# The Yagita Invariant of Symplectic Groups of Large Rank

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## Abstract

Fix a prime *p*, and let \({\mathcal {O}}\) denote a subring of \({\mathbb {C}}\) that is either integrally closed or contains a primitive *p*th root of 1. We determine the Yagita invariant at the prime *p* for the symplectic group \({\mathrm{Sp}}(2n,{\mathcal {O}})\) for all \(n\ge p-1\).

## Mathematics Subject Classification

20J05 (primary) 57T10 (secondary)## 1 Introduction

The Yagita invariant \({p^\circ }(G)\) 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 \({p^\circ }(G)\) was computed for \(G={\mathrm{GL}}(n,{\mathcal {O}})\) for \({\mathcal {O}}\) any integrally closed subring of \({\mathbb {C}}\) and for sufficiently large *n* (depending on \({\mathcal {O}}\)).

In [2], one of us computed the Yagita invariant for \({\mathrm{Sp}}(2(p+1),{\mathbb {Z}})\). 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 \({\mathrm{Sp}}(2(p+1),{\mathbb {Z}})\). In the current paper, we compute \({p^\circ }({\mathrm{Sp}}(2n,{\mathcal {O}}))\) for each \(n\ge p-1\) for each \({\mathcal {O}}\) for which \({p^\circ }({\mathrm{GL}}(n,{\mathcal {O}}))\) was computed in [3]. By using a greater range of finite subgroups, we avoid having to consider mapping class groups.

*p*. Before stating our main result, we recall the definitions of the symplectic group \({\mathrm{Sp}}(2n,R)\) over a ring

*R*, and of the Yagita invariant \({p^\circ }(G)\), which depends on the prime

*p*as well as on the group

*G*. The group \({\mathrm{Sp}}(2n,R)\) is the collection of invertible \(2n\times 2n\) matrices

*M*over

*R*such that

*M*, and as usual \(I_n\) denotes the \(n\times n\) identity matrix. Equivalently, \(M\in {\mathrm{Sp}}(2n,R)\) if

*M*defines an isometry of the antisymmetric bilinear form on \(R^{2n}\) defined by \(\langle x,y\rangle :=x^\mathrm {T}Jy\). If

*C*is cyclic of order

*p*, then the group cohomology ring \(H^*(C;{\mathbb {Z}})\) has the form

*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 \(H^*(G;{\mathbb {Z}})\rightarrow H^*(C;{\mathbb {Z}})\) is contained in the subring \({\mathbb {Z}}[x^n]\). Now, define

## 2 Results

In the following theorem statement and throughout the paper, we let \(\zeta _p\) be a primitive *p*th root of 1 in \({\mathbb {C}}\) and we let \({\mathcal {O}}\) denote a subring of \({\mathbb {C}}\) with \(F\subseteq {\mathbb {C}}\) as its field of fractions. We assume that either \(\zeta _p\in {\mathcal {O}}\) or that \({\mathcal {O}}\) is integrally closed in \({\mathbb {C}}\). We define \(l:=|F[\zeta _p]:F|\), the degree of \(F[\zeta _p]\) as an extension of *F*. For \(t\in {\mathbb {R}}\) with \(t\ge 1\), we define \(\psi (t)\) to be the largest integer power of *p* less than or equal to *t*.

### Theorem 1

With notation as above, for each \(n\ge p-1\), the Yagita invariant \({p^\circ }({\mathrm{Sp}}(2n,{\mathcal {O}}))\) is equal to \(2(p-1)\psi (2n/l)\) for *l* even and equal to \(2(p-1)\psi (n/l)\) for *l* odd.

*l*is even and \({p^\circ }({\mathrm{Sp}}(2n,{\mathcal {O}}))={p^\circ }({\mathrm{GL}}(n,{\mathcal {O}}))\) when

*l*is odd. By definition \({\mathrm{Sp}}(2n,{\mathcal {O}})\) is a subgroup of \({\mathrm{GL}}(2n,{\mathcal {O}})\) and there is an inclusion \({\mathrm{GL}}(n,{\mathcal {O}})\rightarrow {\mathrm{Sp}}(2n,{\mathcal {O}})\) defined by

*n*, \({p^\circ }({\mathrm{GL}}(n,{\mathcal {O}}))\) divides \({p^\circ }({\mathrm{Sp}}(2n,{\mathcal {O}}))\), which in turn divides \({p^\circ }({\mathrm{GL}}(2n,{\mathcal {O}}))\).

Before we start, we recall two standard facts concerning symplectic matrices that will be used in the proof of Corollary 3: if *M* is in the symplectic group, then \(\det (M)=1\) and *M* is conjugate to the inverse of its transpose \((M^{-1})^\mathrm {T}=(M^\mathrm {T})^{-1}\). We shall use the notation \({\mathbb {F}}_p^\times \) to denote the multiplicative group of units in the field \({\mathbb {F}}_p\).

### Proposition 2

Let *f*(*X*) be a polynomial over the field \({\mathbb {F}}_p\) and suppose that 0 is not a root of *f*, but that *f* factors as a product of linear polynomials over \({\mathbb {F}}_p\). 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 integer \(q\ge 0\). If *p* is odd and for each \(i\in {\mathbb {F}}_p^\times \), 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 \((1-iX)(1+iX)=1-i^2X^2\) 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 \({\mathbb {F}}_p\) it is now easy to show that *m* divides \(p-1\). \(\square \)

### Corollary 3

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

### Proof

As in [3, Cor. 7], for each \(C\le G\) of order *p*, we use the total Chern class to give an upper bound for the number *n*(*C*) occurring in the definition of \({p^\circ }(G)\). If *C* is cyclic of order *p*, then *C* has *p* distinct irreducible complex representations, each one dimensional. If we write \(H^*(C;{\mathbb {Z}})={\mathbb {Z}}[x]/(px)\), then the total Chern classes of these representations are \(1+ix\) for each \(i\in {\mathbb {F}}_p\), 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 \({\mathbb {F}}_p[x]=H^*(C;{\mathbb {Z}})\otimes {\mathbb {F}}_p\), the total Chern class of any faithful representation \(\rho :C\rightarrow {\mathrm{GL}}(2n,{\mathbb {C}})\) is a non-constant polynomial of degree at most 2*n* all of whose roots lie in \({\mathbb {F}}_p^\times \). Now, let *F* be a subfield of \({\mathbb {C}}\) with \(l=|F[\zeta _p]:F|\) 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 *l*th roots of unity in \({\mathbb {F}}_p\). In particular, the total Chern class of any representation \(\rho :C\rightarrow {\mathrm{GL}}(2n,F)\le {\mathrm{GL}}(2n,{\mathbb {C}})\) is a polynomial in \(x^l\) whose *x*-degree is at most 2*n*. If \(\rho \) has image contained in \({\mathrm{Sp}}(2n,{\mathbb {C}})\), then it factors as \(\rho = \iota \circ {\widetilde{\rho }}\) with \({\widetilde{\rho }}:C\rightarrow {\mathrm{Sp}}(2n,{\mathbb {C}})\) and \(\iota \) is the inclusion of \({\mathrm{Sp}}(2n,{\mathbb {C}})\) in \({\mathrm{GL}}(2n,{\mathbb {C}})\). 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 \(\rho =\iota \circ {\widetilde{\rho }}\) is a polynomial in \(x^2\). If \(p=2\) (which implies that \(l=1\)), then the total Chern class of any representation \(\rho :C\rightarrow {\mathrm{GL}}(2n,{\mathbb {C}})\) has the form \((1+x)^i\), where *i* is equal to the number of non-trivial irreducible summands. Since \({\mathrm{Sp}}(2n,{\mathbb {C}})\le {\mathrm{SL}}(2n,{\mathbb {C}}),\) 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 \({\widetilde{\rho }}\) be a faithful representation of *C* in \({\mathrm{Sp}}(2n,F)\). In the case when *l* is odd, then the total Chern class of \({\widetilde{\rho }}\) is a non-constant polynomial \({\tilde{f(y)=f(x)}}\) in \(y=x^{2l}\) such that *f*(*x*) has degree at most 2*n*, \({\tilde{f}}(y)\) has degree at most *n* / *l*, and all roots of \(f,{\tilde{f}}\) lie in \({\mathbb {F}}_p^\times \). In the case when *l* is even, the total Chern class of \(\rho \) is a non-constant polynomial \({\tilde{f}}(y)=f(x)\) in \(y=x^l\) such that *f*(*x*) has degree at most 2*n*, \({\tilde{f}}(y)\) has degree at most 2*n* / *l*, and all roots of both lie in \({\mathbb {F}}_p^\times \). By Proposition 2, it follows that each *n*(*C*) is a factor of the number given for \({p^\circ }({\mathrm{Sp}}(2n,{\mathcal {O}}))\), and hence the claim. \(\square \)

### Lemma 4

Let \(H\le G\) with \(|G:H|=m\), and let \(\rho \) be a symplectic representation of *H* on \(V={\mathcal {O}}^{2n}\). The induced representation \(\mathrm{Ind}_H^G(\rho )\) is a symplectic representation of *G* on \(W:={\mathcal {O}}G\otimes _{{\mathcal {O}}H}V\cong {\mathcal {O}}^{2mn}\).

### Proof

*V*is given by

*H*in

*G*, so that \({\mathcal {O}}G= \oplus _{i=1}^m t_i{\mathcal {O}}H\) as right \({\mathcal {O}}H\)-modules. Define a bilinear form \(\langle \,\, ,\,\,\rangle _W\) on

*W*by

*W*, fix \(g\in G\) and define a permutation \(\pi \) of \(\{1,\ldots ,m\}\) and elements \(h_1,\ldots , h_m\in H\) by the equations \(gt_i=t_{\pi (i)}h_i\). Now for each

*i*,

*j*with \(1\le i,j\le m,\)

*W*by the equations

*W*, the bilinear form \(\langle \,\, ,\,\,\rangle _W\) is the standard symplectic form. \(\square \)

### Proposition 5

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

### Proof

To give lower bounds for \({p^\circ }({\mathrm{Sp}}(2n,{\mathcal {O}})),\) we use finite subgroups. Firstly, consider the semidirect product \(H=C_p {\rtimes }C_{p-1}\), where \(C_{p-1}\) acts faithfully on \(C_p\); equivalently, this is the group of affine transformations of the line over \({\mathbb {F}}_p\). It is well known that the image of \(H^*(G;{\mathbb {Z}})\) inside \(H^*(C_p;{\mathbb {Z}})\cong {\mathbb {Z}}[x]/(px)\) is the subring generated by \(x^{p-1}\). It follows that \(2(p-1)\) divides \({p^\circ }(G)\) 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 \({\mathrm{GL}}(p-1,{\mathbb {Z}})\), where \({\mathbb {Z}}^{p-1}\) is identified with the kernel of the *H*-equivariant map \({\mathbb {Z}}\{1,\ldots , p\}\rightarrow {\mathbb {Z}}\). Since \({\mathrm{GL}}(p-1,{\mathbb {Z}})\) embeds in \({\mathrm{Sp}}(2(p-1),{\mathbb {Z}}),\) we deduce that *H* embeds in \({\mathrm{Sp}}(2n,{\mathcal {O}})\) for each \({\mathcal {O}}\) and for each \(n\ge p-1\).

To give a lower bound for the *p*-part of \({p^\circ }({\mathrm{Sp}}(2n,{\mathcal {O}})),\) 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 \({\mathrm{GL}}(3,{\mathbb {F}}_p)\).) For \(m\ge 2\), 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 \({p^\circ }(E(p,m))=2p^m\) 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 one-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 one-dimensional representation \(H\rightarrow C_p\rightarrow {\mathrm{GL}}(1,{\mathbb {C}})\).

According to Bürgisser, \(C_p\) embeds in \({\mathrm{Sp}}(2l,{\mathcal {O}})\) (resp. in \({\mathrm{Sp}}(l,{\mathcal {O}})\) when *l* is even) provided that \({\mathcal {O}}\) is integrally closed in \({\mathbb {C}}\) [1]. Here as usual, \(l:=|F[\zeta _p],F|\) and *F* is the field of fractions of \({\mathcal {O}}\). If instead \(\zeta _p\in {\mathcal {O}}\), then \(l=1\) and clearly \(C_p\) embeds in \({\mathrm{GL}}(1,{\mathcal {O}})\) and hence also in \({\mathrm{Sp}}(2,{\mathcal {O}})={\mathrm{Sp}}(2l,{\mathcal {O}})\). Taking this embedding of \(C_p\) and composing it with any homomorphism \(H\rightarrow C_p,\) we get a symplectic representation \(\rho \) of *H* on \({\mathcal {O}}^{2l}\) for any *l* (resp. on \({\mathcal {O}}^l\) for *l* even). For a suitable homomorphism, we know that \(\mathrm{Ind}_H^G(\rho )\) is a faithful representation of *G* on \({\mathcal {O}}^{2lp^m}\) (resp. on \({\mathcal {O}}^{lp^m}\) for *l* even) and by Lemma 4 we see that \(\mathrm{Ind}_H^G(\rho )\) is symplectic. Hence, we see that *E*(*m*, *p*) embeds as a subgroup of \({\mathrm{Sp}}(2lp^m,{\mathcal {O}})\) for any *l* and as a subgroup of \({\mathrm{Sp}}(lp^m,{\mathcal {O}})\) in the case when *l* is even. Since \({p^\circ }(E(m,p))=2p^m\), this shows that \(2p^m\) divides \({p^\circ }({\mathrm{Sp}}(2lp^m,{\mathcal {O}}))\) always and that \(2p^m\) divides \({p^\circ }({\mathrm{Sp}}(lp^m,{\mathcal {O}}))\) in the case when *l* is even. \(\square \)

Corollary 3 and Proposition 5 together complete the proof of Theorem 1.

We finish by pointing out that we have not computed \({p^\circ }({\mathrm{Sp}}(2n,{\mathcal {O}}))\) for general \({\mathcal {O}}\) when \(n<p-1\); to do this one would have to know which metacyclic groups \(C_p{\rtimes }C_k\) with *k* coprime to *p* admit low-dimensional symplectic representations.

## Notes

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