STABLE SPLITTINGS FOR CLASSIFYING SPACES
OF ALTERNATING, SPECIAL ORTHOGONAL
AND SPECIAL UNITARY GROUPS
by
Hans-Werner Henn and Huynh Mui
Abstract
Let G(n) denote either the symmetric group (n), the orthogonal group O(n) or*
* the
unitary group U(n) and let SG(n) denote either the alternating group A(n), t*
*he spe-
cial orthogonal group SO(n) or the special unitary group SU(n). The classif*
*ying
Wn
spaces BG(n) are known to split stably as BG(n) ' BG(`)=BG(` - 1): We con-
`=1
sider the case of BSG(n) and prove that, after localizing at any prime p, th*
*ere are
similar although somewhat coarser splittings. E.g. we get a stable 2-local*
* splitting
n-1W
BA(4n) ' BA(4n)=BA(4n - 2) _ BA(4` + 2)=BA(4` - 2) . A crucial ingredient*
* in
`=1
our proof is a careful study, for finite p- groups P, of the morphism sets m*
*orA(P; SG(n))
in the "Burnside category" A , and in particular the effect of transfers on *
*these sets.
Introduction
Stable splittings of classifying spaces have received a lot of attention in the*
* last
decade. A good introductory exposition is given in [P]. Perhaps the first known
Wn
splittings were those for the symmetric groups, B(n) ' B(`)=B(` - 1),
`=1
which can be obtained via a geometric version of Dold's proof [D] of Nakaoka's
[Na] decomposition theorem for the homology of symmetric groups (cf. [KP]).
(As usual here and in the following quotients like B(`)=B(` - 1) are to be
interpreted as cofibres of the maps B(` - 1) ! B(`) induced by the inclusion
(` - 1) ! (`).) In [MP] Mitchell and Priddy show how to extend these
_____________________________
1991 Mathematics Subject Classification Primary 55R35, Secondary 55P42, 55R12
This paper is in final form and no version of it will be submitted for publicat*
*ion elsewhere
1
2 HANS-WERNER HENN and HUYNH MUI
Wn
methods and they produce stable splittings BG(n) ' BG(`)=BG(` - 1) with
`=1
G(`) = O(`), U(`), Sp(`) or GL(`; Fq). In the last case one has to invert p if
q = pk.
In this paper we prove the following
THEOREM 0. For each natural number n there are p-local stable splittings of
the form
_n
BSG(n) '(p) BSG(`)=BSG(` - ffi(`))
`=1
in each of the following cases:
(0.1.) p = 2, SG(n)8= A(n), the alternating group on n letters, and
< 4 if ` 2 mod 4
ffi(`)= : 0 if ` 6 2 mod 4 and ` 6= n
k if ` 2 + k mod 4; k 2 {1; 2; 3} and ` = n
(0.2.) p > 2, SG(n)8= A(n), and
< p if ` 2 mod p
ffi(`)= : 0 if ` 6 2 mod p and ` 6= n,
k if ` 2 + k mod p; k 2 {1; 2; : :;:p - 1} and ` = n
(0.3.) p = 2, SG(n) = SO(n), the special orthogonal group of real
n x n-matrices,(and
2 if ` 1 mod 2 ,
ffi(`)= 0 if ` 0 mod 2 and ` 6= n,
1 if ` 0 mod 2 and ` = n
(0.4.) p any prime, SG(n) = SU(n), the special unitary group of complex
n x n-matrices, and
( 2 if ` 1 mod p ,
ffi(`)= 0 if ` 0 mod p and ` 6= n,
1 if ` 6 0; 1 mod p; or ` 0 mod p and ` = n
(By convention SG(m) will be the trivial group if m 0.)
EXAMPLES
n-1W
1) BA(4n) ' BA(4n)=BA(4n - 2) _ BA(4` + 2)=BA(4` - 2), after local-
`=1
ization at 2.
Wn
2) BSO(2n + 1) ' BSO(2` + 1)=BSO(2` - 1), after localization at 2. After
`=1
inverting 2, Snaith [Sn] and Mitchell and Priddy [MP] constructed also stab*
*le
Wn
splittings BSO(2n + 1) ' BSO(2` + 1)=BSO(2` - 1). Is there a global
`=1
splitting of this form?
STABLE SPLITTINGS FOR ALTERNATING GROUPS 3
REMARKS
a) The proof of Theorem 0 will actually show that the natural "filtration" of
BSG(n) by the BSG(`) with ffi(`) 6= 0 splits (stably and after localization)
and hence we may include the limiting case n = 1 in Theorem 0.
b) The splittings of Theorem 0 are coarser than those for (n), O(n) and U(n),
where ffi(`) was always 1. The following observations show that this is to *
*be
expected, at least for A(n) and SO(n).
The restriction maps H*(BA(4n + 2); F2) ! H*(BA(4n); F2) and
H*(BA(pn + 2); Fp) ! H*(BA(pn); Fp) are not epi if n 1 (cf. Remark
4.2.(b) below).
The restriction map H*(BSO(2n + 1); F2) ! H*(BSO(2n); F2) is not split
epi as a homomorphism of modules over the Steenrod algebra (for this con-
sider the action of Sq1 on the Stiefel-Whitney class w2n).
c) It can be shown (cf. [B]) that the inclusion A(n) ! A(n + 1) induces an
isomorphism in mod p cohomology if n 6 1; 3 mod 4 and p = 2, or n 6 1,
p - 1 mod p and p > 2. Then BA(n) and BA(n + 1) are stably homotopy
equivalent at p and hence there are other choices of ffi(`) which still des*
*cribe
the same splittings.
d) If n 6 0; 1 mod p, p > 2, then our splitting of BA(n) is an easy consequenc*
*e of
the splitting of B(n) and the fact that the inclusion A(n) ! (n) induces
a p-local stable equivalence in this case (cf. [B]).
The paper is organized as follows. In Section 1 we construct the splitting maps
Wn
h : BSG(n) ! BSG(`)=BSG(` - ffi(`)) by a modification of the construction
`=1
in the case of BG(n). Our proof of Theorem 0 then depends on studying the
composition of h with (suitable stable) maps BP ! BSG(n) where P runs
through the class of finite p-groups. The splitting map itself is a Section 2 we
will describe a convenient algebraic set up for transfer calculations (cf. [AGM*
*],
[Ma], [Ni] and [HLS]) and state an algebraic analogue (Theorem 2.7) of Theorem
0 in this set up. In Section 3 we show how this leads to a proof of Theorem 0
and in Section 4 we give a proof of Theorem 2.7.
The authors would like to thank the Forschungsschwerpunkt Geometrie at the
University of Heidelberg for enabling them to work together. The second author
is particularly grateful to Prof. D.Puppe for making his visit to Heidelberg (in
late 1988) possible.
A preliminary version of this work was available in preprint form in mid 1989.
Its final publication has been delayed by the difficulties the authors experien*
*ced
in trying to exchange information since then.
4 HANS-WERNER HENN and HUYNH MUI
1. Construction of the splitting maps
In the following all spaces and maps are to be considered in the stable category
of CW -spectra. We denote X with a disjoint base point by X+ . Stably X+ and
X_S0 are equivalent. Let (G(n); SG(n)) be either ((n); A(n)) or (O(n); SO(n))
or (U(n); SU(n)), and let in denote the canonical inclusion SG(n) ! G(n) and
Bin the induced map BSG(n)+ ! BG(n)+ .
Let det : G(n) ! G(ffl) denote either the signum or the determinant map, i.e.
ffl = 2 in case G(n) = (n) and ffl = 1 in the other two cases. Then there is
another canonical inclusion ei: G(n) x G(ffl) ! G(n + ffl) given by juxtapositi*
*on
and the homomorphism
(id;det-1) ei
G(n)---------! G(n) x G(ffl) -! G(n + ffl)
has image in SG(n+ffl). We denote the resulting homomorphism G(n) ! SG(n+
ffl) and the corresponding stable map by dn+fflresp. Bdn+ffl.
Furthermore we write trn;m for the transfer belonging to the inclusion G(n) x
G(m) ! G(n + m) given again by juxtaposition; trn;m is a stable map BG(n +
m)+ ! (BG(n) x BG(m))+ . Finally, the projection map G(n) x G(m) ! G(n)
resp. (BG(n) x BG(m))+ ! BG(n)+ will be denoted by ssn;m resp. Bssn;m or
simply by ss resp. Bss.
We will now define the components of a splitting map
_n
h : BSG(n) ! BSG(`)=BSG(` - ffi(`))
`=1
(with ffi(`) as in Theorem 0) as the composition of a map hn;`: BSG(n)+ !
BSG(`)+ with the inclusion BSG(n) ! BSG(n)+ and the canonical projection
q` of BSG(`)+ to the quotient BSG(`)+ =BSG(` - ffi(`))+ = BSG(`)=BSG(` -
ffi(`)). The map hn;`is defined as
_ identity if n = `,
_ as the following composition if ffl < ` < n.
tr`-ffl;n+ffl-`
BSG(n)+ -Bin--!BG(n)+ ---------! (BG(` - ffl) x BG(n + ffl - `))+
-Bss--!BG(` - ffl)+ -Bd`--!BSG(`)+ :
_ any map if ` ffl. (Then SG(`) and the corresponding wedge summand are
trivial.)
It will be convenient to consider the map
n !
_
h+ : BSG(n)+ ! BSG(`)=BSG(` - ffi(`))_ S0
`=1
STABLE SPLITTINGS FOR ALTERNATING GROUPS 5
with components q`hn;`, 1 ` n, and BSG(n)+ ! B{e}+ ' S0, induced by
the trivial homomorphism SG(n) ! {e}. Clearly, h+ is a p-local equivalence if
and only if h is one.
2. A(P; G)
We will study h+ by composing it with maps BP + ! BSG(n)+ of the form
BP +--tr-!BQ+ -B'--!BSG(n)+ , where Q is a subgroup of P , tr = trQP the
transfer associated to the inclusion Q P and ' : Q ! SG(n) a group homo-
morphism.
In this section we will describe the appropriate algebraic set up for dealing w*
*ith
such compositions (cf. [AGM], [Ma], [Ni] and [HLS]). The crucial input is the
definition of a transfer homomorphism which is modelled after the double coset
formula.
2.1. For a fixed finite group P and a compact Lie group G let A(P; G) be the
free abelian group with basis the equivalence classes of pairs (Q; ') consistin*
*g of
a subgroup Q of P and a homomorphisms ' : Q ! G. Here the pairs (Q; ') and
(Q0; '0) are called equivalent if and only if the groups ' = {(q; 'q) | q 2 Q} *
*and
'0= {(q0; 'q0) | q02 Q0} are conjugate as subgroups of P xG. The equivalence
class of (Q; ') will be denoted by [Q; '] (or, more precisely, by [Q; '](P;G)).*
* It is
easy to see, say by a minor modification of the argument given in [Q2, Lemma
6.3], that the set of these equivalence classes is finite. Therefore, G 7! A(P;*
* G)
defines a functor from the category GcL of compact Lie groups to the category
Ab fof finitely generated free abelian groups. Furthermore, for a closed subgro*
*up
H of G, there is a transfer homomorphism o = oHG : A(P; G) ! A(P; H) which
is defined on a basis element [Q; '] as follows.
Q acts via ' on the homogeneous space G=H, and the double coset space Q\G=H
may be decomposed as a disjoint union of "orbit type manifold components" Mi
([F]). More precisely, if Q0 runs through a set of representatives of conjugacy
classes of subgroups of Q and if G=H is broken up into the disjoint union of the
subspaces (G=H)(Q0), consisting of those points in G=H whose Q-orbits are iso-
morphic to Q=Q0, then the Miare the connected components of the corresponding
orbit spaces Q\(G=H)(Q0)._Let O#_(Mi) be the_"internal_Euler characteristic" of
Mi, i.e. O# (Mi) = O(M i) - O(M i- Mi) with M idenoting the closure of Mi in
Q\G=H. Finally choose a representative gi2 G of any element in Miso that the
isotropy subgroup of giH is Qi. Then we define
X
(2:2:) oHG([Q; ']) = O# (Mi)[Qi; g-1i'gi]:
i
6 HANS-WERNER HENN and HUYNH MUI
(Note that g-1i'gi maps Qi into H !) It is easy to see that the right hand side
depends neither on the particular choice of gi (e.g. because the space of equiv*
*a-
lence classes is finite, in particular discrete) nor on the chosen representati*
*ve of
the equivalence class [Q; '].
We remark that for G a finite group A(P; Q) agrees with the set of morphisms
from P to G in the "Burnside category" as defined in [AGM] .
2.3. This formula for the transfer is fairly complicated. We will now focus
on the "leading term" of the transfer and show that it has a more manageable
description.
Subconjugation defines a partial order on the set of conjugacy classes (Q) of
subgroups of P . From this partial order we get a filtration of A(P; G) by defi*
*ning
F(Q)A(P;_G) as the subgroup generated by the classes [Q0; '] with (Q0) (Q).
We let F(Q)A(P; Q) denote the_quotient of F(Q)A(P; Q) by the subgroups of lower
filtration. Obviously G ! F (Q)A(P; G) is also a functor from_GcL to Ab fand
(2.2) shows that the transfer for A(P; ?) induces one for F(Q)A(P; ?) which will
be denoted by __o.
__
Next we will give a different description of F(Q). Let Rep(Q; G) denote the set
of G-conjugacy classes of homomorphisms from Q to G. The class of a homo-
morphism ' will be denoted by ['] (or more precisely by [']G ). The normalizer
NP (Q) of Q in P acts on Rep(Q; G) via conjugation. We denote the quotient of
Rep(Q; G) with respect to this action by RepP (Q; G).
Then the map Rep(Q; G) ! A(P;_G), ['] 7! [Q; '] induces a natural isomorphism
between Z[Rep P(Q; G)] and F (Q)A(P; G). (Here and in the following the free
abelian group on a set S will be denoted by Z[S]). Thus we get a transfer __oon
Z[Rep P(Q; ?)] which we will now describe.
For homomorphisms ' : Q ! G and : Q ! H let (G=H)' = {gH | [g-1'g]H =
[ ]H } consist of those cosets gH which conjugate [']G into [ ]H and let [' : ]
denote the Euler characteristic of (G=H)'. Define o = oHG : Z[Rep (Q; G)] !
Z[Rep (Q; H)] via
X
(2:4:) ['] 7-! [' : ][ ]:
[ ]2Rep(Q;H)
(cf. [HLS]). Clearly, oHG induces a homomorphism o0HG : Z[Rep P(Q; G)] !
Z[Rep P(Q; H)].
PROPOSITION 2.5. __oHGand o0HGagree.
Proof:_Consider __oHG. By (2.2) we only have to consider the contributions comi*
*ng
from the components of the fixed points (G=H)Q . These components are closed,
STABLE SPLITTINGS FOR ALTERNATING GROUPS 7
hence O# (Mi) and`O(Mi) agree. Now the proposition follows from the observation
that (G=H)' = Mi.
[g-1i'gi]=[ ]
___
|__|
The following Lemma will be useful in Section 4 for computing the coefficients
[' : ] in concrete cases. The proof is straightforward and left to the reader.
LEMMA 2.6. Suppose (G=H)' is not empty and let g0 2 G be any element
satisfying g-10'g0 = . Then the map
CG ( )=CG ( ) \ H-! (G=H)'
induced by g7-! g0gH
is a homeomorphism. (Here CG ( ) denotes the centralizer of the image of in
G, i.e. CG ( ) = {g 2 G | g (q) = (q)g for all q 2 Q}.)
___
|__|
We will finish this section by stating the algebraic analogue of Theorem 0. For
this let P be a finite p-group and p, SG(`) and ffi(`) be as in Theorem 0.
We will see in 4.1 that for such P the homomorphism A(P; SG(` - ffi(`)) !
A(P; SG(`)) induced by inclusion is mono. In Theorem 2.7 below we will identify
A(P; SG(` - ffi(`)) with its image in A(P; SG(`)).
As in Section 1 we introduce homomorphisms hn;`: A(P; SG(n)) ! A(P; SG(`)).
For ` = n we define hn;n = id. For 0 < ` ffl we let hn;`be trivial, and for
ffl < ` < n we define hn;`as composition d`sso`-ffl;n+ffl-`in where d`, ss and *
*in are
as in Section 1 and o`-ffl;n+ffl-`denotes the transfer oG(`-ffl)xG(n+ffl-`)G(n).
THEOREM 2.7. For each natural number n and each finite p-group P the ho-
momorphism
n !
M
h+ : A(P; SG(n)) ! A(P; SG(`))=A(P; SG(` - ffi(`))) A(P; {e}) ;
`=1
with components induced by hn;`respectively the trivial homomorphism P ! {e},
becomes an isomorphism after tensoring with Z=p.
The proof of Theorem 2.7 will be given in Section 4. In the following the tar-
get of h+ will be denoted by C(P; SG(n)), and the component A(P; SG(n)) !
A(P; {e}) = A(P; SG(0)) by hn;0.
8 HANS-WERNER HENN and HUYNH MUI
3. Proof of Theorem 0
The following folklore proposition gives the justification for Section 2. As th*
*ere
P will be a finite group, G a compact Lie group and H a closed subgroup of G.
PROPOSITION 3.1. There are homomorphisms = P;G from A(P; G) to the
group {BP +; BG+ } of homotopy classes of stable maps from BP + to BG+ ,
defined by P;G[Q; '] = B' O trQP: BP +! BQ+ ! BG+ .
These homomorphisms are natural in P and Q and commute with transfers, i.e.
trHGO P;G = P;H O oHG.
Proof:_Inner automorphisms induce selfmaps on BP +and BG+ which are homo-
topic to the identity maps; hence P;Q is well defined. Naturality is trivial and
compatibility with the transfer follows from the double coset formula [F]. (For
the existence of transfers BG+ ! BH+ in the compact Lie group case we refer
to [C].)
___
|__|
We remark that Lewis, May and McClure [LMM] have shown that P;Q induces
an isomorphism between a suitable completion of A(P; G) and {BP +; BG+ } if
G is finite. However, we will not need this in the sequel.
We begin with the proof of Theorem 0, assuming Theorem 2.7. For the remainder
of this section let p, SG(`) and ffi(`) be as in Theorem 0.
3.2. If P` is any finite p-group and '` : P` ! SG(`), 0 ` n, are any
homomorphisms (recall that SG(0) = {e} by convention), then Theorem 2.7
implies that there are elements x` 2 A(P`; SG(n)) and x0`2 C(P`; SG(n)) such
that ______
[P`; '`]= h+ (x`) + px0`;
______
where [P`; '`]denotes the class of [P`; '`] 2 A(P`; SG(`)) in C(P`; SG(n)). Then
Proposition 3.1 gives a diagram
+ nW
BSG(n)+ --h-! BSG(`)=BSG(` - ffi(`)) _ S0
`=1 x
nW ? nW
(3:3:) (x`) ?? q`OB'`
`=0 `=0
nW
BP`+
`=0
which commutes after passing to mod p-cohomology. Now Lemma 3.4 below im-
plies easily that h+ induces an isomorphism in mod p-cohomology and Theorem
0 follows.
STABLE SPLITTINGS FOR ALTERNATING GROUPS 9
LEMMA 3.4.
nW
a) The mod p cohomology of BSG(n)+ and BSG(`)=BSG(` - ffi(`)) _ S0
`=1
are of finite type and isomorphic as graded vector spaces.
Wn
b) There are finite p-groups P` and homomorphisms '` such that q`O B'`
`=0
induces a monomorphism in mod p-cohomology.
Proof:_We claim that q` : BSG(`) ! BSG(`)=BSG(`-ffi(`)) induces a monomor-
phism in reduced mod p-cohomology or, equivalently, that the map BSG(` -
ffi(`))--Bi-!BSG(`) with i denoting the inclusion SG(` - ffi(`)) ! SG(`) induces
an epimorphism in mod p-cohomology. This is clear for SO(`) and SU(`) and
will be proved below for A(`). Now part a) follows immediately.
For b) we let P` be either a p-Sylow subgroup of SG(`), if SG(`) = A(`), or
the maximal p-torus of SG(`), if SG(`) = SO(`) or SU(`), consisting of all
diagonal matrices of order p and determinant 1. For '` we take the inclusion of
P` into SG(`). Part b) follows from the well-known fact that these '` induce
monomorphisms in mod p-cohomology. ___
|__|
It remains to prove the following
LEMMA 3.5. For m ` the restriction map H*(BA(`); Fp) ! H*(BA(m); Fp)
is onto provided that m 6 0; 1 mod 4 if p = 2 resp. m 6 0; 1 mod p if p > 2.
Proof:_If we replace A(`) and A(m) by (`) and (m) then the restriction map
is onto for all m ` by Nakaoka [Na]. Therefore it suffices to show that the
restriction map H*(B(m); Fp) ! H*(BA(m); Fp) is onto if m 6 0; 1 mod 4
resp. m 6 0; 1 mod p.
For odd primes a p-Sylow subgroup of A(m) is also one of (m) and an easy
argument (essentially the same as in the proof of Lemma 4.1 below) with stable
elements [CE] shows that H*(B(m); Fp) ! H*(BA(m); Fp) is an isomorphism
if m 6 0; 1 mod p. For another proof we refer to [B].
For p = 2 the Gysin sequence of the S0-bundle BA(m) ! B(m) tells us
that it is enough to show that multiplication with the mod 2 Euler class e :
H*(B(m); F2) ! H*(B(m); F2) is mono. Now H*(B(m); F2) is detected
by the mod 2-cohomology of its maximal elementary abelian 2-groups E [Q1]
and hence it suffices to show that e restricts nontrivially to each H*(BE; F2),*
* E
maximal. Finally, e 2 H1(B(m); F2) corresponds to the signum map (m) !
Z=2, and for m 6 0; 1 mod 4 each maximal E contains a single transposition (cf.
[Mu,p.346f]), hence e restricts nontrivially to H*(BE; F2).
___
|__|
10 HANS-WERNER HENN and HUYNH MUI
4. Proof of Theorem 2.7
We start by showing that the inclusion j = jm;n : SG(m) ! SG(n) induces
an injection j = jm;n : A(P; SG(m)) ! A(P; SG(n)), at least for those m with
ffi(m) 6= 0. Then we will identify A(P; SG(m)) with its image and show that
the filtration of A(P; SG(n)) by the subspaces A(P; SG(m)) (for those m with
ffi(m) 6= 0) splits via h+ after tensoring with Z=p (cf. Lemma 4.4 and Proposit*
*ion
4.5 below). This will prove Theorem 2.7 and gives, via the argument of Section
3, the strengthening of Theorem 0 mentioned in Remark a) of the introduction.
Throughout this section P will denote a finite p-group, Q a subgroup of P and
p, SG(m) and ffi(m) are as in Theorem 0.
LEMMA 4.1 The inclusion SG(m) -j!SG(n) induces injections
Rep(Q; SG(m)) -j!Rep (Q; SG(n)), RepP(Q; SG(m)) -j!Rep P(Q; SG(n)) and
A(P; SG(m)) -j!A(P; SG(n)) provided ffi(m) 6= 0 (or m = 0).
Proof:_The cases m = 0 and m = n are trivial, so we may assume n > m > 0 .
It suffices to prove the statement for Rep(Q; ?). The other cases are immediate
consequences.
It is clear that the inclusions from G(m) to G(n) induce injections because ele-
ments in Rep(Q; G(m)) correspond to isomorphism classes of Q-sets of cardinal-
ity m (if G(m) = (m)) resp. m-dimensional real or complex representations
of Q (if G(m) = O(m) or U(m)) and inclusion corresponds to adding a trivial
Q-set of cardinality n - m resp. a trivial representation of dimension n - m.
Therefore it suffices to show that the inclusion im : SG(m) ! G(m) induces
an injection if ffi(m) 6= 0. For this suppose that '1 and '2 represent elements
in Rep(Q; SG(m)) and that there is an element g 2 G(m) \ SG(m) such that
im '1 = g(im '2)g-1.
If SG(m) = A(m) we think of im 'k, k = 1; 2, as Q-set of cardinality m. Then
m < n and ffi(m) 6= 0 imply that m 2 mod 4 (if p = 2) or m 2 mod p (if
p > 2), hence these Q-sets contain either two fixed points or an orbit of length
2. Hence there is a transposition o which commutes with im '1 and then we have
im '1 = (og)(im '2)(og)-1 with og 2 A(m), i.e. '1 and '2 represent the same
element in Rep(Q; A(m)).
For SG(m) = SO(m), m < n and ffi(m) 6= 0, we have that m is odd and then the
matrix (-id) commutes with im '1 and has determinant -1. If SG(m) = SU(m),
then . id commutes with i`'1 for each complex number and we can choose
such that det( . id) = det(g)-1 . In both cases we can continue as in the case *
*of
A(m) and conclude that '1 and '2 agree in Rep(Q; SG(m)).
___
|__|
STABLE SPLITTINGS FOR ALTERNATING GROUPS 11
REMARKS 4.2.
a) The proof shows that the restriction ffi(m) 6= 0 in 4.1 is actually unneces*
*sary
if SG(m) = SU(m).
b) On the other hand one can show that there are even elementary abelian p-
groups Q for which the map Rep(Q; A(m)) ! Rep(Q; (m)) is not injective
if m > 0, m 0 mod 4, p = 2, or m > 0, m 0 mod p, p odd.
This and the proof of 4.1 imply that for such m and all ` m + 2 the map
Rep(Q; A(m)) ! Rep(Q; A(`)) is not injective and therefore (cf. [HLS, Sec-
tion 4.2]) the restriction map H*(BA(`); Fp) ! H*(BA(m); Fp) is not even
an F - epimorphism in the sense of [Q2], in particular not an epimorphism
(cf. Remark b) after Theorem 0).
4.3. Now we introduce filtrations on A(P; SG(n)) and C(P; SG(n)) by defining
Em A(P; SG(n))= Im(A(P; SG(m))) -j!A(P; SG(n))
and !
Mm
Em C(P; SG(n)) = A(P; SG(`))=A(P; SG(` - ffi(`))) A(P; SG(0)):
`=1
By 4.1 we may identify Em A(P; SG(n)) with A(P; SG(m)) whenever ffi(m) 6= 0.
Furthermore, in the definition of Em C, we have used 4.1 in order to identify
A(P; SG(` - ffi(`))) with Im(A(P; SG(` - ffi(`))) -j! A(P; SG(`))) for all `. (*
*By
convention we take Em equal to A(P; SG(0)) in both filtrations whenever m 0.)
The maps A(P; SG(m)) -j! A(P; SG(n)) map basis elements [Q; '] to basis
elements [Q; j'] and hence j Z=p is mono whenever j is mono. Therefore
Theorem 2.7 is clearly implied by the following two results.
LEMMA 4.4. The homomorphism h+ : A(P; SG(n)) ! C(P; SG(n)) maps
Em A(P; SG(n)) to Em C(P; SG(n)) for all m n.
PROPOSITION 4.5. The map h+ Z=p induces isomorphisms on E0 and also
on filtration quotients Em Zp=Em-ffi(m) Z=p for all m with 0 < m n and
ffi(m) 6= 0.
In the proofs of 4.4 and 4.5 we will interpret homomorphisms Q -ae! G(m)
as Q-sets of cardinality m (if G(m) = (m)) resp. as real or complex Q-
representationsPof dimension m (if G(m) = O(m) or U(m)) and we will write
ae ~= c with nonnegative numbers c and running through a set of represen-
tatives of the isomorphism classes of irreducible Q-sets resp. Q-representation*
*s.
If we denote the cardinality of resp. the dimension of by ||, then clearly
12 HANS-WERNER HENN and HUYNH MUI
P
c || = m. The trivial Q-set resp. representation will be denoted by = 1.
P
We will abbreviate c || by kaek.
6=1
4.6. Proof_of_Lemma_4.4:_The case m = 0 is trivial. For the other cases it suff*
*ices
to show that the homomorphisms hn;`: A(P; SG(n)) ! A(P; SG(`)) (cf. Section
2) map Em A(P; SG(n)) into A(P; SG(m) (considered as subspace of A(P; SG(`))
via 4.1) for all 0 m < ` n. This is trivial for ` = n and for ` ffl.
In the other cases we get from (2.2) that hn;`is given on a class [Q; '] by a l*
*inear
combination of the form
X
hn;`[Q; '] = Oi[Qi; d`ss i]:
i
Here the Oi are suitable integers, the Qi suitable subgroups of Q and there are
elements gi2 G(n) such that i:= g-1i(in')gimaps Qito G(`-ffl)xG(n+ffl-`).
WePinterpretP i as a pair of Qi-sets resp. Qi-representations and write i ~=
( ai; bi).
P
If [Q; '] is in Em A(P; SG(n)), then (ai + bi)|| m. It is clear that
6=1 P
[Qi; d`ss i] is in Ek+fflA(P; SG(`)), if kss ik = ai|| k, so we may con-
6=1
centrate on the case that m - ffl < kss ik m. But then we find bi = 0 for all
6= 1 (note that || ffl for all 6= 1) and therefore ss i factors through SG(m*
*).
Finally, d` maps SG(m) clearly to itself and this finishes the proof.
___
|__|
4.7. For the proof of Proposition 4.5 we need some preparations. First we obser*
*ve
that the statement about E0 is trivial.
Then we define filtrations eEmon Z[Rep (Q; SG(n))] as in 4.3, i.e.
EemZ[Rep (Q; SG(n))] = Im(Z[Rep (Q; SG(m))] ! Z[Rep (Q; SG(n))]) :
The proof of 4.4 shows also that the maps
hn;`: Z[Rep (Q; SG(n))] ! Z[Rep (Q; SG(`))]
preserve eEmfor all 0 m ` n. (Of course, the definition of these maps is as
in section 2.)
Now fix ` with 0 < ` n and ffi(`) 6= 0. For the proof of 4.5 it suffices now
by 4.4 and 2.3-2.5 to show that hn;`induces an isomorphism on the filtration
quotient eE` Z=p=Ee`-ffi(`) Z=p for all subgroups Q of P . We will show that
STABLE SPLITTINGS FOR ALTERNATING GROUPS 13
hn;`induces an isomorphism on filtration quotients eEm Z=p=Eem-1 Z=p for
` - ffi(`) < m ` and this is clearly enough.
Again the cases ` = n or ` ffl are trivial, so we will assume ffl < ` < n from*
* now
on.
In these cases we get for a basis element ['] 2 Rep(Q; SG(n)) by 2.4
X
hn;`['] = [in' : ][d`ss ]
[ ]
withP[ ] runningPthroughPRep(Q; G(` - ffl) x G(n + ffl - `)). Now we write in' *
*~=
c , ~=( a ; b ). Then [in' : ] = 0 unless a + b = c for all .
In particular, if we write k'k instead of kin'k we have either k'k = kss k or
kss k k'k - ffl for such . If k'k = kss k then this [ ] is unique and we will
also denote it by [ 0].
Together with the fact that d` maps terms [ss ] with kss k k to eEk+ffl(cf. pr*
*oof
of 4.4) this implies
X
(4:8) hn;`['] [in' : 0][d`ss 0] + [in' : ][d`ss ] mod Eek'k-1:
[ ]
kss k=k'k-ffl
(By convention the term involving 0 will be dropped if k'k 6= kss k for all [ *
*].)
It will be enough to show that hn;`['] C['] mod eEk'k-1for all ' with `-ffi(`)*
* <
k'k ` where C is an integer which is not divisible by p and ['] on the right
hand side is regarded as element in Rep(Q; SG(`)) (cf. 4.1).
In order to evaluate (4.8) we need the following Lemma. But first note that for
those [ ] with a + b = c and kss k = k'k - ffl there is a unique 06= 1 with
k0k = ffl and a0 = c0 - 1, a = c for all other 6= 1.
LEMMA 4.9. For ['] 2 Rep(Q; SG(n)) and [ ] 2 Rep(Q; G(` - ffl) x G(n + ffl - `))
with ffi(`) 6= 0 and ` - ffi(`) < k'k ` we get
a) If kss k = k'k or kss k = k'k - ffl, and a + b = c for all , then
[d`ss ] = [']. (Here ' is considered as element in Rep(Q; SG(`)) by using
4.1.)
b) If kss k = k'k and a + b = c for all (i.e. = 0), then
[in' : ] = O ________G(n_-_k'k)________G(`:- k'k - ffl) x G(n *
*+ ffl - `)
c) If kss k = k'k - ffl, and if a + b = c for all and 0is as above, then
0
[in' : ] = O _____G(c__)_____G(cx _____G(n_-_k'k)_____:
0 -G1)(x`G(1)- k'k) x G(n - `)
14 HANS-WERNER HENN and HUYNH MUI
We postpone the proof of 4.9 and give now the proof of 4.5.
4.10. Proof_of_Proposition_4.5:_We evaluate (4.8) in the different cases sepa-
rately.
Case_SG(n)_=_A(n)_and_p_odd:_We write ` = ps + 2 (recall that we discuss the
case ffi(`) 6= 0). Now ` - ffi(`) < k'k ` implies k'k = ps. Furthermore there
are no terms with kss k = k'k - ffl because kss k is divisible by p. Therefore *
*4.9
implies
hn;`['] nn-+k'k2 -[`'] ['] mod eEk'k-1:
Case_SG(n)_=_A(n)_and_p_=_2:_We write ` = 4s + 2 (Recall that ffi(`) 6= 0). Now
` - ffi(`) < k'k ` implies k'k = 4s + 2 or k'k = 4s.
If k'k = 4s + 2 then there are only terms with kss k = k'k - ffl and we get from
4.9 X
hn;`['] c n -nk'k- `['] mod Eek'k-1:
kk=2
P
Again the binomial coefficient is 1 and k'k = 4s + 2 implies that c is od*
*d.
kk=2
If k'k = 4s we find
8 9
< n - k'k X n - k'k =
hn;`['] : n + 2 - ` + c n - ` ['] mod Eek'k-1:
kk=2 ;
P n - k'k n - k'k
Because of k'k = 4s we get c 0 mod 2 and =
kk=2 n + 2 - ` n - k'k
= 1 and we are done again.
The_cases_SG(n)_=_SO(n)_or_SU(n):_If k'k = `, then there are only summands
with kss k = k'k - ffl and 4.9 gives
X
(4:11:) hn;`['] O(G(c )=G(c - 1) x G(1))['] mod Eek'k-1:
kk=1
6=1
The space G(c )=G(c - 1) x G(1) may be identified with the projective space
KP c -1where K denotes R in the case of SO(n) and C in the case of SU(n).
For the Euler characteristic we get O(RP c -1) c mod 2 and O(CP c -1) = c .
ThereforePwe find that the coefficientPof ['] in (4.11) is
c for SU(n) and congruent to c mod 2 for SO(n).
kk=1 kk=1
6=1 6=1
Now Q is a p-group and the dimension of an irreducible Q-representation is eith*
*er
1 or divisible by p (cf. [Se, Chap. 8.1] for the case of complex representation*
*s;
STABLE SPLITTINGS FOR ALTERNATING GROUPS 15
the case of real representationsPand p = 2 is easily reduced to the complex cas*
*e.).
Therefore we find c k'k ` mod p, and hence the coefficient of ['] is
kk=1
6=1
not divisible by p (we have assumed that ffi(`) 6= 0 and ` < n).
It remains to consider the case k'k 6= `. Then ` - ffi(`) < k'k < ` implies
k'k = ` - 1 and ` 1 mod p. Here 4.9 yields
X
hn;`['] ['] + O(KP c -1) . O(KP n-`)['] mod Eek'k-1:
kk=1
6=1
The coefficientsPin the sum are equal to c .(n-`+1), at least mod p, and as abo*
*ve
we see c k'k ` - 1 0 mod p. Therefore hn;`['] C['] mod eEk'k-1
kk=1
6=1
with C 1 mod p. ___
|__|
4.12. It remains to give the
Proof_of_4.9:_Via 4.1 we consider ['] as element in Rep(Q; SG(`)). If kss k = k*
*'k
and a + b = c for all (i.e. = 0) then ss factors through SG(` - ffl) and
hence i`d` adds only trivial Q-orbits resp. representations to ss . Therefore
i`[d`ss ] = i`['] where i` denotes the homomorphism induced by the inclusion
SG(`) ! G(`).
If kss k = k'k - ffl and a + b = c for all , then ss does not factor through
SG(` - ffl)and i`d` adds the(Q-setdresp.eQ-representationtcorresponding)to-the1
homomorphism Q--ss-!G(`-ffl)-- -! G(ffl). It is easy to see that this correspon*
*ds
just to the unique 0 with 0 6= 1, k0k = ffl, a0 = c0 - 1 and a = b for all
other 6= 1. Therefore i`[d`ss ] = i`['].
Now i` is mono if ffi(`) 6= 0 (cf. proof of 4.1) and hence a) follows.
b) and c): The proofs of b) and c) are similar and are just an application of
Lemma 2.6. If wePabbreviatePG(n) byPG, G(` - ffl) x G(n + ffl - `) by H and wri*
*te
as before in' ~= c , ~=( a ; b ), then (G=H)in' is nonempty if and
only if a + b = c for all .
Furthermore, it is not hard to see that
Y
CG ( ) ~= AutQ(c )
with AutQ (c ) G(c ||) denoting the automorphism group of the Q-set re-
spectively Q-representation c ; similarly we have
Y
CG ( ) \ H ~= AutQ(a ) x AutQ(b ):
16 HANS-WERNER HENN and HUYNH MUI
Q
Hence (G=H)in'is homeomorphic to AutQ (c )=Aut Q(a )xAut Q(b ) and
b) and c) follow easily.
For example, in case c) we have c = a for all except for = 0 as above
and = 1. Now AutQ (c ) is isomorphic to G(c ) if || = 1, respectively to the
wreath product AutQ () o G(c ) if G(n) = (n) and is any irreducible Q-set.
Hence we get in this case
CG ( )=CG ( ) \ H ~=
~=(G(c0 )=G(c0 - 1) x G(1))x (G(n - k'k)=G(` - k'k) x G(n - `)):
___
|__|
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Hans-Werner Henn Huynh Mui
Mathematisches Institut Department of Mathematics
der Universit"at University of Hanoi
Im Neuenheimer Feld 288 Dai Hoc Tong Hop
D-6900 Heidelberg Hanoi
Federal Republic of Germany Vietnam