Computing in exceptional groups by Bill Casselman for Tom Hales - - PowerPoint PPT Presentation

Computing in exceptional groups

by Bill Casselman for Tom Hales’ 60th, University of Pittsburgh, June 2018 These slides can be found at

http://www.math.ubc.ca/~cass/slides/hales-bday.pdf

!

This is potentially a very dull and exceedingly technical topic. I should say that it is an improved version of material I sent to Tom in email a few years ago, and which I believe he found useful. I hope to exhinit some interesting problems that deserve to be better known. Some have already been solved, but there are also some that have not yet been finished off. My main, if somewhat eccentric, claim is that we do not yet really under- stand the structure of semi-simple groups, or even semi-simple Lie alge-

• bras. Whether this is important or not remains to be seen, but it seems to

me that sometimes these old questions come close to main stream prob- lems. 2/38

Contents

• 1. Introduction ................................................................ 4
• 2. Chevalley’s contribution ................................................. 8
• 3. Primary references ...................................................... 21
• 4. Kottwitz’ contribution ................................................... 22
• 5. Tits’ contribution ......................................................... 30
• 6. What about the group? ................................................. 36
• 7. More references .......................................................... 38

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• 1. Introduction

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A complex Lie group is called reductive if every continuous finite-dimensional representation decomposes into a direct sum of irreducible ones. Some reductive groups are defined very simply in terms of matrices. For example the group SLn(C) which is the group of all n × n complex matri- ces of determinant 1. Or the complex symplectic group Sp2n(C) of all of all 2n × 2n complex matrices X such that

tXJX = J

• J =
• −I

I

• .

But in the late nineteenth century a finite number of exceptional reductive

groups were found to be characterized in more abstract terms. For exam-

ple, the group G2 of dimension 14, which is rather sparsely embedded in

GL7(C). There are altogether five exceptional types, and for each of them

certain useful matrix realizations are known, but these are difficult to work

• with. If I understand correctly, the classification of their Lie algebras came
• first. Although the terminology was different, this was in terms of what

we now call root systems•. 5/38

It seems to have been L. E. Dickson who first realized that nearly all of these complex matrix groups should have analogues defined over finite

• fields. In particular, in a real tour de force he managed to show that there

was an analogue of the smallest exceptional group, now called type G2. He constructed it in terms of its embedding into GL7, and the size of

G(Fq) is q6(q6 − 1)(q2 − 1). The other four exceptional groups remained a

mystery. Dickson had much more trouble with fields Fq when q = 2n. It was in- evitable that in his approach small finite fields will cause difficulties. How to distinguish Sp from SO? This line of inquiry was initiated by Galois, looking for simple groups. 6/38

The situation was completely changed about 1955 when Chevalley discov- ered how to deal uniformly with reductive groups over any field, in terms

• f what is now called root data, a variant of the notion of root system.

Chevalley’s discovery amounted to a revolution. His discovery is many years behind us, and much of the subject has by now become very familiar. There are parts of his work, however, which are now almost forgotten—partly because its more technical aspects are not necessary to work productively in the field. These aspects have be- come part of the machinery behind the curtain, so to speak. But with the possibility of using computers to carry out computations in arbitrary reductive groups, questions raised by Chevalley have come into light again, and that’s what I’ll talk about. I have to confess, however, that I do not know if this material will ever be- come part of the main stream of the subject. 7/38

• 2. Chevalley’s contribution

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Let G be a split reductive group defined over any field F in which one knows how to do arithmetic.

• How can one compute in G?
• For that matter, how can one specify elements of G?
• How can one multiply them? Find inverses?

There are two general methods to deal with these questions. (1) The clas- sical approach is to find a good embedding of G into somemain GLn, so every element of G is represented by a matrix. (2) Chevalley’s approach is to represent elements of G more directly in terms of the root datum that defines it. From the root datum one can define Borel subgroups B = TU and the normalizer NG(T). Given representives ˙

w of elements of W = NG(T)/T , every element of G can be factored as b ˙ wu in B ˙

• wU. With a

suitable restriction on u this becomes a unique expression with which

• ne can work.

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The first method works well for the so-called classical groups, which are characterized very nicely in terms of matrices. It has been used also for exceptional groups, although awkwardly. Finding products and inverses is easy, but relating the answer to the structure of G is not simple. In this talk I’ll discuss the second method, in particular a recent contribu- tion due to Robert Kottwitz, building on an old idea due to Jacques Tits, and implemented in programs by myself. This method is closely related to the structure of G, but it is not so easy to perform group operations. 10/38

In truth, I have already deceived you. It turns out that computing in the group reduces quickly to computing in its Lie algebra. (This was already known to Chevalley.) I shall probably say little about the group, and a lot about the Lie algebra. It is in the Lie algebra that the principal and inter- esting difficulties arise. 11/38

The most interesting part of the subject originates with Chevalley. Suppose g to be a semi-simple Lie algebra over C. Chevalley explained how to assign a Z-structure to g, one which led (famously) in turn to his construction of split groups over arbitrary fields. Choose a frame (´ epinglage) for g, which is to say a triple (b, t, {eα}α∈∆). All choices are equivalent. For each α there exists a good copy of sl2, mapping

• 1
• → eα,
• −1
• → e−α,
• 1

0 −1

• → hα .

There exists a unique involution θ acting as −I on t and mapping each eα to e−α. For classical Lie algebras this takes x to −tx. One can find elements eλ in gλ such that eθ

λ = e−λ. These are unique up

to sign. They make up part of an invariant integral basis. 12/38

If λ, µ, λ + µ are all roots, then

[eλ, eµ] = Nλ,µeλ+µ

for some structure constant Nλ,µ.

• Theorem. (Chevalley) For an invariant integral basis

Nλ,µ = ±(pλ,µ + 1) .

Here pλ,µ is the string constant:

λ µ pλ,µ = 0 λ µ pλ,µ = 0 1 λ µ pλ,µ = 0 1 2 λ µ pλ,µ = 0 1 2 3

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Chevalley’s theorem is (literally) wonderful. Corresponding to λ is an em- bedding of Q into G:

t − → exp(teλ) .

Chevalley’s theorem implies that

exp(adteλ)eµ =

• m≥0

tm · Nλ,µNλ,λ+µ . . . Nλ,(m−1)λ+µ m! · emλ+µ

makes sense. It is a finite series, and equal to

exp(teλ)eµ exp(−teλ) .

With a little work, this makes the string lattice spanned by the emλ+µ into a representation of SL2(Z), and eventually allows you to define G(Z). 14/38

Already, I can bring up my first mystery. Chevalley’s results eventually lead to a construction of a smooth group scheme defined over Z asso- ciated to every root datum. This was begun by Chevalley himself, elab-

• rated by Demazure and Grothendieck in SGA 3, and has most recently

been redone in some very thorough lecture notes by Brian Conrad (based

• n his lectures at a summer school in Luminy, 2011 that was devoted to

updating SGA 3). One part of this business is existence—constructing a split reductive group scheme over Z corresponding to a given root datum. At the AMS Boulder conference of 1965, Kostant proposed an elegant and direct construction of the affine ring of this scheme. Unfortunately, he did not give details. In his Yale lecture notes Robert Steinberg appar- ently used Kostant’s ideas in order to construct the group G as an alge- braic group. (Curiously, as far as I can see, neither Kostant nor Steinberg refers to the other.) I find Steinberg’s discussion unsatisfying, and Conrad doesn’t say much about it.

Is Kostant’s construction correct? Conrad tells me this is highly unlikely,

but I don’t think anybody really knows one way or the other. 15/38

I can even bring up my second mystery, although it lies further afield. The involution θ determines a Z-structure on G, but also a maximal compact subgroup K of G(R). In some sense, the groups K and Γ = G(Z) are hence tied together. (This is a fact that is transparently true for p-adic groups.) There should be a strong version of this assertion in terms of Arthur’s partition of Γ\G/K. There are questions, too, about the fine structure of this partition when

G is not split. Some relationship with ramification and discriminants,

strongly suggested by work of Ulrich Stuhler and Dan Grayson on arith- metic groups and stability of lattices. 16/38

But now suppose you want to compute things, in both g and G. You must (i) explain how to specify a particular invariant basis eλ, and then (ii) find those constants Nλ,µ such that

[eλ, eµ] = Nλ,µeλ+µ

whenever λ, µ, λ + µ are all roots. In addition, you will want to (iii) specify representatives ˙

w of elements w

• f W and then (iv) find the constants c(w, λ) such that

˙ weλ = c(w, λ)ewλ .

According to Chevalley’s theorem, it is just a matter of signs. However, even a small amount of experimentation will convince you that it is not a trivial problem. Changing just one eλ to −eλ can force a lot of changes in the structure constants. There is no obvious pattern to the changes. 17/38

The first problem encountered, already tricky, is how to specify an integral

• basis. We start with the nilpotent elements eα in the frame. Then assign

an order to ∆. (Carter) For every root λ not in ∆, there exists a minimal α such that µ =

λ − α has smaller height. Given eµ, choose eλ so that [eα, eµ] = (pα,µ + 1)eλ .

This is the method used in the paper of Cohen, Murray, and Taylor. Find- ing other structure constants depends on the complicated graph defined by the additive structure of the roots. This is a highly unintuitive process. 18/38

(Tits) The frame determines homorphisms from SL2 into the group of au- tomorphisms of g. Let ˙

sα be the image of

• 1

−1

• .

Its image in W is sα. Every W -orbit of roots intersects ∆, so for every λ there exists a chain

β = β0 → β1 = sα1β → . . . → λ = βm = sαmβm−1 .

Given a root λ, find the minimum α such that λ = ˙

sαµ. Define eλ = Ad ˙

sαeµ = ˙

sα♦eµ .

In this method, one has to work with the graph whose edges are links

λ → sαλ. This method is conceptually more elegant than the previous

• ne, and apparently of about the same efficiency.

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In effect, both methods rely on choosing a spanning tree in a certain graph. There is clearly something highly arbitrary about this. Both these methods work also with Kac-Moody algebras, but the second is theoretically much more interesting. It suggests some structure in the degree of arbitrariness involved. 20/38

• 3. Primary references
• Roger Carter, Simple groups of Lie type, Wiley, 1972.
• Bill Casselman, ‘Stability of lattices and the partition of arithmetic quo-

tients’, jourAsian Journal of Mathematics 8 (2004), 607–637.

• —–, ‘Structure constants of Kac-Moody Lie algebras’, 55–83 in Symme-

try: representation theory and its applications, Birkh¨ auser, 2014.

• —–, ‘On Chevalley’s formula for structure constants’, Journal of Lie The-
• ry 25 (2015), 431–441.
• Claude Chevalley, ‘Sur certains groupes simples’, Tˆ
• hoku Mathematics

Journal 48 (1955), 14–66.

• Arjeh Cohen, Scott Murray, and Don Taylor, ‘Computing in groups of Lie

type’, Mathematics of Computation 73 (2004), 1477–1498.

• Brian Conrad, Reductive group schemes, available at

http://math.stanford.edu/ conrad/papers/luminysga3.pdf

• Jacques Tits, ‘Sur les constants de structure et le th´

eor` eme d’existence des alg` ebres de Lie semi-simple’, Publications de l’I. H. E. S. 31 (1966), 21–58. 21/38

• 4. Kottwitz’ contribution

22/38

A few years ago, Kottwitz suggested to me a new way to find an invariant basis. The group T(Z) is isomorphic to {±1}∆. The element t acts on gλ by

λ(t), and µ∨(x) by xλ,µ∨.

The extension

1 − → T(Z) − → NG,T (Z) − → W − → 1

is well understood. It does not generally split, but the given frame deter- mines a valuable section w → ˙

w found by Tits. His basic observation is

that if w has the reduced expression w = sα1 . . . sαn then

˙ w = ˙ sα1 . . . ˙ sαn

depends only on w. Furthermore (as in SL2)

˙ s2

α = α∨(−1) .

Langlands and Shelstad (!) found an explicit formula for the associated cocycle in H2(W, T(Z)). 23/38

Let Σ be the set of all roots and define the lattice

VZ = ⊕ gλ(Z) .

Let S(Z) be the group {±1}Σ, which may be identified with Aut(VZ). There is a natural homomorphism from T(Z) to S(Z), taking

µ∨(x) − → (xλ,µ∨)λ .

A new extension is defined by the diagram

1 − → T(Z) − → NG,T (Z) − → W − → 1 ↓ ↓ ↓ 1 − → S(Z) − → Next(Z) − → W − → 1

• Theorem. (Kottwitz) There exists a splitting w → ¨

w of this sequence with

the property that if wλ = λ then ¨

weλ = eλ.

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We have ¨

w = ˙ wτw for some τw in S(Z) with a relatively simple formula.

Explicitly, first set

• β, α

= β, α∨

if it is positive

pα,β

if β, α∨ = 0

• therwise.

and then

F(w, λ) =

• wβ<0
• β, λ

.

Kottwitz’ definition:

(τw)β = (−1)F (w,β) .

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• Corollary. Choose for each W-orbit in Σ a representative α in ∆. The spec-

ification ewα = ¨

w ♦eα determines an integral basis of VZ.

It is not invariant under the involution θ, but it fails in a simple way. If λ >

0 then eθ

λ = (−1)ht(λ)−1e−λ .

Therefore

fλ =

• eλ

if λ > 0

(−1)ht(−λ)−1eλ

if λ < 0 is invariant. It turns out that there is an especially simple way to calculate the struc- ture constants Nλ,µ and c(sα, λ) for the fλ. There is in fact a simple for- mula for the second, and for the first there is an algorithm that is both simple and fast. 26/38

Let’s look at an example. Take g = sl3. For this Lie algebra xθ = −tx. Start with the frame

eα =   1   , eβ =   1  

and choose eα as origin element. Then

eα =   1   , eβ =   0 −1   , eα+β =   −1   .

So there are some odd features of Kottwitz’ basis. The frame you wind up with is rarely the one you start with. 27/38

The W -orbits of roots correspond to the lengths of roots. There are at most two, and each corresponds to a segment of the Dynkin diagram. Fix

• ne base root in ∆ on each segment. As a consequence of the example
• f SL3

eβ = cβeβ (cβ = (−1)d(β))

if d(β) is the distance in the Dynkin diagram of β from its base. Because

• f this phenomenon, Tit’s section w
• for the eα differs from the section ˙

w

for eα, but not by much. Adjust e to f. 28/38

Why is Kottwitz’ basis such a good thing? The main point is the formula for the constant c(sα, λ) such that

˙ sαfλ = c(sα, λ) fsαλ .

Recall

• β, α

= β, α∨

if it is positive

pα,β

if β, α∨ = 0

• therwise.
• Theorem. For λ > 0 and α in ∆

c(sα, λ) = (−1)

λ,α cλ,α∨ α

.

This follows immediately from Kottwitz’ theorem and the definition of ¨

sα.

29/38

• 5. Tits’ contribution

30/38

The constants c(sα, λ) play an important role in computing the other struc- ture constants Nλ,µ. This is where some sophisticated ideas of Tits come

• in. I used these same ideas in an earlier paper on computing the Nλ,µ. I

want now to sketch how Tit’s ideas apply to computation. Tit’s own motivation is not so clear to me. One application he has in mind is the construction of semi-simple Lie algebras, but it is not all that suc- cessful. From now on, suppose given an arbitrary invariant integral basis (fλ). 31/38

Central features of Tits’ approach are two types of symmetries. One of them we have already seen—the choice of basis elements invariant under the opposition involution. Another is a rotational symmetry in expressing the Nλ,µ. I call (λ, µ, ν) a Tits triple if λ + µ + ν = 0. Here is one pleasant consequence of this notion:

pλ,µ + 1 ν2 = pµ,ν + 1 λ2 = pν,λ + 1 µ2 .

• I. e. knowing one of the p tells you the others.

32/38

We know that at least two of a Tits triple are of the same length. In a way, the entire basis of Tits’ approach is the fact that in a Tits triple this com- mon length cannot be larger than the length of the third root. The follow- ing configuration is forbidden: I define an ordered Tits triple to be one in which

λ ≥ µ = ν .

Equivalent: sλµ = −ν, µ, λ∨ = −1. Hence we can list easily all ordered triples (α, µ, ν) with α in ∆. 33/38

Every Tits triple can be rotated to become an ordered triple.

This is important because rotation preserves the sign of structure con- stants:

• Theorem. Suppose given an ordered triple (λ, µ, ν). The following are

equivalent:

[eλ, eµ] = ε (pλ,µ + 1) e−ν [eµ, eν] = ε (pµ,ν + 1) e−λ [eν, eλ] = ε (pν,λ + 1) e−µ .

If λ = α lies in ∆, these are also equivalent to

˙ sα♦eµ = ε(−1)pα,µ e−ν .

I repeat: knowing c(sα, µ) determines certain structure constants. 34/38

One convenient fact about invariant bases is that N−λ,−µ = Nλ,µ. In order to compute arbitary structure constants it hence suffices to com- pute them for ordered triples with λ > 0. But every positive root is at the end of one starting at some α in ∆. So we can compute Nλ,µ by following up the chain. If

[eλ, eµ] = Nλ,µ e−ν

then

[ ˙ sαeλ, ˙ sαeµ] = Nλ,µ ˙ sαe−ν .

So knowing how to compute . . .α eλ allows you to determine all structure constants. 35/38

• 6. What about the group?

36/38

Every element in G can be expressed uniquely as u1t ˙

wu2 with u1 in U, u2

in U ∩ ˙

w−1U ˙ w.

Given an order on the positive roots, every element of U many be ex- pressed as uniquely as an ordered product

• exp(tλeλ) .

Chevalley gave explicit formulas

uµuλ = uλuµ

• ukλ+ℓµ ,

which allows you to find product expressions explicitly. Implementing this is not impossibly slow, since most uλ and uµ commute. But it is some- what slow—primarily because the length of product expressions builds up rapidly (in a process known as collection). 37/38

• 7. More references
• Armand Borel and Dan Mostow (editors), Algebraic groups and dis-

continuous subgroups, Proceedings of Symposia in Pure Mathematics IX, American Mathematical Society, 1966.

• Bill Casselman, ‘A simple way to compute structure constants of semi-

simple Lie algebras’,

http:www.math.ubc.ca/~cass/research/pdf/KottwitzConstants.pdf

• Bertram Kostant, ‘Groups over Z’, pp. 90–98 in [Borel-Mostow:1966].
• Steinberg, Lectures on Chevalley groups, Yale, 1967.

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