The Yagita invariant of symplectic groups of large rank

Cornelia M. Busch^{∗}  Ian J. Leary^{†}

March 26, 2018

Abstract

Fix a prime p, and let O denote a subring of ℂ that is either integrally closed or contains a primitive pth root of 1. We determine the Yagita invariant at the prime p for the symplectic group Sp(2n, O) for all n ≥ p − 1.

1 Introduction

The Yagita invariant p^{∘}(G) of a discrete group G is an invariant that generalizes the period of the p-local Tate-Farrell cohomology of G, in the following sense: it is a numerical invariant defined for any G that is equal to the period when the p-local cohomology of G is periodic. Yagita considered finite groups [6], and Thomas extended the definition to groups of finite vcd [5]. In [3] the definition was extended to arbitrary groups and p^{∘}(G) was computed for G = GL(n, O) for O any integrally closed subring of ℂ and for sufficiently large n (depending on O).

In [2], one of us computed the Yagita invariant for Sp(2(p+1), ℤ). Computations from [3] were used to provide an upper bound and computations with finite subgroups and with mapping class groups were used to provide a lower bound [4]. The action of the mapping class group of a surface upon the first homology of the surface gives a natural symplectic representation of the mapping class group of a genus p + 1-surface inside Sp(2(p + 1), ℤ). In the current paper, we compute p^{∘}(Sp(2n, O)) for each n ≥ p − 1 for each O for which p^{∘}(GL(n, O)) was computed in [3]. By using a greater range of finite subgroups we avoid having to consider mapping class groups.

Throughout the paper, we fix a prime p. Before stating our main result we recall the definitions of the symplectic group Sp(2n, R) over a ring R, and of the Yagita invariant p^{∘}(G), which depends on the prime p as well as on the group G. The group Sp(2n, R) is the collection of invertible 2n × 2n matrices M over R such that

∗The first author acknowledges support from ETH Zürich, which facilitated this work

†The second author would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme Non-positive Curvature, Group Actions and Cohomology, when work on this paper was undertaken. This work was supported by EPSRC grant no. EP/K032208/1 and by a grant from the Leverhulme Trust.

M^{T}JM = J, where J := (\begin{matrix}0 & I_{n} \\ −I_{n} & 0 \end{matrix}) .

Here M^{T} denotes the transpose of the matrix M, and as usual I_{n} denotes the n × n identity matrix. Equivalently M ∈ Sp(2n, R) if M defines an isometry of the antisymmetric bilinear form on R^{2n} defined by ⟨x, y⟩ := x^{T}Jy. If C is cyclic of order p, then the group cohomology ring H^{∗}(C; ℤ) has the form

H^{∗}(C; ℤ) ≅ ℤ[x]/(px), x ∈ H^{2}(C; ℤ).

If C is a cyclic subgroup of G of order p, define n(C) a positive integer or infinity to be the supremum of the integers n such that the image of H^{∗}(G; ℤ) → H^{∗}(C; ℤ) is contained in the subring ℤ[x^{n}]. Now define

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

It is easy to see that if H ≤ G then p^{∘}(H) divides p^{∘}(G) [3, Prop. 1].

In the following theorem statement and throughout the paper we let ζ_{p} be a primitive pth root of 1 in ℂ and we let O denote a subring of ℂ with F ⊆ ℂ as its field of fractions. We assume that either ζ_{p} ∈ O or that O is integrally closed in ℂ. We define l := |F [ζ_{p}] : F |, the degree of F [ζ_{p}] as an extension of F. For t ∈ ℝ with t ≥ 1, we define ψ(t) to be the largest integer power of p less than or equal to t.

Theorem 1. With notation as above, for each n ≥ p−1, the Yagita invariant p^{∘}(Sp(2n, O)) is equal to 2(p − 1)ψ(2n/l) for l even and equal to 2(p − 1)ψ(n/l) for l odd.

By the main result of [3], the above is equivalent to the statement that p^{∘}(Sp(2n, O)) = p^{∘}(GL(2n, O)) when l is even and p^{∘}(Sp(2n, O)) = p^{∘}(GL(n, O)) when l is odd. By definition Sp(2n, O) is a subgroup of GL(2n, O) and there is an inclusion GL(n, O) → Sp(2n, O) defined by

A ↦ (\begin{matrix}A & 0 \\ 0 & (A^{T})^{−1} \end{matrix}) ,

and so for any n, p^{∘}(GL(n, O)) divides p^{∘}(Sp(2n, O)), which in turn divides p^{∘}(GL(2n, O)).

Lemma 2. If M is in the symplectic group, then M is conjugate to (M^{−1})^{T} = (M^{T})^{−1}, and det(M) = 1.

Proof. The matrix J defining the symplectic form satisfies J^{4} = I, and so in particular it is invertible. The equation M^{T}JM = J implies the equation JMJ^{−1} = (M^{T})^{−1}.

The usual way to show that the determinant of M is equal to 1 is via the Pfaffian. The Pfaffian is a function A ↦ pf(A) on the set of skew-symmetric matrices, which is polynomial in the matrix coefficients and is a square root of the determinant, i.e., pf(A)^{2} = det(A) for each skew-symmetric matrix A. Given these properties, it is easy to verify that the identity pf(M^{T}AM) = det(M)pf(A) holds for all matrices M and all skew-symmetric matrices A. Since J is invertible, pf(J) ≠ 0, and if M is symplectic, the equations

pf(J) = pf(M^{T}JM) = det(M)pf(J)

imply that det(M) = 1.

Proposition 3. 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 an integer n so that f(X) = g(X^{n}), then n has the form n = mp^{q} for some m dividing p − 1 and some positive integer q. If p is odd and for each i ∈ 𝔽_{p}^{×}, the multiplicity of i as a root of f is equal to that of −i, then m is even.

Proof. The only part of this that is not contained in [3, Prop. 6] is the final statement. Since (1 − iX)(1 + iX) = 1 − i^{2}X^{2} is a polynomial in X^{2}, the final statement follows. For the benefit of the reader, we sketch the rest of the proof. If n = mp^{q} where p does not divide m, then g(X^{n}) = g(X^{m})^{p^{q}}, so we may assume that q = 0. If g(Y) = 0 has roots y_{i}, then the roots of g(X^{m}) = 0 are the roots of y_{i} − X^{m} = 0. Since p does not divide m, these polynomials have no repeated roots; since their roots are assumed to lie in 𝔽_{p} it is now easy to show that m divides p − 1.

Corollary 4. With notation as in Theorem 1, let G be a subgroup of Sp(2n, F). Then the Yagita invariant p^{∘}(G) divides the number given for p^{∘}(Sp(2n, O)) in the statement of Theorem 1.

Proof. As in [3, Cor. 7], for each C ≤ G of order p, we use the total Chern class to give an upper bound for the number n(C) occuring in the definition of p^{∘}(G). If C is cyclic of order p, then C has p distinct irreducible complex representations, each 1-dimensional. If we write H^{∗}(C; ℤ) = ℤ[x]/(px), then the total Chern classes of these representations are 1 + ix for each i ∈ 𝔽_{p}, where i = 0 corresponds to the trivial representation. The total Chern class of a direct sum of representations is the product of the total Chern classes, and so when viewed as a polynomial in 𝔽_{p}[x] = H^{∗}(C; ℤ)⊗𝔽_{p}, the total Chern class of any faithful representation ρ : C → GL(2n, ℂ) is a non-constant polynomial of degree at most 2n all of whose roots lie in 𝔽_{p}^{×}. Now let F be a subfield of ℂ with l = |F [ζ_{p}] : F | as in the statement. The group C has (p−1)/l non-trivial irreducible representations over F, each of dimension l, and 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}. In particular, the total Chern class of any representation ρ : C → GL(2n, F) ≤ GL(2n, ℂ) is a polynomial in x^{l} whose x-degree is at most 2n. If ρ has image contained in Sp(2n, ℂ), then it factors as ρ = ι ∘ ρ̃ with ρ̃: C → Sp(2n, ℂ) and ι is the inclusion of Sp(2n, ℂ) in GL(2n, ℂ). In this case the matrix representing a generator for C is conjugate to the transpose of its own inverse; in particular it follows that the multiplicities of the irreducible complex representations of C with total Chern classes 1 + ix and 1 − ix must be equal for each i. Hence in this case, if p is odd, the total Chern class of the representation ρ = ι ∘ ρ̃ is a polynomial in x^{2}. If p = 2 (which implies that l = 1) then the total Chern class of any representation ρ : C → GL(2n, ℂ) has the form (1 + x)^{i}, where i is equal to the number of non-trivial irreducible summands. Since Sp(2n, ℂ) ≤ SL(2n, ℂ) it follows that for symplectic representations i must be even, and so for p = 2, the total Chern class is a polynomial in x^{2}.

In summary, let ρ̃ be a faithful representation of C in Sp(2n, F). In the case when l is odd, then the total Chern class of ρ̃ is a non-constant polynomial f̃(y) = f(x) in y = x^{2l} such that f(x) has degree at most 2n, f̃(y) has degree at most n/l, and all roots of f, f̃ lie in 𝔽_{p}^{×}. In the case when l is even, the total Chern class of ρ is a non-constant polynomial f̃(y) = f(x) in y = x^{l} such that f(x) has degree at most 2n, f̃(y) has degree at most 2n/l, and all roots of both lie in 𝔽_{p}^{×}. By Proposition 3, it follows that each n(C) is a factor of the number given for p^{∘}(Sp(2n, O)), and hence the claim.

Lemma 5. Let H ≤ G with |G : H| = m, and let ρ be a symplectic representation of H on V = O^{2n}. The induced representation Ind_{H}^{G}(ρ) is a symplectic representation of G on W := OG ⊗_{OH} V ≅ O^{2mn}.

Proof. Let e_{1}, …, e_{n}, f_{1}, …, f_{n} be the standard basis for V = O^{2n}, so that the bilinear form ⟨v, w⟩ := v^{T}Jw on V is given by

⟨e_{i}, e_{j}⟩ = 0 = ⟨f_{i}, f_{j}⟩, ⟨e_{i}, f_{j}⟩ = −⟨f_{i}, e_{j}⟩ = δ_{ij}.

The representation ρ is symplectic if and only if each ρ(h) preserves this bilinear form.

Let t_{1}, …, t_{m} be a left transversal to H in G, so that OG = ⊕_{i=1}^{m}t_{i}OH as right OHmodules. Define a bilinear form ⟨ , ⟩_{W} on W by

⟨∑_{i=1}^{m} t_{i} ⊗ v^{i}, ∑_{i=1}^{m} t_{i} ⊗ w^{i}⟩_{W} := ∑_{i=1}^{m}⟨v^{i}, w^{i}⟩.

To see that this bilinear form is preserved by the OG-action on W, fix g ∈ G and define a permutation π of {1, …, m} and elements h_{1}, …, h_{m} ∈ H by the equations gt_{i} = t_{π(i)}h_{i}. Now for each i, j with 1 ≤ i, j ≤ m

⟨Ind(ρ(g))t_{i} ⊗ v, Ind(ρ(g))t_{j} ⊗ w⟩_{W} = ⟨t_{π(i)} ⊗ ρ(h_{i})v, t_{π(j)} ⊗ ρ(h_{j})w⟩_{W}
= δ_{π(i)π(j)}⟨ρ(h_{i})v, ρ(h_{j})w⟩
= δ_{ij}⟨ρ(h_{i})v, ρ(h_{i})w⟩
= δ_{ij}⟨v, w⟩
= ⟨t_{i} ⊗ v, t_{j} ⊗ w⟩_{W} .

To see that ⟨ , ⟩_{W} is symplectic, define basis elements E_{1}, …, E_{mn}, F_{1}, …, F_{mn} for W by the equations

E_{n(i−1)+j} := t_{i} ⊗ e_{j}, and F_{n(i−1)+j} := t_{i} ⊗ f_{j}, for 1 ≤ i ≤ m, 1 ≤ j ≤ n.

It is easily checked that for 1 ≤ i, j ≤ mn

⟨E_{i}, E_{j}⟩_{W} = 0 = ⟨F_{i}, F_{j}⟩_{W} , ⟨E_{i}, F_{j}⟩_{W} = −⟨F_{i}, E_{j}⟩_{W} = δ_{ij},

and so with respect to this basis for W, the bilinear form ⟨ , ⟩_{W} is the standard symplectic form.

Proposition 6. With notation as in Theorem 1, the Yagita invariant p^{∘}(Sp(2n, O)) is divisible by the number given in the statement of Theorem 1.

Proof. To give lower bounds for p^{∘}(Sp(2n, O)) we use finite subgroups. Firstly, consider the semidirect product H = C_{p}⋊C_{p−1}, where C_{p−1} acts faithfully on C_{p}; equivalently this is the group of affine transformations of the line over 𝔽_{p}. It is well known that the image of H^{∗}(G; ℤ) inside H^{∗}(C_{p}; ℤ) ≅ ℤ[x]/(px) is the subring generated by x^{p−1}. It follows that 2(p − 1) divides p^{∘}(G) for any G containing H as a subgroup. The group H has a faithful permutation action on p points, and hence a faithful representation in GL(p − 1, ℤ), where ℤ^{p−1} is identified with the kernel of the H-equivariant map ℤ{1, …, p} → ℤ. Since GL(p − 1, ℤ) embeds in Sp(2(p − 1), ℤ) we deduce that H embeds in Sp(2n, O) for each O and for each n ≥ p − 1.

To give a lower bound for the p-part of p^{∘}(Sp(2n, O)) we use the extraspecial p-groups. For p odd, let E(p, 1) be the non-abelian p-group of order p^{3} and exponent p, and let E(2, 1) be the dihedral group of order 8. (Equivalently in each case E(p, 1) is the Sylow p-subgroup of GL(3, 𝔽_{p}).) For m ≥ 2, let E(p, m) denote 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}. Yagita showed that p^{∘}(E(p, m)) = 2p^{m} for each m and p [6]. The centre and commutator subgroup of E(p, m) are equal and have order p, and the abelianization of E(p, m) is isomorphic to C_{p}^{2m}. The irreducible complex representations of E(p, m) are well understood: there are p^{2m} distinct 1-dimensional irreducibles, each of which restricts to the centre as the trivial representation, and there are p − 1 faithful representations of dimension p^{m}, each of which restricts to the centre as the sum of p^{m} copies of a single (non-trivial) irreducible representation of C_{p}. The group G = E(p, m) contains a subgroup H isomorphic to C_{p}^{m+1}, and each of its faithful p^{m}- dimensional representations can be obtained by inducing up a 1-dimensional representation H → C_{p} → GL(1, ℂ).

According to Bürgisser, C_{p} embeds in Sp(2l, O) (resp. in Sp(l, O) when l is even) provided that O is integrally closed in ℂ [1]. Here as usual, l := |F [ζ_{p}], F | and F is the field of fractions of O. If instead ζ_{p} ∈ O, then l = 1 and clearly C_{p} embeds in GL(1, O) and hence also in Sp(2, O) = Sp(2l, O). Taking this embedding of C_{p} and composing it with any homomorphism H → C_{p} we get a symplectic representation ρ of H on O^{2l} for any l (resp. on O^{l} for l even). For a suitable homomorphism we know that Ind_{H}^{G}(ρ) is a faithful representation of G on O^{2lp^{m}} (resp. on O^{lp^{m}} for l even) and by Lemma 5 we see that Ind_{H}^{G}(ρ) is symplectic. Hence we see that E(m, p) embeds as a subgroup of Sp(2lp^{m}, O) for any l and as a subgroup of Sp(lp^{m}, O) in the case when l is even. Since p^{∘}(E(m, p)) = 2p^{m}, this shows that 2p^{m} divides p^{∘}(Sp(2lp^{m}, O)) always and that 2p^{m} divides p^{∘}(Sp(lp^{m}, O)) in the case when l is even.

Corollary 4 and Proposition 6 together complete the proof of Theorem 1.

We finish by pointing out that we have not computed p^{∘}(Sp(2n, O)) for general O when n < p−1; to do this one would have to know which metacyclic groups C_{p}⋊C_{k} with k coprime to p admit low-dimensional symplectic representations.

References

[1] B. Bürgisser, Elements of finite order in symplectic groups. Arch. Math. (Basel) 39 (1982), no. 6, 501–509.

[2] C. M. Busch, The Yagita invariant of some symplectic groups Mediterr. J. Math. 10 (2013), no. 1, 137–146.

[3] H. H. Glover, I. J. Leary, and C. B. Thomas, The Yagita Invariant of general linear groups. Algebraic topology: new trends in localization and periodicity (Sant Feliu de Guíxols, 1994), Progr. Math. 136 (1996), 185–192.

[4] H. H. Glover, G. Mislin, and Y. Xia, On the Yagita invariant of mapping class groups. Topology 33 (1994), no. 3, 557–574.

[5] C. B. Thomas, Free actions by p-groups on products of spheres and Yagita’s invariant po(G). Transformation groups (Osaka, 1987), Lecture Notes in Math. 1375 (1989), 326–338.

[6] N. Yagita, On the dimension of spheres whose product admits a free action by a nonabelian group. Quart. J. Math. Oxford Ser. (2) 36 (1985), no. 141, 117–127.

Authors’ addresses:

cornelia.busch@math.ethz.ch

Department of Mathematics ETH Zürich Rämistrasse 101 8092 Zürich Switzerland

i.j.leary@soton.ac.uk

School of Mathematical Sciences, University of Southampton, Southampton, SO17 1BJ United Kingdom