The Yagita invariant of general linear groups

H. H. Glover, I. J. Leary and C. B. Thomas

We give a definition of the Yagita invariant p^{∘}(G) of an arbitrary group G, and compute p^{∘}(GL_{n}(O)) for each prime p, where O is any integrally closed subring of the complex numbers C (i.e., O is integrally closed in its field of fractions F, which is also a subring of C). We also show that p^{∘}(SL_{n}(O)) is equal to p^{∘}(GL_{n}(O)) for n ≥ 2 except that possibly p^{∘}(SL_{n}(O)) is 1/2p^{∘}(GL_{n}(O)) for some small n and ‘small’ rings O. Our definition of p^{∘}(G) extends both Yagita’s original definition for G finite [9] and the definition given by one of us for G of finite virtual cohomological dimension (or vcd) [8]. For G of finite vcd such that the Tate-Farrell cohomology Ĥ^{∗}(G) of G is p-periodic, p^{∘}(G) is equal to the p-period. Hence our results may be viewed as a generalisation of those of Bürgisser and Eckmann [1,2], who compute the p-periods of GL_{n}(O) and SL_{n}(O) for various O such that these groups have finite vcd and for all n such that these groups are p-periodic. The methods we use are similar to those used in [1].

Notation. Throughout this paper p shall be a fixed prime number, O an integrally closed subring of C, and F the field of fractions of O. ζ_{p} shall be a primitive pth root of unity in C, and l shall be the degree of F [ζ_{p}] as an extension of F. In [1,2,4], the notation ϕ_{F} (p) is used for l.

Definition. Let G be a group. If C is an order p subgroup of G, then

H^{∗}(BC; Z) = Z[x]/(px)

for x a generator of H^{2}. Define n(C) (either a positive integer or infinity) to be the supremum of the integers n such that the image of H^{∗}(BG) in H^{∗}(BC) is contained in the subalgebra generated by x^{n}. Now define the Yagita invariant p^{∘}(G) to be

p^{∘}(G) = l.c.m.{2n(C) : C ≤ G, |C| = p},

where the least common multiple of the empty set is equal to 1, and the least common multiple of an unbounded set of integers, or of a set containing infinity, is infinity. The invariant p^{∘}(G) depends on the prime p as well as on the group G, but we have resisted the temptation to write, for example, 2^{∘}(G) for p^{∘}(G) in the case when p = 2.

The following result is immediate from the definition.

Proposition 1. Let f: H → G be a group homomorphism whose kernel contains no element of order p. (For example, f might be injective, or the kernel of f might be torsion-free.) Then p^{∘}(H) divides p^{∘}(G).

It is easy to see that p^{∘}(G) is finite if G is finite, and Proposition 1 applied in the case when G is the quotient of H by a torsion-free normal subgroup of finite index implies that p^{∘}(H) is finite if H is a group of finite vcd (see [5] for a more detailed proof of this fact). Using Chern classes, we shall show in Corollary 7 that if G admits a faithful finite-dimensional representation over C, then p^{∘}(G) is finite.

Some remarks concerning the origin of the definition of p^{∘}(G) and its geometric motivation are in order. N. Yagita made the above definition for finite groups [9], and proved that if a finite group G acts freely on a (finite) product S^{m−1} ×…×S^{m−1} of spheres of the same dimension m−1, with trivial action on their homology, then p^{∘}(G) divides m. This result may be viewed as a generalisation of the theorem bounding the dimension of a sphere with a free G-action in the case when the cohomology of G is periodic. One of us suggested extending the definition to groups of finite vcd in [8]. The connection with actions on products of spheres extends to this case too. If G is a group of finite vcd acting freely, properly discontinuously and with trivial action on homology, on a finite product S^{m−1} × … × S^{m−1} × R^{n}, then p^{∘}(G) divides m, by the same argument as used in [9] for the case when G is finite. For arbitrary groups, the connection with the geometry of group actions is less clear. One reason for extending the definition to arbitrary groups is that this makes our results easier to state and no harder to prove.

To motivate our work we start by stating the main theorems and some corollaries. The proofs of the main theorems will occupy the remainder of the paper.

Theorem 2. As explained under the title ‘Notation’, let O be an integrally closed subring of C with field of fractions F, and let l = |F [ζ_{p}] : F |. Define ψ(t), for t a positive real, to be the greatest integer power of p less than or equal to t. The Yagita invariant p^{∘}(GL_{n}(O)) is given by the following table.

p^{∘}(GL_{n}(O)) = \{ \begin{matrix}1 & for n < l \\ 2l.c.m.\{m : l|m, m|(p − 1), m ≤ n\} & for l ≤ n ≤ p − 1 \\ 2(p − 1)ψ(n/l) & for n ≥ p \end{matrix}

Remark. The above theorem holds for all primes p, but the statement can be simplified slightly for p = 2. In this case l = 1, and p^{∘}(GL_{n}(O)) = 2ψ(n) for all n ≥ 1. Note that p^{∘}(GL_{n}(O)) depends only on F, so that if R is any subring of C such that O ≤ R ≤ F, then p^{∘}(GL_{n}(R)) = p^{∘}(GL_{n}(O)).

Replacing GL_{n}(O) by SL_{n}(O) does not change the Yagita invariant, except for small values of n and ‘small’ rings O, when it may be necessary to introduce a factor of 1/2. Thus one obtains the following.

Theorem 3. With notation as above, for n ≥ 2, p^{∘}(SL_{n}(O)) is equal to p^{∘}(GL_{n}(O)) except that p^{∘}(SL_{n}(O)) may be equal to \frac{1}{2}p^{∘}(GL_{n}(O)) in the following cases.

a) p = 2, n = 2, and O contains no square root of −1. b) p is odd, n has the form n = 2^{r}l for 2^{r} dividing (p − 1)/l, and O contains no nth root of −1.

Corollary 4. Define the function ψ as in the statement of Theorem 2. Then for n ≥ 2 and any prime p, the Yagita invariant of GL_{n}(Z) is as follows.

p^{∘}(GL_{n}(Z)) = \{ \begin{matrix}1 & for n < p − 1, \\ 2(p − 1)ψ(n/(p − 1)) & for n ≥ p − 1. \end{matrix}

For all n ≥ 2 and all p, p^{∘}(SL_{n}(Z)) = p^{∘}(GL_{n}(Z)), except that for p odd, p^{∘}(SL_{p−1}(Z)) = p − 1, and for p = 2, p^{∘}(SL_{2}(Z)) = 2.

Proof. The statement for the general linear groups is a special case of Theorem 2. The only cases when p^{∘}(SL_{n}(Z)) is not completely determined by Theorem 3 are those when n = p − 1 and when n = p = 2, but in these cases SL_{n}(Z) is p-periodic and the p-period is determined in [1].

The following statement may be proved in the same way as Theorems 2 and 3, and could be considered as a corollary but for the weaker conditions imposed on the subring R of C.

Theorem 5. Let R be any subring of C containing ζ_{p}, a primitive pth root of 1, and let ψ be as in the statement of Theorem 2. Then the Yagita invariant of GL_{n}(R) is given by the following table.

p^{∘}(GL_{n}(R)) = \{ \begin{matrix}2l.c.m.\{m : m|(p − 1), m ≤ n\} & for n ≤ p − 1 \\ 2(p − 1)ψ(n) & for n ≥ p \end{matrix}

For R as above, p^{∘}(SL_{n}(R)) = p^{∘}(GL_{n}(R)) if any of the following conditions is satisfied:

a) n ≥ max{p, 3},

b) n ≥ 2, p is odd, and R contains a (p − 1)st root of −1,

c) n = 2, p = 2, and R contains a square root of −1.

Our upper bound for p^{∘}(GL_{n}(O)) uses the Chern classes of the natural representation in GL_{n}(C), and relies on the following proposition.

Proposition 6. Let f(X) be a polynomial over the field F_{p}, all of whose roots lie in F_{p}^{×}. If there is a polynomial g and integer n such that f(X) = g(X^{n}), then n has the form mp^{q} for some m dividing p−1 and some positive integer q.

Proof. Let n = mp^{q}, where p does not divide m. It remains to prove that m divides p − 1. Now g(X^{n}) = g(X^{m})^{p^{q}}, so without loss of generality we may assume that q = 0. Now if g(Y) = 0 has roots y_{1}, …, y_{k}, then the roots of g(X^{m}) = 0 are the roots of y_{i} − X^{m} = 0 for each i, and because p does not divide m these polynomials have no repeated roots. If y_{i} is not an element of F_{p}^{×} then y_{i} − X^{m} can have no roots in F_{p}. It has exactly m roots in F_{p} if and only if the inverse image of y_{i} under the map x ↦ x^{m} from F_{p}^{×} to itself has order m, which can only happen if m divides p − 1.

We will use the following Corollary in the proof of Theorem 2.

Corollary 7. With notation as in the statement of Theorem 2, let G be a subgroup of GL_{n}(F). Then the Yagita invariant p^{∘}(G) divides the number given for p^{∘}(GL_{n}(O)) in Theorem 2.

Proof. For each order p subgroup C of G, we bound the number n(C) occuring in the definition of p^{∘}(G) by considering the image of the cohomology H^{∗}(BGL_{n}(C)) of the Lie group GL_{n}(C) in H^{∗}(BC) under the inclusion of C in GL_{n}(C). Recall (or see for example [7]) that H^{∗}(BGL_{n}(C)) is a free polynomial algebra on generators c_{1}, …, c_{n}, the universal Chern classes, where c_{i} has degree 2i. For a representation ρ of a group H in GL_{n}(C), the element ρ^{∗}(c_{i}) is usually written c_{i}(ρ), and called the ith Chern class of the representation ρ. The sum c.(ρ) of all the c_{i}(ρ) is known as the total Chern class of ρ, and has the property that c.(ρ ⊕ ρ′) = c.(ρ)c.(ρ′).

If H^{∗}(BC) = Z[x]/(px), then the total Chern classes of the p distinct 1dimensional complex representations of C are 1 + ix for i ∈ F_{p}, where the case i = 0 corresponds to the trivial representation. If we fix a generator of C such that the 1-dimensional representation of C sending this generator to ζ_{p} has total Chern class 1 + x, then the representation sending this same generator to ζ_{p}^{i} has total Chern class 1+ix. For F as in the statement, there are 1+(p−1)/l irreducible F -representations of C, the trivial representation of dimension one and (p − 1)/l others of dimension l. Over C the faithful representations split as a direct sum of one copy each of the representations sending a fixed generator of C to ζ_{p}^{ri^{j}}, for some r, where i generates a subgroup of F_{p}^{×} of order l and 0 ≤ j < l. It follows that the total Chern classes of these representations have the form 1 − ix^{l}, where i ranges over the (p − 1)/l distinct lth roots of unity in F_{p}.

Now the ring H^{∗}(BC)⊗F_{p} is isomorphic to the free polynomial ring F_{p}[x], and the total Chern class of the inclusion of C in GL_{n}(C) is a polynomial f(x) of degree less than or equal to n satisfying the hypotheses of Proposition 6. Moreover, since this inclusion factors through GL_{n}(F), the total Chern class can be viewed as a polynomial f̃(y) in y = x^{l} of degree less than or equal to n/l satisfying the hypotheses of the Proposition. This polynomial is not the trivial polynomial because the inclusion of C is not the trivial representation. The integer n(C) (resp. n(C)/l) is the greatest integer m such that f(x) (resp. f̃(y)) can be expressed as a polynomial in x^{m} (resp. y^{m}), and so by Proposition 2 each of n(C) and n(C)/l is a divisor of p − 1 multiplied by a power of p. Note also that n(C) is at most n. This implies that for each C, the number n(C) divides the bound given in Theorem 2, and hence so does the l.c.m. of all such n(C).

Remark. The discussion of the total Chern classes of F -representations of the cyclic group of order p could be avoided by quoting the general bounds on the orders of Chern classes of F -representations given in [4]. Corollary 7 also holds for groups G having a representation in GL_{n}(F) which is faithful on every order p subgroup, by the same proof. Corollary 7 may be compared with Lemma 1.2 of [5].

Our lower bounds for p^{∘}(GL_{n}(O)) and p^{∘}(SL_{n}(O)) use various finite subgroups, which we define below.

Definition. For p an odd prime, and m a divisor of p−1, let G_{1}(p, m) be the split metacyclic group C_{p}: C_{m}, where C_{m} acts faithfully on C_{p}, and let G_{2}(p, m) be the split metacyclic group C_{p}: C_{2m}, where C_{2m} acts via the faithful action of its quotient C_{m}. For p an odd prime let E(p, 1) be the non-abelian group of order p^{3} and exponent p, and let E(2, 1) be the dihedral group of order eight. For any p, and any posititve integer m, let E(p, m) be the central product of m copies of E(p, 1), so that E(p, m) is one of the two extraspecial groups of order p^{2m+1}. In ATLAS notation [3], the group E(p, m) is called p_{+}^{1+2m}. Since the symbol p is already overworked in this paper we shall not use the ATLAS notation here.

Lemma 8. For p an odd prime the Yagita invariants of the groups defined above are as follows.

p^{∘}(G_{1}(p, m)) = p^{∘}(G_{2}(p, m)) = 2m, p^{∘}(E(p, m)) = 2p^{m}

For p = 2, let Q_{8} be the quaternion group of order 8. Then

p^{∘}(E(2, m)) = 2^{m+1} and p^{∘}(Q_{8}) = 4.

Proof. The groups G_{1}(p, m) and G_{2}(p, m) are p-periodic, and their p-periods are shown to be as claimed in Proposition 3.1 and Section 3.3 of [2]. The Yagita invariant for the extraspecial groups was computed in Example B of Section 2 of [9], which in turn used results of [6].

Lemma 9. For p an odd prime, and m a divisor of p − 1, let n = ml/(m, l). The group G_{1}(p, m) embeds in GL_{n}(O), and in SL_{n+1}(O). If either l/(m, l) is even or m is odd (and greater than 1) then G_{1}(p, m) embeds in SL_{n}(O). If m is even and O contains an mth root of −1, then G_{2}(p, m) embeds in SL_{n}(O). The group E(p, m) embeds in SL_{lp^{m}}(O).

For p = 2 and m > 1 the group E(2, m) embeds in SL_{2^{m}}(Z). The group E(2, 1) (the dihedral group of order 8) embeds in GL_{2}(Z) and in SL_{3}(Z). The quaternion group of order eight embeds in SL_{2}(Z[i]).

Proof. First consider the case when p is odd. For G_{1}(p, m) and G_{2}(p, m) this is essentially contained in [2] Section 3.2. If m divides m′, then G_{1}(p, m) is a subgroup of G_{1}(p, m′), so without loss of generality we may assume that m is divisible by l and show that in this case G_{1}(p, m) embeds in GL_{m}(O). As an O-module, O[ζ_{p}] is free of rank l, which is closed under multiplication by ζ_{p} and under the action of Gal(F [ζ_{p}]/F), which together generate a group of O-linear automorphisms of O[ζ_{p}] isomorphic to G_{1}(p, l). If m is a proper multiple of l, view G(p, l) as a subgroup of G(p, m), and the induced representation V coming from the above representation is an m-dimensional faithful representation of G(p, m) over O.

It is easy to check that the determinant of the action of an element of order p of G_{1}(p, m) on V is equal to 1, and that V ⊗ F restricts to a cyclic subgroup of G_{1}(p, m) of order m as a sum of l/(m, l) copies of the regular representation (use the normal basis theorem). Thus if either m is odd or l/(m, l) is even, the determinant of an element of order m acting on V is 1, and so G_{1}(p, m) is contained in SL(V) ≅ SL_{n}(O). In any case, the determinant of the action of G_{1}(p, m) on V has image contained in {±1} ⊆ O^{×}, so the action of G_{1}(p, m) on V ⊕ Λ^{n}(V) gives an embedding from G_{1}(p, m) into SL_{n+1}(O).

If m is even and l/(m, l) is odd, then O contains an mth root of −1 if and only if O contains an nth root of −1. In this case, if μ is an nth root of −1 in O, and A of order p and B of order m with det(B) = −1 generate a subgroup of GL_{n}(O) isomorphic to G_{1}(p, m), then A and μB generate a subgroup of SL_{n}(O) isomorphic to G_{2}(p, m).

Now consider the extraspecial group E(p, m) for p odd. Note that the centre Z of E(p, m) is cyclic of order p, and that E(p, m) has a subgroup of index p^{m} containing Z as a direct factor. This subgroup has an l-dimensional Orepresentation which is faithful on Z, and the corresponding induced E(p, m)- module is an lp^{m}-dimensional representation which must be faithful (since any nontrivial normal subgroup of a p-subgroup meets the centre nontrivially). Over C, E(p, m) has (p−1) faithful irreducible representations, each of dimension p^{m} (arising as induced modules in the above way for different choices of 1-dimensional modules for Z). Using characters it is easy to see that these representations restrict to any non-central subgroup of order p as a sum of copies of the regular representation and to the centre as a sum of copies of a single irreducible representation, and hence that they have image in SL_{p^{m}}(C) (note that E(p, m) has exponent p). The representations over O constructed above split over C into a sum of some of the faithful irreducibles, so have determinant 1.

For p = 2, there is a unique faithful irreducible complex representation of E(2, m), which has dimension 2^{m}, and an argument similar to the above shows that it is realisable over Z. Using characters one can show that the restriction of this representation to any cyclic subgroup of order four lies in SL_{2^{m}}(Z), and that the restriction to any non-central subgroup of order two is isomorphic to a sum of 2^{m−1} copies of the regular representation, so lies in SL_{2^{m}}(Z) provided that m ≥ 2.

The left action of the quaternion group Q_{8} on Z[i, j, k] commutes with the right action of Z[i], giving a faithful representation of Q_{8} which has image in SL_{2}(Z[i]).

Proof of Theorem 2. Corollary 7 gives an upper bound for p^{∘}(GL_{n}(O)). For n ≤ p − 1, for each m ≤ n such that l divides m and m divides p − 1, Lemma 9 tells us that G_{1}(p, m) occurs as a subgroup of GL_{n}(O), and so by Proposition 1 and Lemma 8, p^{∘}(GL_{n}(O)) is divisible by p^{∘}(G_{1}(m, p)) = 2m. This gives the bound for n ≤ p − 1, and shows that for n ≥ p, p^{∘}(GL_{n}(O)) is divisible by 2(p − 1). Now (for any p) the group E(p, m) is a subgroup of GL_{n}(O) for each n ≥ 2lp^{m}, and has Yagita invariant 2p^{m}. This gives the p-part of the bound for n ≥ p.

Proof of Theorem 3. This is similar to the proof of Theorem 2. In the case when n ≤ p − 1, the l.c.m. occuring in the expression given for p^{∘}(GL_{n}(O)) is clearly equal to the following expression.

l.c.m.{m : m ≤ n, m = lq^{r} for some prime q, q^{r}|(p − 1)/l}

In other words, we need only consider those m of the form lq^{r} for some prime q such that q^{r} divides (p−1)/l. If q is an odd prime and q divides l exactly s times, then G_{1}(p, q^{r+s}) is a subgroup of SL_{n}(O) for n = lq^{r}, and has Yagita invariant 2q^{r+s}. If 2 divides l exactly s times, and 2^{r}l divides p − 1, then G_{1}(p, 2^{r+s}) is a subgroup of SL_{n+1}(O) for n = l2^{r} and has Yagita invariant 2^{1+r+s}. From these examples it already follows that p^{∘}(SL_{n}(O)) is divisible by 2(p − 1) for n ≥ p, and that for n ≤ p − 1, p^{∘}(SL_{n}(O)) is equal to p^{∘}(GL_{n}(O)) except possibly if n is of the form 2^{r}l and is a factor of p − 1, when the Yagita invariant for SL_{n}(O) might be half the Yagita invariant for GL_{n}(O). If O contains a 2^{r+s}th root of −1, or equivalently an nth root of −1 (where n = l2^{r}), then G_{2}(p, 2^{r+s}) is a subgroup of SL_{n}(O), and has Yagita invariant 2^{1+r+s}, so that in this case too p^{∘}(SL_{n}(O)) = p^{∘}(GL_{n}(O)).

Proof of Theorem 5. The groups GL_{n}(R) and SL_{n}(R) are subgroups of GL_{n}(C), so their Yagita invariants are bounded above by p^{∘}(GL_{n}(C)). By the hypothesis on R, the cyclic group C_{p} admits a faithful representation in GL_{1}(R). As in Lemma 9 one may use induced representations of the groups G_{i}(p, m) and E(p, m) to give lower bounds equal to the above upper bounds. We leave the details as an exercise.

Remark. The methods that we use also gives some information concerning the Yagita invariant of the groups G(O) for other algebraic groups G. We hope to address this question in a future publication.

Acknowledgements. The work of the first named author was partially funded by the Centre de Recerca Matemàtica, and the work of the second named author was funded by a DGICYT research fellowship at the Centre de Recerca Matemàtica.

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H. H. Glover, Ohio State University, Columbus, Ohio 43210.

I. J. Leary, Centre de Recerca Matemàtica, Institut d’Estudis Catalans, E-08193, Bellaterra.

C. B. Thomas, DPMMS, University of Cambridge, Cambridge CB2 1SB.