The Yagita invariant of symplectic groups of large rank

Fix a prime $p$, and let $R$ be any subring of the complex numbers that is either integrally closed or contains a primitive $p$th root of 1. For each $n\geq p-1$ we compute the Yagita invariant at the prime $p$ for the symplectic group $Sp(2n,R)$.


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 * 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. 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 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 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; 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 and so for any n, Proof. The matrix J defining the symplectic form satisfies J 4 = I, and so in particular it is invertible. The equation 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 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 • ρ with ρ : 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 • ρ 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 ρ be a faithful representation of C in Sp(2n, F ). In the case when l is odd, then the total Chern class of ρ is a non-constant polynomialf (y) = f (x) in y = x 2l such that f (x) has degree at most 2n,f (y) has degree at most n/l, and all roots of f,f lie in F × p . In the case when l is even, the total Chern class of ρ is a non-constant polynomial f (y) = f (x) in y = x l such that f (x) has degree at most 2n,f (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.
The representation ρ is symplectic if and only if each ρ(h) preserves this bilinear form.
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 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 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 2m p . 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 m+1 p , and each of its faithful p mdimensional 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 G H (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 G H (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.