Commutator width of Chevalley groups over rings of stable rank 1

1 Introduction

The study of commutators in linear groups over fields has a rich history, culminating in the celebrated proof of the Ore conjecture [11, 15], stating that every element of a finite non-abelian simple group is a commutator.

With this definition the Ore conjecture can be reformulated as follows: all finite simple groups have commutator width 1.

When G is not perfect, one must distinguish the commutator width of G from the commutator width of [G,G], the latter sometimes being greater.

Linear groups over rings, on the other hand, received much less attention. It was shown in [25] that for an associative ring R of stable rank 1 the group GL⁢(n,R), n⩾3, has the commutator width ⩽2. It was then generalized (with somewhat worse bounds and with slightly stronger assumptions on the ring) in [2] to symplectic, orthogonal and unitary groups in even dimension in the context of hyperbolic unitary groups [4].

Recall that a commutative ring R is said to have stable rank 1 if for any a,b∈R such that they generate R as an ideal there exists c∈R such that a+b⁢c∈R* is invertible. Basic examples of rings of stable rank 1 are fields, semilocal rings, boolean rings, the ring of all algebraic integers, the ring of entire functions, the disc-algebra.

The goal of the present paper is to provide a similar result for exceptional groups, namely we prove:

The estimate for 𝖤6 seems to be non-optimal, but as with the other cases, we do not see at present how to improve these bounds. It is tricky even over specific local rings such as p-adic integers, where only partial results are available [3]. It is shown in [3] that there exists a finite local ring R such that SL⁢(n,R) has commutator width at least 2. In fact, a central element is not necessarily a commutator even over a field (if it has characteristic zero), see [22]. Over a local ring the role of the center is played by the full congruence subgroups C⁢(Φ,R,𝔪), the preimage of the center under the reduction homomorphism R→R/𝔪. If the residue field is large enough, then every element outside C⁢(Φ,R,𝔪) is a single commutator [12, Theorem 3], and therefore the commutator width is at most 2. See also [3, Theorem 3.5] for a proof in a particular case of SLn with a slightly better estimate on the size of the base field.

It should be noted that there is no way to extend this result to groups over rings of large dimension, as it can fail even over Euclidean rings (which all have stable rank at most 2). Namely, it is shown in [10] that SL⁢(n,ℂ⁢[x]) has infinite commutator width.

This reflects the fact that in general Chevalley groups have very few commutators. For every root system Φ of rank at least 2 there exists a natural number N, depending only on Φ, such that each commutator [x,y] of elements x∈G⁢(Φ,R) and y∈E⁢(Φ,R) is a product of at most N elementary root unipotents (see, e.g., [19, 13]). And a classical result [23] shows that the width of SL⁢(n,ℂ⁢[x]) with respect to elementary generators is unbounded.

Over a ring R of rank 2 one has to use arithmetic properties of R, as is done in [8, 1, 20, 21, 30] for various Dedekind rings of arithmetic type. The case of a polynomial ring F⁢[x], F is a field of finite transcendence degree over its prime subfield, is wide open.

The proof of Theorem 1 follows the line of those in [25, 2], but tries to avoid explicit matrix calculations, thus giving a simpler and (almost) uniform treatment for Chevalley groups of all normal types. Apart from exceptional groups, it also covers Spin and odd-dimensional orthogonal groups, which were not considered in the previous papers.

It was a surprise for the author that the case of the special linear group over a ring of stable rank 1 is not presented in [25] or anywhere else. Apparently, the reason is that one has to do some additional considerations as in Lemma 5 and Remarks 3 and 4 (compare with [25, Proposition 8]), leading to certain (insignificant) complications in the proof of Theorem 1 for SL4⁢k+2.

Let us now fix some notation. All rings are assumed to be commutative with 1. For a ring R we denote by R* its multiplicative group. All group commutators are left-normed: [x,y]=x⁢y⁢x-1⁢y-1.

Let Φ be a reduced irreducible root system, α1,…,αℓ its fundamental roots (numbered as in Bourbaki), W⁢(Φ) the corresponding Weyl group, generated by the simple reflections σ1,…,σℓ. We always assume Φ to be of rank ⩾2.

For a commutative ring R with 1 by the Chevalley group G⁢(Φ,R) we mean the group of points of the corresponding Chevalley–Demazure group scheme G⁢(Φ,-). Unless specified otherwise, all groups are assumed to be simply connected.

To each root α∈Φ there correspond elementary root unipotents xα⁢(t), t∈R. They are subject to the following relations, additivity and Chevalley commutator formula (α,β∈Φ, α≠-β):

The integers Nα⁢β⁢i⁢j do not depend on r,s. For simply-laced root systems (𝖠ℓ, 𝖣ℓ, 𝖤ℓ) these numbers all equal ±1, while for doubly-laced systems (𝖡ℓ, 𝖢ℓ, 𝖥4) they can also be equal to ±2 (and ±2 or ±3 for 𝖦2).

The elementary Chevalley group E⁢(Φ,R) is a subgroup of G⁢(Φ,R), generated by all elementary root unipotents:

In the particular case Φ=𝖠ℓ we write E⁢(ℓ+1,R)⩽SL⁢(ℓ+1,R) for E⁢(Φ,R). We will also use the group G⁢E⁢(n,R)⩽GL⁢(n,R), which is generated by E⁢(n,R) together with {diag⁡(t,1,…,1)∣t∈R*}

The elementary subgroup is usually strictly smaller than the ambient group, the difference is measured by the non-stable K1-functor (the elementary subgroup is normal):

If R has stable rank 1, then K1⁢(𝖠ℓ,R)=S⁢K1⁢(ℓ+1,R) is trivial, and for groups of other types one has to impose some stronger assumptions, such as absolute stable rank 1 or semilocality. However, K1⁢(Φ,R) is always trivial for Euclidean rings, but not for PID in general.

The elementary subgroup E⁢(Φ,R) is always perfect [18], with the following additional assumptions for groups of small ranks:

1. Φ=𝖠1 and elements t2-1 for t∈R* generate R as an ideal,

2. Φ=𝖢2,𝖦2 and R has no residue field with two elements.

The ambient group G⁢(Φ,R) is not always perfect, and its commutator subgroup can be strictly bigger that its elementary subgroup [24], so that K1 is not necessary abelian. Moreover, there exist finite-dimensional subrings R of the rings of continuous functions on some topological space such that K1⁢(n,R), n⩾3, has arbitrary large nilpotency class.

We extensively use weight diagrams, see [16, 26, 27]. Weight diagrams allow one to visualize the action of the elementary root unipotents xα⁢(t). Below is the weight diagram for the natural vector representation of SLℓ+1, the nodes correspond to the weights of the representation, and the edges correspond to the fundamental roots. Note that xi⁢j⁢(t) acts on a vector (vk)k=1ℓ+1 by adding t⁢vj to vi, or, in terms of the diagram, along the chain connecting j to i.

Figure 1

Weight diagram for (𝖠ℓ,ϖ1).

In all of the weight diagrams that appear below we only show the positive part of the adjoint module. The rightmost column of nodes corresponds to the rk⁡(Φ) zero weights and serves as the symmetry axis for the complete diagram.

We call a set of roots S⊆Φ closed if for any α,β∈S, if α+β∈Φ is a root, then α+β∈S. Two important examples of closed sets of roots are the following. Let mi⁢(α), i=1,…,ℓ, denote the coefficients in the expansion of α as an integer linear combination of the fundamental roots. Then put

The sets Σk are unipotent (i.e. S∩-S=∅), and Δk are symmetric (i.e. S=-S). The notation Σk=n, Σk⩽n and Σk⩾n is self-explanatory.

The sets of all positive and negative roots Φ+ and Φ- are also both closed and unipotent.

To a closed set of roots S we associate a subgroup

The ring R is often clear from the context and thus omitted in the notation. If S is unipotent, we sometimes write U⁢(S) instead of E⁢(S). The unitriangular subgroups U⁢(Φ±) are denoted by U±.

An important fact about U⁢(S) for a closed unipotent S is that it decomposes into a product U⁢(S)=∏α∈SXα of its root subgroups in any prescribed order.

U⁢(Σk) is the unipotent radical of the corresponding parabolic subgroup Pk (or E⁢(Δk∪Σk)). Levi decomposition states that E⁢(Δk∪Σk) is the semi-direct product of its elementary Levi subgroup E⁢(Δk) and its normal subgroup U⁢(Σk).

2 Commutators and companion matrices

Note that Φ+=Θ∪⋃k⩾0Ωk and the union is disjoint.

We define, as usual

We write H⁢(Φ,R)=〈hα⁢(s)∣α∈Φ,s∈R*〉 for the elementary part of the torus. For simply connected groups it coincides with the whole torus.

The above relations hold on the level of Steinberg group, while on the level of the elementary subgroup they imply the following formula (note that the signs cancel out):

The extended Weyl group W~⁢(Φ) is the subgroup of G⁢(Φ,R), generated by wα⁢(1), α∈Φ. If 2≠0 in R, it coincides with N⁢(Φ,ℤ), the group of integer points of the torus normalizer. It is an extension C2ℓ↪N⁢(Φ,ℤ)↠W⁢(Φ), and the action of the generators on the kernel is described by the above formula.

For any Coxeter element wc one has Θwc=∅. Thus with the choice of π~ as above Θπ~=∅ in all cases except 𝖤6, when Θπ~=Σ2.

The above description (for Φ≠𝖤6) is obtained as follows: first, one checks that π~ sends the right-hand side to Φ- and that the number of roots in it equals the rank of Φ. Then it remains to note that |Ω0π~|=rk⁡(Φ). This follows from the fact that all orbits of a Coxeter element wc have the same size, equal to the Coxeter number h (since wc acts by rotation by 2⁢π/h on its Coxeter plane and no root projects to zero) and that |Φ|=h⋅rk⁡(Φ). For Φ=𝖤6 one applies this argument to the 𝖠5-subsystem Δ2.

A lift w of a Coxeter element to the extended Weyl group is essentially a set of signs siw=±1, assigned to the fundamental roots αi.

Since for Φ=𝖠ℓ,𝖣ℓ,𝖤7 we can not use Lemma 3, we have to do some additional calculations in these cases.

Let w0 denote the longest element of the Weyl group. We write w0^ for its obvious lift to the extended Weyl group, obtained by sending each of σi in the reduced expression to wi⁢(1). We will later fix another lift in case Φ=𝖠ℓ.

As a corollary, we see that in case Φ=𝖣ℓ one has w0^⁢π⁢w0^=π, since wℓ⁢(1) and wℓ-1⁢(1) commute.

Let pn denote the n×n peridentity matrix, that is, pn=(δi,n-j+1)i,j=1,…,n. If n≠4⁢k+2, one has either det⁡(pn)=1 or det⁡(-pn)=1, so for SL⁢(n,R)=G⁢(𝖠n-1,R) we specifically fix w0^ to be pn or -pn, which is not the obvious lift of w0. But with this choice one has

which can be written simply as w0^⁢π-1⁢w0^-1=π.

For n=4⁢k+2 neither pn nor -pn lies in SL⁢(n,R). In this case we set w0^ to be an anti-diagonal matrix with 1’s and -1’s alternating. Then

3 Proof of the main result

Proof.

We start with working out the case Φ=𝖠ℓ. Denote Δ=Δℓ; then the Levi factor E⁢(Δ) acts on the unipotent radical U⁢(Σℓ).

Write θ=∏α∈Σxα⁢(ξα). An element g∈E⁢(Δ)≅E⁢(ℓ,R) acts on the vector consisting of ξα exactly as in (𝖠ℓ-1,ϖ1). To avoid confusion the result will be denoted by (ξα)g.

Let g∈E⁢(Δ) be the element, constructed in Lemma 7. Then put

η=[g,∏α∈Σxα(ζα)],where (ζα)=(ξα)(g-1)-1.

Expanding the commutator, one has

g⋅∏α∈Σxα(ζα)⋅g-1=∏α∈Σxα(ζα′) with (ζα′)=(ζα)g

and

η=∏α∈Σxα(ζα′′) with (ζα′′)=(ζα′)-(ζα)=(ζα)(g-1)=(ξα).

Thus in this case θ=η, a commutator.

Figure 2

Weight diagram for (𝖢ℓ,2⁢ϖ1).

If Φ=𝖢ℓ, we first write

θ=xαℓ⁢(aℓ)⋅∏α∈Σ′xα⁢(aα)=xαℓ⁢(aℓ)⋅θ′, where ⁢Σ′=Σ∖{αℓ},

and

xαℓ-1+αℓ⁢(*)⁢xαℓ⁢(t)=[x2⁢αℓ-1+αℓ⁢(1),x-αℓ⁢(±t)].

Since αℓ-1+αℓ∈Σ=Σℓ=1∩Σℓ-1⩽1, we obtain

xαℓ⁢(t)=c⋅xαℓ-1+αℓ⁢(*), where c is a commutator,

θ=c⋅θ′′ for some ⁢θ′′∈U⁢(Σ∖{αℓ}).

If ℓ⩾3, an element g∈E⁢(Δℓ∩Δℓ-1), constructed in Lemma 7, does not use roots from Σℓ-1 and it is clear from the diagram (Figure 2) that it acts on θ′′ as prescribed by Lemma 7, allowing us to repeat the argument we used for Φ=𝖠ℓ. Thus θ is a product of two commutators.

If ℓ=2, the above argument does not work (the chain marked by dashed line has length 1). However, α1+α2 is a short root, so we can use the hyperbolic embedding of G⁢E⁢(2,R), corresponding to that root. Namely, we assume 1=r+s for some r,s∈R* and express xα1+α2⁢(t) as a commutator

xα1+α2⁢(t)=[hα1+α2⁢(s)⁢hα2⁢(s-1),xα1+α2⁢(r-1⁢t)]

in H⁢(𝖢2,R)⋅E⁢(〈α1+α2〉,R) (see Remark 5)

If Φ=𝖡ℓ (Figure 3), we do the same as for 𝖢ℓ. This time we exclude αℓ and note that Δℓ∩Δℓ-1 acts on the chain Σ∖{αℓ} as (𝖠ℓ-2,ϖ1). Again,

xαℓ⁢(t)=[xαℓ-1+αℓ⁢(1),x-αℓ-1⁢(±t)]⋅xαℓ-1+2⁢αℓ⁢(*).

Figure 3

Weight diagram for (𝖡ℓ,ϖ2).

Figure 4

Weight diagram for (𝖣ℓ,ϖ2).

If Φ=𝖣ℓ (Figure 4), we divide Σ into the following two parts: Σ1∩Δℓ-1 and {α=α1+…+αℓ-1}. The first one is subject to the action of Δℓ-1 (of type 𝖠ℓ-1), while xα⁢(t)=[xα+αℓ⁢(t),x-αℓ⁢(1)].

The very same method works for Φ=𝖤7,𝖤8, see Figures 5 and 6.

Figure 5

Weight diagram for (𝖤7,ϖ1).

Figure 6

Weight diagram for (𝖤8,ϖ8).

Figure 7

Weight diagram for (𝖥4,ϖ1).

If Φ=𝖥4 (Figure 7), the subsystem subgroup E⁢(〈α1,α3〉)≅E⁢(𝖠1)×E⁢(𝖠1) acts on U⁢(Σ), and the edges labelled 1 and 3 only meet each other outside Σ.

Figure 8

Weight diagram for (𝖤6,ϖ2).

If Φ=𝖦2, we choose the structure constants as in [9, Section 12.4] and write

xα1⁢(t)=[hα1⁢(s)⁢h2⁢α1+α2⁢(s-1),xα1⁢(-r-1⁢t)],

x3⁢α1+α2⁢(t)=[x3⁢α1+2⁢α2⁢(t),x-α2⁢(-1)].

In the remaining case Φ=𝖤6 (Figure 8) we split Σ into three parts, marked by solid, dashed and dotted outlines on the diagram, acted on by E⁢(Δ2), E⁢(〈α5,α6〉) and E⁢(〈α1,α2,α3〉) (of type 𝖠5, 𝖠2 and 𝖠2×𝖠1) correspondingly. ∎