computers work exceptionally on lie groups

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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


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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


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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.


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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 ?


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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


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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)


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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


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An example: Chow ring

An example of invariants:The Chow ring A∗(G)


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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)


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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.


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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


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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


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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


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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)


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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


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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)


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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


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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


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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


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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


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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


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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,

· · ·


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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


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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


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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.


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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 ?


computers work exceptionally on lie groups

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