Covering Numbers for Simple Algebraic Groups

Let G be a simple algebraic group over an algebraically closed field, and let C be a noncentral conjugacy class of G. The covering number cn(G,C) is defined to be the minimal k such that G = Ck, where Ck = {c1c2⋯ck : ci ∈ C}. We prove that cn(G,C)≤cdimGdimC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$cn(G,C) \le c \frac {\dim G}{\dim C}$\end{document}, where c is an explicit constant (at most 120). Some consequences on the width and generation of simple algebraic groups are given.

with the discussion of the case where G is a finite simple group and C a nontrivial conjugacy class, the main result of [19] shows that there is an absolute constant c such that cn(G, C) ≤ c log |G| log |C| for all such G, C. (Obviously log |G| log |C| is a lower bound for the covering number.) In [22], it is suggested that an analogous result should hold for algebraic groups: namely, there should be a constant c such that cn(G, C) ≤ c dim G dim C for all non-central conjugacy classes C of simple algebraic groups G. Such a bound is established in [22,Theorem C] for the case where C is a unipotent conjugacy class in good characteristic. In this paper we prove such a result in general. Note that dim G dim C is an obvious lower bound for cn(G, C).
Theorem 1 There is a constant c such that for any simple algebraic group G over an algebraically closed field K, and any non-central conjugacy class C of G, we have Moreover, this holds with c = 120.
In fact, our proof gives smaller values of the constant c in most cases: specifically, we may take for G of type A l , 36 for G of type C l , 120 for G of type B l , D l , 18 for G of exceptional type. These are certainly not the lowest possible values of c, but more effort would be required to reduce them substantially.
Theorem 1 follows from [7] for groups of bounded rank (including exceptional groups). For G = SL n (K), it is a consequence of a more general result in [21]: from Theorem 2.8 together with the proof of Proposition 2.10 of that paper, it follows that if C 1 , . . . , C k are conjugacy classes of G = SL n (K) such that dim C i ≥ 12 dim G, then G = C 1 · · · C k . In particular this implies the conclusion of Theorem 1 with c = 12. A version of this result for arbitrary simple algebraic groups G, but with an unspecified constant in place of 12, follows from [13,Theorem 8.11].
Our proof of Theorem 1 is rather short and elementary, and runs along very different lines to those in [13,21].
We now give some consequences of Theorem 1. The first concerns the involution width of a simple algebraic group G-that is, the minimal positive integer t such that every element of G can be expressed as a product of t or fewer involutions. Continue to let K denote an algebraically closed field. Corollary 2 Let G be an adjoint simple algebraic group over K. The involution width of G is at most 2c, where c is the constant in Theorem 1.
This is a simple consequence of Theorem 1, together with the fact that there is a conjugacy class of involutions in G of dimension at least 1 2 dim G (see for example [18, §4]). Again, the bound of 2c in the corollary is far from best possible-in fact, as shown to us by the referee, using the main theorem of [20] one can quickly prove that the correct bound is 4, as follows. Let V be the variety of involutions in G. If K is the algebraic closure of a finite field, then V 4 = G by [20]. The same conclusion then follows for K an arbitrary algebraically closed field of prime characteristic, for example by the statement (1) in the proof of [9, Proposition 1.1]. Finally, one can extend this to characteristic zero by a standard compactness argument, along the lines of [9, §7].
The corollary generalizes to elements of arbitrary order: Corollary 3 Let r ≥ 2 be an integer, and let G be an adjoint simple algebraic group over K such that G contains an element of order r. Then every element of G can be expressed as a product of at most 2c elements of order r, all of which can be chosen from the same conjugacy class of G.
This again follows from Theorem 1, together with [14], which shows that G has a conjugacy class of elements of order r of dimension at least 1 2 dim G. Again one can substantially reduce the constant 2c in the conclusion using further arguments, but we do not do this here, as our main point in stating these corollaries is to show some consequences of Theorem 1, rather than obtaining best possible bounds.
The next consequence concerns generation properties of simple algebraic groups G over K. We say that a finite set S of elements generates G topologically, if the subgroup S is Zariski dense in G. Of course, if K is algebraic over a finite field, then G is locally finite, hence is not topologically generated by any finite subset. So assume that K is not algebraic over a finite field. For a non-central conjugacy class C of G, define the generation number gn(G, C) to be the minimal number d such that G can be topologically generated by d elements of C. It is proved in [10, §8] that gn(G, C) ≤ 2 rank(G) + 1 for all classes C. Our final corollary provides a bound that reflects the dependence of the generation number on C: Corollary 4 Let G be a simple algebraic group over K, and assume that K is not algebraic over a finite field. If C is a non-central conjugacy class of G, then where c is the constant in Theorem 1.
This can be quickly justified using Theorem 1, so we do this here. Let x ∈ C. By [8,Theorem 3.3], there is an element y ∈ G such that G is topologically generated by x, y. And by Theorem 1, we can write y = y 1 · · · y m , where each y i ∈ C and m ≤ c dim G dim C . Thus G is topologically generated by the m + 1 elements x, y 1 , . . . , y m , all of which lie in C.
Again, the bound c in the conclusion of Corollary 4 can be greatly improved-in fact the result follows from [6] for SL n (K) with c = 2, and from [1,2] in general with c = 5.
The rest of the paper is devoted to the proof of Theorem 1. After some preliminaries in Section 2, we present the proof in Section 3.

Preliminaries
Throughout the rest of the paper, K denotes an algebraically closed field of characteristic p ≥ 0.
We begin with a brief discussion of unipotent classes in symplectic and orthogonal algebraic groups, taken from Section 3.3.2 and Chapter 4 of [17]. In Sp 2m (K) and SO 2m (K), there is a subgroup GL m (K) stabilizing a pair of totally singular m-spaces, and we denote by W (m) a unipotent element that acts as a single Jordan block in this subgroup GL m (K). Also, in each of Sp n (K) (n even), SO n (K) (n odd, p = 2) and O n (K) (n even, p = 2), there is a unipotent element that acts as a single Jordan block, which we denote by V (n). Every unipotent element of a symplectic or orthogonal group where for (G, p) = (SO(V ), 2), the number of summands V (n j ) is even. The regular unipotent elements u reg ∈ G are: The next lemma shows that regular classes have very small covering numbers.

Lemma 2.1
Let G be a simple algebraic group over K, and let C = x G be a conjugacy class.
(ii) Let UT be a Borel subgroup of G with unipotent radical U and maximal torus T , and let x = us be regular, with unipotent part u ∈ U and semisimple part s ∈ T . Let U − denote the maximal unipotent subgroup of G opposite to U . By We shall need to extend Lemma 2.1 to some further unipotent classes of orthogonal groups in characteristic 2. For this we define the following notation. As noted above, there is a subgroup X = GL m (K) of G = SO 2m (K). For β 1 , . . . , β m ∈ K * , let d β 1 ,...,β m = diag(β 1 , . . . , β m ) ∈ X; on the natural module for G this acts as a diagonal matrix with eigenvalues β ±1 i for 1 ≤ i ≤ m.
Proof Let q = 2 f , and let F q be a Frobenius endomorphism of G such that the fixed point group We shall use character theory, via the well-known fact that for any d ≥ 2 and any element z ∈ G(q), the number of solutions (x 1 , . . . , x d ) ∈ C(q) × · · · × C(q) to the equation where the sum is over the set of all irreducible characters Irr(G(q)) of G(q). (For example, this follows easily by induction using [11,Theorem 30.4].) Let z ∈ G(q) be regular semisimple. We shall show that for d = 8 and for sufficiently large q, the dominant term in the sum in (2) comes from the trivial character, proving that the sum is positive. From this it will follow that z ∈ C(q) 8 , as required for the lemma.
In what follows, c 1 , c 2 , . . . are positive absolute constants. In order to analyse the sum in (2), we marshall a few facts: (a) By [12], From (c) and (d), it follows that for any nontrivial χ ∈ Irr(G(q)), It follows that 1 =χ ∈Irr(G(q)) For m ≥ 3 this tends to 0 as q → ∞; hence for large q, the sum in (2) is positive, showing as noted above that z ∈ C(q) 8 .
Regarding G as SL 2 ⊗ SL 2 , we have u = J 2 ⊗ J 2 , the tensor product of two Jordan blocks of size 2. Hence from Lemma 2.1 we have C 2 = G in this case.
To conclude this section, it is convenient to deal with some low rank cases of Theorem 1 here:

Lemma 2.3
Suppose that G is a simple algebraic group that is either of exceptional type, or of classical type of rank less than 5. Then cn(G, C) ≤ 18 dim G dim C for all non-central conjugacy classes C of G.
Proof This is immediate from [7, Theorem 2], which implies that for the groups G in the hypothesis we have cn(G, C) ≤ 18 for all non-central classes C.

Proof of Theorem 1
Let G be a simple algebraic group of rank l over an algebraically closed field K of characteristic p ≥ 0, and let C = x G be a non-central conjugacy class of G.
In view of Lemma 2.3, we can assume that G is a classical group of rank l ≥ 5. We shall prove Theorem 1 for G of simply connected type, from which it follows for all isogeny types.
Before starting the proof we describe our strategy, very roughly. First, we write x in Jordan form, and collect blocks together into larger blocks, in such a way that we can apply Lemma 2.1 to each block and obtain a supply of semisimple elements in a small power C r .
We then argue that a further suitable power of C r contains a regular semisimple element, and deduce the theorem using Lemma 2.1(i).

The Case G = SL n (K )
We prove Theorem 1 for G = SL n (K) with the constant c = 27. Let V = V n (K) be the natural module for G. Let x ∈ G be non-central and C = x G . Write x in Jordan normal form as where D is diagonal (or empty), the α i are scalars, and each J r i (1) is a unipotent Jordan block of size r i ≥ 2. Let where the scalars λ 1 , . . . , λ b are distinct and n 1 ≥ · · · ≥ n b . (In the case where D is empty, we treat all n i as 0 in the following discussion.) For 1 We can conjugate this by elements of the subgroup SL r 1 × · · · × SL n 2 −n 3 2 and use Lemma 2.1(i,iii) to see that C 3 contains diagonal matrices diag β 1 , . . . , β s , λ 3 1 I n 1 −n 2 , where s = n − n 1 + n 2 and the β i are arbitrary, subject to the determinantal conditions β 1 · · · β r 1 = α 3r 1 1 , etc. There is a product of n s such diagonal matrices that is regular semisimple in G. Hence by Lemma 2.1(i) we have Now C G (x) contains GL n 1 , so dim C ≤ n 2 − n 2 1 − 1 and Also s ≥ n − n 1 , and a routine calculation shows that Therefore n s ≤ 3 dim G dim C , and it follows from (3) that as required.

The Case G = Sp 2m (K )
Here we prove Theorem 1 for G = Sp(V ) = Sp 2m (K) with the constant c = 36. Let x ∈ G be non-central and C = x G . Write x = su, where s, u are the semisimple and unipotent parts of x. Then with respect to a suitable basis, we have  (1): where all m i , m i ≥ 2 and all n j , n j are even. Now consider the projection of u to a factor GL c i of C G (s). Write this as a sum of Jordan blocks J 1 (1) e + r j ≥2 J r j (1). Then each block J r j (1) is regular unipotent in a subgroup GL r j ; and for each block J 1 (1) ∈ GL 1 < Sp 2 , x acts on the corresponding 2-space as λ i , λ −1 i , which is regular semisimple in Sp 2 . From all this we conclude that where each x i is one of: We need to further refine the decomposition (4) when p = 2. Replacing x by −x if necessary, we can assume that c ≥ d. Let y = (I 2 , −I 2 ) ∈ Sp 4 , and write x as There is a product of m s such diagonal matrices that is regular semisimple in G. Hence by Lemma 2.1(i) we have Now Also s ≥ m − c, and one checks that Therefore m s ≤ 3 dim G dim C , and it follows from (5) that as required.

The Case Where G is Orthogonal
Here we prove Theorem 1 for G orthogonal, with the constant c = 120. The simply connected version is a spin groupĜ = Spin n (K). However, it is more convenient for us to work with the orthogonal group G = SO n (K). As above, our method of proof is to show that for any non-central class C of G, some suitable power C r contains a regular semisimple element of G, hence C 3r = G by Lemma 2.1. IfĈ is any class ofĜ that maps to C, thenĈ r also contains a regular semisimple element ofĜ, henceĈ 3r =Ĝ. Thus our proof of Theorem 1 for G will also give the result forĜ. Assume then that G = SO(V ) = SO n (K). The proof is quite similar to the symplectic case, but there are a number of extra complications at various points. It is convenient to deal separately with the cases where p = 2 and p = 2.

The Case p = 2
Suppose first that p = 2. Let x ∈ G be non-central and C = x G . Write x = su, where s, u are the semisimple and unipotent parts of x. With respect to a suitable basis, we have where a + b + 2 c i = n and λ i = ±1 for all i. As in the symplectic case, letting V a , V b be the ±1-eigenspaces of s, write where all m i , m i ≥ 2, and all n j , n j are odd and ≥ 3; also d and the number of blocks V n j are either both even or both odd (as det(x) = 1). Let the projection of u to each factor GL c i be a sum of e i Jordan blocks of size 1 and the rest of size ≥ 2. Thus where and each x i is a scalar multiple (by some λ i = ±1) of a regular unipotent element of a subgroup GL s i , s i ≥ 2. (Note that we have singled out the ±W (2) summands in D, as W (2) lies in one of the SL 2 factors in SL 2 ⊗ SL 2 = SO 4 , so does not have finite covering number in SO 4 .) As in the symplectic case, observe that for each summand contains regular semisimple elements of G i ; and for each summand y j = ±V (n j ), and G j = SO n j , we have y Next, we need to refine the decomposition of D in (7). We seek to combine all the terms ±W in O 5 or O 3 ; the class of such a block has 8th power containing SO 5 or SO 3 .
Assuming that (c, d) = (0, 0) (we postpone the (0, 0) case until later), the process in the preceding paragraph can be carried out to write and so by (6),

The Case p = 2
This case follows along very similar lines. Assume that p = 2. As in the previous subsection, write x = su, where Let V a denote the 1-eigenspace of s. By (1), we can write where all the n j are even, and the number of summands V (n j ) is even. Again let the projection of u to each factor GL n i be a sum of e i Jordan blocks of size 1 and the rest of size ≥ 2. Then and each x i is a scalar multiple (by some λ i = 1) of a regular unipotent element of a subgroup GL s i , s i ≥ 2. Next, we refine the decomposition of D just as in the p = 2 case: assuming c > 2, we can write where c ≤ c and each summand z i is non-central in the corresponding orthogonal subgroup SO 6 or SO 8 . Thus x = I c ⊕ z i ⊕ x i ⊕ V (n j ). Now using Lemma 2.2, we see that C 10 contains diagonal matrices diag I c , β 1 , β −1 1 , . . . , β s , β −1 s , where 2s = n − c , and the β i ∈ F q , any q ≥ q 0 , are arbitrary subject to the relevant determinantal conditions. As in the previous subsection (see (10)), it follows that C 30 n 2s = G. Now 2s = n − c ≥ n − c, while dim G dim C ≥ n 2 −n n 2 −n−c 2 +c . We check that n n − c + 1 ≤ 3(n 2 − n) n 2 − n − c 2 + c and hence n 2s ≤ 3 dim G dim C . It follows that cn(G, C) ≤ 90 dim G dim C , giving the result. Finally, for the case where c ≤ 2, either D can be written in the form (12), or it has one of the forms in (11), or it has the form (I 2 , λ 1 , λ −1 1 ) ⊕ z i . Now the conclusion follows by a very similar argument to the one given after (11).
The proof of Theorem 1 is now complete.
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