The Yagita invariant of symplectic groups of large rank
arXiv:1803.09561v1 [] 26 Mar 2018

Cornelia M. Busch∗

Ian J. Leary†

March 28, 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.

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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.
†

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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


0 In
T
.
M JM = J, where J :=
−In 0
Here M T denotes the transpose of the matrix M, and as usual In denotes the n × n identity
matrix. Equivalently M ∈ Sp(2n, R) if M defines an isometry of the antisymmetric bilinear
form on R2n defined by hx, yi := xT 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[xn ]. 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 ζp 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 ζp ∈ O or that O is integrally closed in C. We define l := |F [ζp ] : F |,
the degree of F [ζp ] 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
0
,
A 7→
0 (AT )−1
◦

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 7→ 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
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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) 6= 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 Fp , all of whose roots lie in F×
p . If
n
there is a polynomial g and an integer n so that f (X) = g(X ), then n has the form n = mpq
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 − i2 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 = mpq where p does not divide
q
m, then g(X n ) = g(X m )p , so we may assume that q = 0. If g(Y ) = 0 has roots yi , then
the roots of g(X m) = 0 are the roots of yi − X m = 0. Since p does not divide m, these
polynomials have no repeated roots; since their roots are assumed to lie in Fp 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 ∈ Fp , 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 Fp [x] = H ∗ (C; Z) ⊗Fp , 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 [ζp ] : 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 − ixl , where i ranges over the (p − 1)/l
distinct lth roots of unity in Fp . In particular, the total Chern class of any representation
ρ : C → GL(2n, F ) ≤ GL(2n, C) is a polynomial in xl whose x-degree is at most 2n. If ρ has
image contained in Sp(2n, C), then it factors as ρ = ι ◦ ρe with ρe : 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 ◦ ρe is a polynomial in x2 . 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
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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 x2 .
In summary, let ρe be a faithful representation of C in Sp(2n, F ). In the case when l is
odd, then the total Chern class of ρe is a non-constant polynomial f˜(y) = f (x) in y = x2l
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 = xl 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.
Lemma 5. Let H ≤ G with |G : H| = m, and let ρ be a symplectic representation of H
on V = O2n . The induced representation IndG
H (ρ) is a symplectic representation of G on
2mn
∼
W := OG ⊗OH V = O .
Proof. Let e1 , . . . , en , f1 , . . . , fn be the standard basis for V = O2n , so that the bilinear form
hv, wi := v T Jw on V is given by
hei , ej i = 0 = hfi , fj i, hei , fj i = −hfi , ej i = δij .
The representation ρ is symplectic if and only if each ρ(h) preserves this bilinear form.
Let t1 , . . . , tm be a left transversal to H in G, so that OG = ⊕m
i=1 ti OH as right OHmodules. Define a bilinear form h , iW on W by
+
* m
m
m
X
X
X
i
i
hv i , w i i.
:=
ti ⊗ w
ti ⊗ v ,
i=1

i=1

W

i=1

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 h1 , . . . , hm ∈ H by the equations gti = tπ(i) hi .
Now for each i, j with 1 ≤ i, j ≤ m
hInd(ρ(g))ti ⊗ v, Ind(ρ(g))tj ⊗ wiW = tπ(i) ⊗ ρ(hi )v, tπ(j) ⊗ ρ(hj )w

W

= δπ(i)π(j) hρ(hi )v, ρ(hj )wi
= δij hρ(hi )v, ρ(hi )wi
= δij hv, wi
= hti ⊗ v, tj ⊗ wiW .
To see that h , iW is symplectic, define basis elements
equations
E n(i−1)+j

:= ti ⊗ ej , and

F n(i−1)+j

:= ti ⊗ fj ,

E 1 , . . . , E mn , F 1 , . . . , F mn

for W by the

for 1 ≤ i ≤ m, 1 ≤ j ≤ n.

It is easily checked that for 1 ≤ i, j ≤ mn
hE i , E j iW = 0 = hF i , F j iW , hE i , F j iW = −hF i , E j iW = δij ,
and so with respect to this basis for W , the bilinear form h , iW is the standard symplectic
form.
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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 = Cp ⋊Cp−1 , where Cp−1 acts faithfully on Cp ; equivalently this
is the group of affine transformations of the line over Fp . It is well known that the image
of H ∗ (G; Z) inside H ∗ (Cp ; Z) ∼
= Z[x]/(px) is the subring generated by xp−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 Zp−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 p3 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, Fp ).) 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 p2m+1 . Yagita showed that
p◦ (E(p, m)) = 2pm 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 Cp2m . The
irreducible complex representations of E(p, m) are well understood: there are p2m distinct
1-dimensional irreducibles, each of which restricts to the centre as the trivial representation,
and there are p − 1 faithful representations of dimension pm , each of which restricts to the
centre as the sum of pm copies of a single (non-trivial) irreducible representation of Cp . The
group G = E(p, m) contains a subgroup H isomorphic to Cpm+1 , and each of its faithful pm dimensional representations can be obtained by inducing up a 1-dimensional representation
H → Cp → GL(1, C).
According to Bürgisser, Cp 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 [ζp], F | and F is the field of
fractions of O. If instead ζp ∈ O, then l = 1 and clearly Cp embeds in GL(1, O) and
hence also in Sp(2, O) = Sp(2l, O). Taking this embedding of Cp and composing it with
any homomorphism H → Cp we get a symplectic representation ρ of H on O2l for any l
(resp. on Ol for l even). For a suitable homomorphism we know that IndG
H (ρ) is a faithful
2lpm
lpm
representation of G on O
(resp. on O
for l even) and by Lemma 5 we see that IndG
H (ρ)
m
is symplectic. Hence we see that E(m, p) embeds as a subgroup of Sp(2lp , O) for any l
and as a subgroup of Sp(lpm , O) in the case when l is even. Since p◦ (E(m, p)) = 2pm , this
shows that 2pm divides p◦ (Sp(2lpm , O)) always and that 2pm divides p◦ (Sp(lpm , 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 Cp ⋊Ck with k coprime
to p admit low-dimensional symplectic representations.

5

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(2013), no. 1, 137–146.
[3] H. H. Glover, I. J. Leary, and C. B. Thomas, The Yagita Invariant of general linear
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Guı́xols, 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),
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[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

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