′
w
4 we obtain the ring structure of H∗(G/B; Z)
5 hence we obtain A(G) ∼= H∗(G/B; Z)/H2(G/B; Z)
S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 17 / 1
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An example: Chow ring Optimization
Computational complexity
characteristic map:
c(f) =∑
l(w)=k
∆w(f)Zw
Giambelli formula:
Zw = c
(∆w−1w0
(∏α∈∆+ α
|W |
))For G = E8,
|W | = 696729600 = 8064 ∗ 24 ∗ 3600
l(w0) = |∆+| = 12 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
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An example: Chow ring Optimization
Borel presentation
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 groupelements polynomials Schubert classes
geometry no algebraic cycles
ring structure easy hard (main theme of Schubert calculus)
Borel presentationcharacteristic map⇔Giambelli formula
Schubert presentation
S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 19 / 1
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An example: Chow ring Optimization
Results from algebraic topology
By spectral sequence argument, Borel presentation for each H∗(G/B; Z) wascomputed by Borel, Toda-Watanabe, Bott-Samelson, and Nakagawa.
They have the following form in general.
H∗(G/B; Z) ∼= Z[ti, γj ]/(ideal), (|ti| = 2, |γj | > 2
Our strategy is:
1 Compute A∗(G) purely algebraically from Borel presentation
A∗(G) ∼= H∗(G/B; Z)/(H2(G/B; Z)) ∼= Z[γj ]/(ideal)
2 Find Schubert varieties representing the generators γj
S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 20 / 1
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An example: Chow ring More optimization for type E
Convenient presentation of H∗(BT ; Z)
Let G be either E6, E7, or E8.
e e e e ee
α1 α3 α4 α5 α6
α2
If we take away α2, then the Dynkin diagram becomes type A.By this observation, We take another set of generators forH∗(BT ; Z) = Z[ω1, ω2, . . . , ωl]:tl = ωl
ti = si+1(ti+1) =
ωi − ωi+1 (4 ≤ i ≤ l − 1)
ωi−1 + ωi − ωi+1 (i = 2, 3)
t1 = s1(t2) = −ω1 + ω2
t = ω2
S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 21 / 1
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An example: Chow ring More optimization for type E
Toda-Watanabe’s magical basis
Let ci = i-th elementary symmetric function in t1, . . . , tl (1 ≤ i ≤ l)
H∗(BT ; Z) = Z[ω1, ω2, . . . , ωl]
= Z[t1, t2, . . . , tl, t]/(c1 − 3t).
si (i 6= 2) act on ti1≤i≤l as permutations and trivially on t.
⇒ For example, for f ∈ Z[t, c2, . . . , cl], ∆if = 0 if i 6= 2
⇒ this reduces the computation
S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 22 / 1
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An example: Chow ring Ingredients from algebraic topology
Theorem (Nakagawa(2001))
H∗(E7/B; Z) ∼= Z[t1, t2, . . . , t7, t, γ3, γ4, γ5, γ9]
/(ρ1, ρ2, ρ3, ρ4, ρ5, ρ6, ρ8, ρ9, ρ10, ρ12, ρ14, ρ18),
ρ1 = c1 − 3t,
ρ2 = c2 − 4t2,
ρ3 = c3 − 2γ3,
ρ4 = c4 + 2t4 − 3γ4,
ρ5 = c5 − 3tγ4 + 2t2γ3 − 2γ5,
ρ6 = γ32 + 2c6 − 2tγ5 − 3t2γ4 + t6,
ρ8 = 3γ42 − 2γ3γ5 + t(2c7 − 6γ3γ4)− 9t2c6
+ 12t3γ5 + 15t4γ4 − 6t5γ3 − t8,
· · ·
S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 23 / 1
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An example: Chow ring Ingredients from algebraic topology
From the result in previous slide, an easy calculation by hand shows
A∗(E7) ∼= A∗(E7/B)/(A1(E7/B))
∼= H∗(E7/B; Z)/(H2(E7/B; Z))
∼= Z[γ3, γ4, γ5, γ9]/(2γ3, 3γ4, 2γ5, γ23 , 2γ9, γ
25 , γ
34 , γ
29)
By using a Maple script, we obtain
γ3 = Z342 + 2Z542
γ4 = Z1342 + 2Z3542 + Z6542
γ5 = Z76542
γ9 = 2Z154376542 + Z654376542
Note that we abbreviate si1si2 · sik∈ W as i1 · · · ik
A∗(E7) = Z[Z542, Z6542, Z76542, Z654376542]/2Z542, 3Z6542, 2Z76542, Z2542, 2Z654376542,
Z276542, Z
36542, Z
2654376542
S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 24 / 1
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An example: Chow ring Final results
Theorem (K-Nakagawa)
A(E6) = Z[Z542, Z6542]/(2Z542, 3Z6542, Z2542, Z
36542),
(Z542 = B(w0s6s5s4s2)B ⊂ G etc.)
A(E7) = Z[X3, X4, X5, X9]
/(2X3, 3X4, 2X5, X23 , 2X9, X
25 , X3
4 , X29 )
(X3 = Z542, X4 = Z6542, X5 = Z76542, X9 = Z654376542)
A(E8) = Z[X3, X4, X5, X6, X9, X10, X15]/2X3, 3X4, 2X5, 5X6, 2X9, X
25 − 3X10,
X34 , 2X15, X
29 , 3X2
10, X83 ,
X215 + X3
10 + 2X56
S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 25 / 1
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An example: Chow ring Computation to theory
We encounter spin-off problems during the process.
Definition
Zw is indecomposable ⇔ Zw 6∈ 〈Zv〉l(v)<l(w)
torsion index and decomposability
t(G) = mint|t · Zw ∈ Im(c),∀w ∈ W
combinatorics of Weyl group and decomposability
Schubert polynomial for exceptional types(Schubert polynomials live in H∗(BT ; Z)⊗ Z[γi], where γi’s correspond toindecomposable 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
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Open problems
Open problems
Cohomology ringI Invariant ring of Weyl group Z[w1, . . . , wl]
W
I Invariant ring of mod p Weyl group Fp[v1, . . . , vm]Wp
I Action of Steenrod operations on H∗(BG; Fp)
Homotopy groupsI Handy free resolutions for algebras which arose as cohomology ringsI 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