HOMOTOPY REPRESENTATIONS OF SO(7) AND Spin(7) AT
PRIME 2
KRZYSZTOF ZIEMIA'NSKI
Abstract.A homotopy (complex) representation of a compact connected Lie
group L at prime p is a map from BL into the p-completion of the classif*
*ying
space of the unitary group BU(n)^p. In this paper we give a partial clas*
*sifica-
tion of homotopy representations of SO(7) and Spin(7) at prime 2. Motivi*
*a-
tion for considering this problem is twofold: first, one may hope that i*
*t would
help to understand maps between classifying spaces. Secondly, constructi*
*on of
a homotopy representation of Spin(7) is a crucial step in the constructi*
*on of
a faithful representation of the 2-compact group DI(4) [14].
Let p be a prime and let L be a compact connected Lie group. A group P
is p-toral if it is an extension of a finite p-group by a torus; a p-toral subg*
*roup
P L is p-stubborn if NL(P )=P is finite and contains no non-trivial normal p-
subgroups. Let Op(L) (resp. Rp(L)) be a category of L-orbits L=P for p-toral P
(resp. p-stubborn P ) and L-equivariant maps. By [7], the map
(hocolimL=P2Rp(L)(EL xL L=P )^p)^p-! BL^p
is a weak homotopy equivalence. The category Rp(L) contains a maximal object
G=N, where N is a p-normalizer of a maximal torus T . Let N1 be a p-discrete
approximation of N, i.e. its dense subgroup such that T \ N1 be a subgroup of *
*all
elements of T having a p-power order. By a version of Dwyer-Zabrodsky theorem
[8] the map
Rep(N1 , U(n)) 3 ' 7! (B')^p! [(EL x L=N)^p, BU(n)^p],
where Rep(N1 , U(n)) := Hom (N1 , U(n))= Inn(U(n)), is an isomorphism. A rep-
resentation ' of N1 is Rp(L)-invariant iff (B')^pdefines a homotopy compatible
family of maps in limL=P2Rp(L)[EL xL (.), BU(n)^p]. Obviously if (B')^pextends
to a homotopy representation of L (i.e. a map BL^p! BU(n)^p), then ' is Rp(L)-
invariant; the question is whether or not an Rp(L)-invariant representations ex-
tends (after completion) to a homotopy representation of L. The main result of *
*the
present paper is the following
Theorem. Let p = 2. If L = Spin(7) or L = SO(7), then every Rp(L)-invariant
representation of N1 extends to a homotopy representation of L.
Throughout the whole paper G = SO(7), "G= Spin(7), ss : "G! G is an obvious
projection and u 2 "Gis a non-trivial lift of unity in G. For any subgroup H G
let "H:= ss-1(H) "G.
Organization of the paper. Section 1 contains generalization of some elemen-
tary facts from representation theory of finite groups onto locally finite grou*
*ps (in
particular onto discrete approximations of p-toral groups). In Section 2 we des*
*cribe
a general method of construction homotopy representations of compact Lie groups.
1
2 KRZYSZTOF ZIEMIA'NSKI
Section 3 contains a classification of 2-stubborn subgroups of orthogonal and s*
*ym-
metric groups, and a construction of a full inclusion R2( n) ! R2(O(n)). The la*
*st
result simplifies calculations of higher limits on R2(O(n)). In Section 4 some *
*facts
about representations of discrete approximations of 2-stubborn subgroups of SO(*
*7)
and Spin(7) are provided. Section 5 contains specific computations of higher li*
*mits
of atomic functors. Finally, the main theorem is proven in Section 6.
1. Representations of countable locally finite groups
A group is countable locally finite if it satisfies the following equivalen*
*t condi-
tions:
o is countable and every finitely generated subgroup of is finite.
o There is an ascending sequence of finite groups
(1.1) {1} = (0) (1) (2) . . .
S
such that s (s)= .
By a complex representation of a group we mean a homomorphism ' : !
GL (V ), where V is a finite dimensional complex vector space. A representation*
* is
unitary if V is equipped with a hermitian scalar product which is preserved by *
*'(g)
for each G 2 .
Throughout this section denotes anyScountable locally finite group and (s)*
*is
a fixed sequence (1.1). Let Rep( ) := n Rep( , GL(Cn)) and let IR( ) Rep( )
be the set of isomorphism classes of irreducible representations of . The foll*
*owing
proposition provides a main reason for similarity of representation theory of l*
*ocally
finite groups with the finite case.
Proposition 1.2. If ' : ! GL (Va)nd _ : ! GL (W )are representations, then
there is s < 1 such that
Hom (V, W ) = Hom (s)(V, W ).
Proof.A sequence
Hom (0)(V, W ) Hom (1)(V, W ) Hom (2)(V, W ) . . .Hom (V, W )
eventually stabilizesTsince the dimensions of these spaces do. Moreover, we ha*
*ve
Hom (V, W ) = 1r=0Hom (r)(V, W ). Thus Hom (V, W ) = Hom (s)(V, W ) for
large enough s.
Here follow immediate corollaries of Proposition 1.2:
Proposition 1.3. Fix representations ' : ! GL (V ) and _ : ! GL (W ).
(1) If ' is irreducible, then for some s res (s)' is irreducible.
(2) If for all s res (s)' and res (s)_ are isomorphic, then ' and _ are iso-
morphic.
(3) (Schur lemma) If ' is irreducible, then End (V ) contains only homotetie*
*s.
If additionally _ is irreducible and non-isomorphic to ', then Hom (V, *
*W ) =
0.
Proof.Fix s such that End (V ) = End (s)(V ) and Hom (V, W ) = Hom (s)(V, W )
(1) If res (s)' is reducible, then there is a projection f 2 End (s)(V ) ont*
*o one
of its irreducible summands. Since f 2 End (V ) it denies irreducibility*
* of
'.
HOMOTOPY REPRESENTATIONS OF SO(7) AND Spin(7) AT PRIME 2 3
(2) If Hom (s)(V, W ) contains an isomorphism, then Hom (V, W ) also does.
(3) By (1) and (2) it follows from the Schur lemma for finite groups.
Proposition 1.4. [12] Every representation of is semisimple, and its decompo-
sition into irreducible summands is unique up to permutation of summands.
Proof.Let ' : L ! GL (V ) be any representation and fix s such that End (s)V =
End (V ). Let iWi be a decomposition of res (s)' onto irreducible summands.
All Wi's are irreducible -subrepresentationsL(since any (s)-projection V ! Wi*
* is
also L-homomorphism).LThen iWi is a decomposition of '.
If iWi and jXj are decompositions of ', then there is a (s)-isomorphism
M M
f : V = Wi- ! Xj = V
i j
permuting irreducible (s)-summands. But f is also -isomorphism.
A decomposition into irreducible summands can be obtained functorially. Pick
a representative oe : ! GL (Woe) of every element of IR( ).
Proposition 1.5. For any representation ' : ! GL (V ) the evaluation
M ev
Woe Hom (Woe, V ) -! V
oe2IR( )
is an isomorphism. Furthermore, the homomorphism
M M
End (Hom (Woe, V )) 3 foe7!
oe2IR( ) oe
iM j
7! ev O IdWoe foe O (ev)-1 2 End (V )
oe
is an isomorphism.
L
Proof.By Proposition 1.4 we can assume that V = oe2IR(W)oloee. By the Schur
lemma Hom (Woe, Woeloe) ~=Hom (Woe, V ). Then the evaluation is a composition
M M M
Woe Hom (Woe, V ) ~= Woe Hom (Woe, Woeloe) ~= Woe Cloe= V.
oe oe oe
Again by the Schur lemma
M
End (V ) ' End (Woe Hom (Woe, V )).
oe2IR( )
Moreover,
End (Woe Hom (Woe, V ))
= Hom (Woe Hom (Woe, V ), Woe Hom (Woe, V ))
= Hom (Hom (Woe, V ), Hom (Woe, Woe Hom (Woe, V )))
= Hom (Hom (Woe, V ), Hom (Woe, Woe) Hom (Woe, V ))
= End(Hom (Woe, V )).
Let Ch( ) C be a vector subspace spanned by all characters of representa-
tions (it does not contain all class functions).
4 KRZYSZTOF ZIEMIA'NSKI
Proposition 1.6. If O, O0 2 Ch( ), then the sequence (O|O0)s := (O| (s)|O0| (s))
stabilizes. In particular,
(O|O0) := lims!1(O|O0)s
is a hermitian product on Ch( ) and characters of irreducible representations f*
*orm
an orthonormal basis of Ch( ).
Proof.By Proposition 1.4 it is enough to prove that the sequnce (O|O0)s stabili*
*zes
for O = O', O0 = O_, where ', _ 2 IR( ). By 1.3.(1) for large enough s both
res (s)' and res (s)_ are irreducible. Then if ' and _ are isomorphic, then
(O'|O_)s = 1; if not (O'|O_)s = 0 (cf. 1.3.(2)).
From Propositions 1.4 and 1.6 we obtain
Corollary 1.7. Two representations of a given locally finite group are isomorph*
*ic
iff their characters are equal.
The next proposition states that representations of products behave similarly*
* to
the finite case:
Proposition 1.8. Let and be countable locally finite groups. The map
~ : IR( ) x IR( ) 3 (', _) 7! res x ' res x _ 2 IR( x )
is a bijection.
Proof.If ' 2 IR( ), _ 2 IR( ) then for some s both representations res (s)'
and res (s)_ are irreducible. Hence res x(s)x (s)' ~_ is irreducible and so is *
*' ~_.
Then the map ~ is well-defined.
Now let ! 2 IR( x ). There is r such that for s r res (s)! is irreducible
and then it is isomorphic to a tensor product of irreducible representations of*
* fac-
tors. Then both res x(s)x{1}! and res{x1}x (s)! are sums of pairwise isomorphic
irreducible representations. As a consequence of Proposition 1.3.(2) we have
res xx{1}! ' ' dim_, res{x1}x! ' _ dim'
for some ' 2 IR( ), _ 2 IR( ). For each s r the characters of ! and ' ~_ are
equal on (s)x (s)and hence they are equal on x . Now Corollary 1.7 implies
that ! is isomorphic to ' ~_.
Up to this point we have considered linear representations when Dwyer-Zabrosky
theorem requires unitary ones. The following proposition states that any complex
representation admits a unique unitary structure.
Proposition 1.9. Let V be a complex linear space with a hermitian scalar produc*
*t.
Then the map Rep( , U(V )) ! Rep( , GL(V )) is a bijection.
Proof.Consider the following (obviously commutative) diagram (lim stands for an
inverse limit)
Rep( , U(V ))__________wRep ( , GL(V ))
| |
(1.10) || ||
|u |u
lims!1Rep( (s), U(V_))__slim!1Rep(w(s), GL(V ))
HOMOTOPY REPRESENTATIONS OF SO(7) AND Spin(7) AT PRIME 2 5
where the vertical maps assign to any representation the collection of its rest*
*rictions
to subgroups (s). The lower horizontal arrow is bijective since groups (s)are
finite. For L = U(V ) or L = GL (V ) let {'s} 2 lims!1 Hom ( (s), L). Define
homomorphisms _s 2 Hom ( (s), L) by induction. For s > 0 let _s(g)S= h-1's(g)h
where h is any element such that _s-1 = h-1's| (s-1)h. Then _ = s _s is a well-
defined homomorphism ! L such that _| (s)is conjugate to 's. This implies
the surjectivity of both vertical maps. Let ', _ 2 Hom ( , U(V )) be non-conjug*
*ate
homomorphisms and put Gs = {g 2 U(V ) : g-1'gT= _}. Gs is a non-increasing
sequence of closed subsets of U(V ). Since rGr = ; and U(V ) is compact, then
there is s such that Gs = ;. Then '| (s)is not conjugate to _| (s)and the map
Rep( , U(V )) ! limsRep( (s), U(Vi))s injective. Injectivity of the right verti*
*cal
arrow follows from Proposition 1.7. Now the conclusion follows.
We conclude with a corollary from Propositions 1.5 and 1.9:
Corollary 1.11. For any unitary representation ' : ! U(V ) we have
Y
CU(V )('( )) = U (V ) ~= U(Hom (Woe, V )).
oe2IR( )
2. Homotopy representations of compact Lie groups
The tool which appears the most appriopriate for calculating the set of homo-
topy representations of L is a subgroup homotopy decomposition due to Jackowski,
McClure and Oliver [7]. Recall that holds the following
Theorem 2.1 ([7, 1.4]). The map
(hocolimL=P2Rp(L)EL xL L=P )^p' BL^p
induced by projection is a weak homotopy equivalence.
Notice that EL xL L=P is homotopy equivalent to BP . The next theorem
describes the mapping space map (BP, BU(n)^p) if P is p-toral.
Theorem 2.2 ([8, Thm. 1.1]). Let P be a p-toral group and let H be a compact
connected Lie group. Then:
(1) The maps
Rep(P 1, H) -B![BP 1, BH^p] - [BP, BH^p]
are bijections.
(2) For any ' : P 1 ! H the pairing BCH ('(P 1)) x BP 1 ! BH induces a
homotopy equivalence
BCH ('(P 1))^p-! map(BP, BH^p)B' .
Hence limL=P[EL xL (.), BU(n)^p] ' limL=PRep(P 1, U(n)). Let N L be a
p-normalizer of a maximal torus and let ' : N1 ! U(V ) be an Rp(L)-invariant
representation. For i > 0 define contravariant functors i from Rp(L) to the
category of groups (abelian if i > 1)
'i(L=P ) := ssimap ((EL=P )^p, BU(n)^p)B'^p|(EL=P)^p.
By the theorem of Wojtkowiak [13], the obstructions to an existence of an exten*
*sion
of (B')^pto (BL)^plie in groups Hi+1(Rp(L); i) for i 1. Luckily functors i
6 KRZYSZTOF ZIEMIA'NSKI
can be interpreted in terms of irreducible representations of restrictions resN*
*1P1'
for p-stubborn P .
For any ! 2 Rep ( , U(n)) let IR( , !) IR( ) be a subset of isomorphism
classes of irreducible subrepresentations of !. Furthermore, let R( ) := Z[IR( *
*)]
and R( , !) := Z[IR( , !)].
Theorem 2.3. For each L=P 2 Rp(L) we have '1(L=P ) = '3(L=P ) = 0. Fur-
thermore, there is a functorial isomorphism
1
'2(L=P ) ' Z^2 R(P 1, resNP1').
Proof.Let W be a1unitary space with a P 1-action which represents some element
of Rep(P 1, resNP1'). Choose a monomorphism j : W ! V sand put
1 s
(2.4) JP,': R(P 1, resNP1') 3 [_] 7! j*(lW ) 2 ss1(UP1 (V )),
where lW 2 ss1(U(W )) is represented by a loop t 7! e2ssit. I. Maps JP,' does
not depend on the choice of j since all inclusions W ! V sdetermine conjugate
homomorphisms UP (W ) ! UP (V s). It also does0not depend on the choice of s
since for s < s0the map UP (V s) ! UP (V s) induces an isomorphism on ss1.
Let a : L=Q ! L=P be a morphism in Rp(L). Commutativity of a diagram
1 JP,'
R(P 1, resNP1')_____wss1(UP1 (V ))
| |
a*| |
| |
|u 1 JQ,' |u
R(Q, resNP1')______wss1(UQ1 (V ))
implies that J' is a natural transformation. Moreover, both groups R(P 1, resN1*
*P1') 1
and ss1(UP (V )) are free and abelian, the generator set of them both is IR(P 1*
*, resNP1')
and the generators are mapped to the corresponding generators. Therefore J' :
R(-, ') ! ss1Aut (-)(V ) is a natural equivalence. Thus the composition
Z^p R(-, ') Id-J'---!Z^p ss1(U(-)(V )) ~=Z^p ss1(CU(V )('(-)))
~= ^ ^
-! Zp ss2(BCU(V )('(-))) -! ss2(BCU(V )('(-)))p
~= ^ ^ '
-! ss2(map (EL=(-), BU(V )p)B'|P)p = 2(L=P ).
(2.2)
is a required natural equivalence. Furthermore, by 1.11 UP1 (V ) = CU(n)('(P 1))
is a product of unitary groups and hence ss0(Aut P1(V )) = ss2(Aut P1(V )) = 0.
For any Rp(L)-invariant representation ' of N1 denote (') := Z^p R(-, ').
The following is an immediate consequence of 2.3:
Theorem 2.5. Let ' : N1 ! U(V ) be an Rp(L)-invariant representation. If
H3(Rp(L); (')) = 0 and Hi(Rp(L); M) = 0 for i > 4 and any contraviariant
functor M : Rp(L) ! Z^p-Mod , then B'^2extends to a map BL ! BU(m)^p .
3. Stubborn subgroups of symmetric and orthogonal groups
In this section we recall results of Oliver [9], who gave the classification *
*of 2-
stubborn subgroups of orthogonal groups and the description of the categories
HOMOTOPY REPRESENTATIONS OF SO(7) AND Spin(7) AT PRIME 2 7
R2(O(n)). Furthermore, we recall the classification of 2-stubborn groups of sym-
metric groups (given by Alperin and Fong [1]) and describe the categories R2( n*
*).
The section is concluded with a constuction of a full inclusion R2( n) ! R2(O(n*
*))
which makes easier calculations of higher limits on R2(O(n)).
2-Stubborn subgroups of orthogonal groups. Each 2-stubborn subgroup of an
orthogonal group is built up out of some number of "pieces" by taking products *
*and
wreath products. (Recall that the wreath product G o H is the semi-direct produ*
*ct
GH n H, where H acts on GH by shifting coordinates). Let In 2 GL n(R) denote
the identity matrix. If M 2 GL m(R) and N 2 GL n(R), then M N 2 GL nm(R is
the matrix with entries
(M N)am+b-m,cm+d-m = Mb,dNa,c.
for a, c = 1, . .,.n, b, d = 1, . .,.m. Let
` ' ` '
(3.1) A := 10 -01 , B := 01 10 ,
and for i < n define the matrices Ani, Bni2 O(2n) by
(3.2) Ani:= I2i A I2n-i-1, Bni:= I2i B I2n-i-1.
Let
(3.3) 2n := <-I2n, An0, . .,.Ann-1, Bn0, . .,.Bnn-1> O(2n)
(3.4) ~ 2n:= <{X I2n-1}X2SO(2), An0, . .,.Ann-1, Bn0, . .,.Bnn-1> O(2n)
The group 2n is an extra-special group of order 22n+1 and ~ 2nis a 2-toral gro*
*up
with 1-dimensional torus. Note that 1 ~={ 1} = O(1) and ~ 2~=O(2).
Definition 3.5. ([9, Def. 2]) Let T (k) Tirr(k) Tprod(k) be the sets of 2-t*
*oral
subgroups of O(k) defined as follows:
o T (1) = { 1 = O(1)},
o T (2) = {~ 2= O(2)},
o T (2n) = { 2n, ~ 2n} for n > 1,
o T (k) = ; for k 6= 2n,
o Tirr(2n) is the set of those wreath products in O(2n) of the form
o Ct12o . .o.Cts2,
where 2 T (2l), n = l + t1 + . .+.ts and t1 > 1 if = 1,
o Tprod(k) is the set of all products in O(k) of the form
P = P1 x . .x.Ps O(k1) x O(k2) x O(ks) O(k),
where Pi2 Tirr(ki) and k = k1 + . .+.ks.
Remark. If O(2l), then the embedding o Ct2 O(2l+t) is chosen as follows:
t l+t l+t l+t l+t
o Ct2~=< 2 , Bl , Bl+1, . .,.Bl+t-1> O(2 ).
By [9, Theorem 8] each 2-stubborn subgroup of O(k) is conjugate to a subgroup
in Tprod(O(k)). Possibly not all elements 2 Tprod(O(k)) are 2-stubborn _ it
depends on the Weyl group of . The following proposition follows immediately
from [9, Theorem 6]:
Proposition 3.6. Let WL(P ) := NL(P )=P . Then
8 KRZYSZTOF ZIEMIA'NSKI
o WO(2n)( 2n) ' O+ (Xn),
o WO(2n)(~ 2n) ' Sp(Vn),
o WO(2n)( o Ct12o . .o.Cts2) ' WO(2m )( ) x GLt1(F2) x . .x.GLts(F2), 2
T (2, O(2m )).
The following theorem determines which elements of Tprod(k) are in fact 2-
stubborn subgroups of O(k):
Theorem 3.7. ([9, Theorem 6]) Let P 2 O(k) be a 2-stubborn subgroup. Then P
is conjugate to an element of Tprod(k). If P 2 Tprod(k) then P is 2-stubborn if*
* and
only if when written as a product
P = P1 x . .x.Ps, (Pi2 Tirr(ki))
there is no factor Pi with WO(ki)(Pi) = 1 which occurs with multiplicity exactl*
*y 2
or 4.
Theorem 3.8. ([9, Proposition 9]) Let P, P 02 Tprod(k). If P 0is conjugate to a
subgroup of P , then x-1P 0x P for some permutation matrix x which permutates
irreducible factors of P 0. If P 0 P then the inclusion is a composite of prod*
*ucts of
the following types:
(a)O(1) x O(1) O(2),
(b)O(1) o C2t12o Ct22o . .o.Cts2 O(2) o Ct12o Ct22o . .o.Cts2,
(c)( o Ct12o . .o.Cts2)ts+1 o Ct12o . .o.Cts2o Cts+12,
(d) 2ko Ct12o . .o.Cts2 ~ 2ko Ct12o . .o.Cts2,
(e) o . .o.Cti2o Cti+12o . .o.Cts2 o . .o.Cti+ti+12o . .o.Cts2,
(f) 2k+t1o Ct22o . .o.Cts2 2ko Ct12o Ct22o . .o.Cts2,
(g)~ 2k+t1o Ct22o . .o.Cts2 ~ 2ko Ct12o Ct22o . .o.Cts2,
where stands for either 2k or ~ 2k.
Corollary 3.9. Each morphism in R2(O(n)) is a composition of automorphisms
and inclusions enlisted in 3.8.
2-Stubborn subgroups of symmetric groups. The following classification of 2-
stubborn subgroups of symmetric groups is due to Alperin and Fong [1] (a subgro*
*up
of a finite group is 2-stubborn iff it is 2-radical).
Note that if G m , H n, then the product G x H is a subgroup of m+n
and the wreath product G o H is a subgroup of mn .
Definition 3.10. For any sequence t1, . .,.ts of positive integers let
B(t1, . .,.ts) := 1 o Ct12o . .o.Cts2 2t,
where t = t1 + . .+.ts (we treat 1 as a subgroup of 1). The groups B(t1, . .,.*
*ts)
will be called basic subgroups of 2t. The set of all basic subgroups of 2twil*
*l be
denoted by Birr(2t).
Definition 3.11. Let Bprod(n) denotes the family of all products of basic subgr*
*oups
in n, i.e.
Bprod(n) = {P1 x . .x.Pr : Pi2 Birr(2ti), n = 2t1+ . .+.2tr}.
Here follow two propositions which are consequences of [1, (2B)]:
Proposition 3.12. If t = t1 + . .+.ts, then
W 2t(B(t1, . .,.ts)) ' GL t1(F2) x . .x.GLts(F2).
HOMOTOPY REPRESENTATIONS OF SO(7) AND Spin(7) AT PRIME 2 9
Proposition 3.13. Let Pi 2ti, for i = 1, .P.,.r, be a collection of pairwise
non-isomorphic basic subgroups, and let k = ri=12tili. Then
_ r !
Y Yr
W k Pili = W 2ti(Pi) o li.
i=1 i=1
Theorem 3.14. Each 2-stubborn subgroup G k is, up to conjugacy, a product
of basic subgroups (i.e. G 2 Bprod(k)). A group P 2 Bprod(k) is stubborn if and
only if written as a product of basic subgroups P = P1 x . .x.Pr there is no fa*
*ctor
isomorphic to B(1, . .,.1) which occurs with multiplicity exactly 2 or 4.
Proof.The first statement is a consequence of [1, (2A)]. Since GL n(F2) has no *
*non-
trivial normal 2-subgroups and n has a non-trivial normal 2-subgroup if and on*
*ly
if n = 2 or n = 4, then the second statement follows immediately from Propositi*
*ons
3.12 and 3.13.
Proposition 3.15. If groups P, Q 2 Bprod(k) are conjugate, then there exists a
conjugacy between them which permutes its basic factors.
Proof.Since each basic subgroup of a symmetric group acts transitively on the
set of letters, basic factors of P (and, similarly, Q) are in bijection with th*
*e set
of P -orbits (Q-orbits). The conjugacy between P and Q permutes the orbits and
therefore it also permutes its basic factors.
Proposition 3.16. Fix collections of subgroups Pi ki for i = 1, . .,.r and
Hj ljfor j = 1, . .,.s. Assume that for each i the group Gi acts transitivel*
*y on
the set of letters, and that n := k1 + . .+.kr = l1 + . .+.ls. If
Q := Q1 x . .x.Qs P := P1 x . .x.Pr n,
then Q = (Q \ P1) x . .x.(Q \ Pr).
Proof.ForSeach i = 1, . .,.r let OPibe an orbit of Pi P . Note that {1, . .,.k*
*} =
r P Q
i=1Oi is a decomposition onto G-orbits. Similarly define Q-sets Oj , for j =
1, . .,.s. Since Q P , then for each j there exists i such that OQj OPi. Hen*
*ce
Qj P \ OPi= Pi and the conclusion follows.
Proposition 3.17. For any subgroup P k let ffi(P ) P denotes the subgroup
generated by all elements g 2 P which have a fixed point. The following holds:
o if P 2 Bprod(k) is a non-trivial product of basic subgroups, then ffi(P *
*) = P .
o ffi(B(t1, . .,.tr)) = B(t1, . .,.tr-1)2tr,
Proof.The product Q x Q0is generated by Q x {1} [t{1} x Q0. It implies that the
first statement holds, and that B(t1, . .,.tr-1)2 r ffi(B(t1, . .,.tr)). On th*
*e other
hand, for Ctr2 2trwe have ffi(Ctr2) = {1}. Then each element g 2 B(t1, . .,.t*
*r) \
B(t1, . .,.tr-1)2tracts freely on the set {1, . .,.2t} (where t = t1 + . .+.tr)*
*. As a
consequence we obtain the second statement.
Theorem 3.18. Let P, Q 2 Bprod(k). Assume that Q P . Then the inclusion
Q P is a composite of products of inclusions of the following types:
(a)B(t1, . .,.tr-1)2tr B(t1, . .,.tr-1, tr),
(b)B(t1, . .,.tj+ tj+1, . .,.tr) B(t1, . .,.tj, tj+1, . .,.tr).
10 KRZYSZTOF ZIEMIA'NSKI
Proof.If Q is reducible, then by Proposition 3.16 the inclusion is the product *
*of in-
clusions (Q\Pi) Pi, where Pi's are irreducible, so assume that P = B(t1, . .,*
*.tr).
If Q is reducible, then by Proposition 3.17 we obtain
tr
Q = ffi(Q) ffi(P ) = B(t1, . .,.tr-1)2 B(t1, . .,.tr-1, tr) = P.
In this case the inclusion is the composition of an inclusion of type (a) with Q
ffi(P ). Finally, if Q is irreducible then Q = B(t01, . .,.t0s). We have
t0s 2tr
ffi(Q) = B(t01, . .,.t0s-1)2 B(t1, . .,.tr-1) = ffi(P ).
Since Q-orbits are contained in P -orbits, then t0s tr. If t0s= tr, then we a*
*re
reduced to the case of smaller inclusion of irreducible subgroups. If t0s> tr, *
*then
the inclusion t0-t
B(t01, . .,.t0s-1)2 s r B(t1, . .,.tr-1)
factors through B(t1, . .,.tr-2)2tr-1. Hence t0s- tr tr-1 and then t0s tr-1+*
* tr.
Finally, we obtain the factorization
Q B(t1, . .,.tr-2, tr-1 + tr) B(t1, . .,.tr-2, tr-1, tr) = G.
As a consequence we obtain
Corollary 3.19. Each morphism in R2(O(n)) is a composition of automorphisms
and inclusions enlisted in 3.18.
A full inclusion R2( n) ! R2(O(n)).
Definition 3.20. For any P 2 Bprod(n) let ~P2 Tprod(n) be given by
P~= { 1} o Ct12o . .o.Ctr2 for P = B(t1, . .,.tr), t1 > 1
~P= O(2) o Ct22o . .o.Ctr2 for P = B(1, t2, . .,.tr)
P~= ~P1x . .x.~Pr for Pi2 Birr(ki)
Remark. For each P 2 Bprod(n) holds
(3.21) ~P\ { 1} o k = { 1} o P.
Theorem 3.22. The formulae
R2( n) 3 n=P 7! O(n)=P~2 R2(O(n))
Mor R2( n)(Q, P ) 3 gP 7! gP~2 MorR2(O(k))(Q~, ~P)
define the functor I : R2( n) ! R2(O(n)) which is an inclusion onto the full
subcategory.
Proof.The functor I is well-defined.
It is sufficient to check that for each generating morphism gP : Q ! P holds
g-1Q~g P~. It is clear for automorphisms (cf. Propositions 3.12 and 3.13), *
*so
assume that g = 1 and Q ! P is a product of the inclusions enlisted in Theorem
3.18. If the inclusion Q P is a non-trivial product of inclusions Q1 P1 and
Q2 P2, then ~Q ~Pif and only if ~Q1 ~P1and ~Q2 ~P2. Hence we are reduced
to the case when the inclusion is of type (a) or type (b) (cf. 3.18). If
tr
Q = B(t1, . .,.tr-1)2 B(t1, . .,.tr-1, tr) = P,
then for t1 > 1 we obtain
~Q= ({ 1} o Ct12o . .o.Ctr-12)2tr { 1} o Ct12o . .o.Ctr2= ~P
HOMOTOPY REPRESENTATIONS OF SO(7) AND Spin(7) AT PRIME 2 11
and for t1 = 1
~Q= (O(2) o Ct22o . .o.Ctr-12)2tr O(2) o Ct22o . .o.Ctr2= ~P.
If
Q = B(t1, . .,.tj+ tj+1, . .,.tr) B(t1, . .,.tj, tj+1, . .,.tr) = P,
then for j > 1 we obtain the inclusion
~Q= K o Ct22o . .o.Ctj+tj+12o . .o.Ctr2 K o Ct22o . .o.Ctj2o Ctj+12o . .o.Ct*
*r2= ~P,
where K = O(2) if t1 = 1 and K = { 1} o Ct12otherwise. Similarly if j = 1 and
t1 > 1, then the inclusion Q~ P~is straightforward. The only non-trivial case
appears when j = t1 = 1. Then
Q~= { 1} o C1+t22o . .o.Ctr2 O(2) o Ct22o . .o.Ctr2= ~P,
since { 1} o C1+t22 { 1} o C2 o Ct22 O(2) o Ct22. Hence I is well-defined.
The functor I is faithful.
By combining Propositions 3.6, 3.12 and 3.13 we see that for each subgroup P 2
Bprod(k) the homomorphism I : AutR2( k)(P ) ! AutR2(O(k))(P~) is actually an
isomorphism. Now fix Q 6= P 2 Bprod(k) and choose morphisms g1P, g2P : Q ! P
in the category of k-orbits. Let us consider the compositions
Q giP--!g-1iQgi-1.P-!P
for i = 1, 2. By Proposition 3.15 g-11Qg1 and g-12Qg2 differ by conjugation by *
*an
element h which permutes irreducible factors. Hence the conjugation by i(h), wh*
*ere
i : k ! O(k) is an obvious inclusion, sends the group g-11~Qg1 onto g-12~Qg2 a*
*nd
also permutes irreducible factors. By (3.21) h 2 P if and only if i(h) 2 ~P. It*
* shows
that g1P and g2P represent the same morphism Q ! P in R2( k) if and only if
they represent the same morphism in R2( k). As a consequence we get that I is
an isomorphism on sets of morphisms.
4. 2-Stubborn subgroups of G, "Gand their representations
2-Stubborn subgroups of O(7). By [9, Prop. 11], [9, Th. 12] and [7, Prop.
1.6.(i)] the functors
(4.1) R2(O(7)) 3 O(7)=P 7! G=(P \ G) 2 R2(G)
R2(G) 3 G=P 7! "G=P"2 R2(G")
are natural equivalences. We have
Tirr(1) = {{ 1}}, Tirr(2) = {O(2)},
Tirr(4) = {O(2) o C2, { 1} o C22, ~ 4, 4}.
Therefore
(4.2) Tprod(7) = {H x O(2) x { 1}, H x { 1}3}H2Tirr(4)
[ {O(2)ix { 1}7-2i}i=0,...,3.
All groups in Tprod(7) but O(2)2 x { 1}3 are 2-stubborn in O(7) (by 3.7). In-
troduce the following notation for 2-stubborn subgroups of G and "G. Let N :=
12 KRZYSZTOF ZIEMIA'NSKI
O(2) o C2, K := { 1} o C22, J := ~ 4and M := 4, and for each H 2 {J, K, M, N}
let
(4.3) H1 := (H x O(2) x { 1}) \ G (O(4) x O(2) x O(1)) \ G G
(4.4) H0 := (H x { 1}3) \ G (O(4) x O(3)) \ G G,
and for i = 0, 1, 3 let
(4.5) Li:= (O(2)ix { 1}7-2i) \ G G.
Then Ob(R2(G)) = {G=P } and Ob(R2(G")) = {G"=P"}, where P is conjugate to one
of Ji Ki, Mi, Ni (i = 0, 1) or Li (i = 0, 1, 3).
Here follows the list of Weyl groups of 2-stubborn subgroups of G:
(4.6) WG (N1) = 1, WG (N0) ~=WG (L3) ~=WG (K1) ~=WG (J1) ~= 3
WG (L1) ~= 5, WG (L0) ~= 7, WG (K0) ~=WG (J0) ~= 3 x 3
WG (M1) ~= 3 o 2, WG (M0) ~= 3 o 2 x 3
The set of morphisms of R2(O(7)) ~=R2(G) ~=R2(G") is generated by automor-
phisms and by inclusions presented on the following diagram:
N1u[^
A AACaeo|[
A A ae | [
A A ae |
L3u N0 K1 J1
u [[AC^ [[^ u
| aeoA|A Aaeo aeo |
| aeA | ae[ ae[ |
| aeAA | ae ae |
L1u K0 [^ J0u M1
| aeo [ | aeo
| ae [ | ae
| ae | ae
L0 M0
Representations. The remaining part of this section contains a partial classifi*
*ca-
tion of complex representations of discrete approximations of 2-stubborn subgro*
*ups
of G and "G. At the beginning let us introduce some notation. Let ` denote a tr*
*ivial
irreducible representation of any group and let ' be a non-trivial irreducible *
*repre-
sentation of an order 2 group. Every irreducible representation of a 2-discrete*
* torus
Z=21 ~=SO(2)1 is 1-dimensional and has the form
` '
(4.7) %k : Z=21 3 n_2t7! exp 2ssikn_2t2 U(1), k 2 Z^2
Finally, let
1 ^
(4.8) ffk:= indO(2)SO(2)1%k for k 2 1 + Z2
1 ^
(4.9) fik:= indO(2)SO(2)1%k for k 2 2Z2 \ {0}.
We have IR(O(2)1 ) = {ffk}k21+Z^2[ {fik}k22Z^2\{0}[ {`, o := det}.
HOMOTOPY REPRESENTATIONS OF SO(7) AND Spin(7) AT PRIME 2 13
Odd representations of subgroups of "G. Let be any locally finite subgroup
of G and let " := ss-1( ). An irreducible representation ' of " is
o even if it is a restriction of some representation of (and O'(u) = dim*
*')
o odd if it is not (in this case O'(u) = - dim')
The following lemma will be applied later to groups M"0and "L0:
Proposition 4.10. Let 2 O(4) be a locally finite group and let
":= ss-1(( x { 1}3) \ G) "G.
Then any odd representation of " is invariant under conjugation by an element
l 2 "Gsuch that ss(l) = -(I5 B) (cf. 3.1).
Proof.Let ' be an odd representation of " and let g 2 ". We have ss(g) = (h, t),
where h 2 O(4), t 2 { 1}3. If t = (1, 1, 1) or t = (-1, -1, -1), then g and l
commute and hence Ol*'(g) = O'(l-1gl) = O'(g). Otherwise l-1gl is conjugate to
l-1glu and g is conjugate to gu. Since ' is odd, we have O'(g) = O'(gu) = -O'(g*
*).
Therefore O'(g) = 0 (and similarly O'(l-1gl) = 0). Then representations ' and
l*' have equal characters and they are isomorphic.
Representations of L0 and L11. Let r be an odd positive integer. For each
sequence (~1, . .,.~r) 2 IR({ 1})r define
r
(4.11) o(~1,...,~r):= res{{1} 1}r\SO(r)~1~ . .~.~r.
L
Let jri:= o(~1,...,~r), where the sum is taken over all sequences in which ' *
*appears
exactly i times. Obviously jri~ jrr-i. Then each r-invariant representation *
*of
{ 1}r \ SO(r) (where r acts by permutations) is isomorphic to a direct sum of
jri's for 0 i < r_2. In particular, holds
Corollary 4.12. Any WG (L0) ' 7-invariant representation of L0 is isomorphic
to a direct sum of j7i's for i = 0, 1, 2, 3.
An isomorphism
O(2)1 x ({ 1} \ SO(5)) 3 (g, h) 7! (g, h . detg) 2 L11
provides a bijection IR(L11) ' IR(O(2)1 ) x IR({ 1}5 \ SO(5)); moreover, the
action of 5 ' WG (L1) on IR(L11) coincides with an action on IR({ 1}5\SO(5))by
permutations. Thus every 5-invariant representation of L11 is isomorphic to a
product of representations of the form ~ ~j5i, where ~ 2 IR(O(2)1 ), i = 0, 1, *
*2.
L Next we calculate restrictions of representations of L11 to L0. Let j7i(j):=
o(~1,...,~7), where the sum is taken over all sequences such that exactly j of
the representations ~1, ~2 are isomorphic to ' and exactly i of the representat*
*ions
~1, . .,.~7 are isomorphic to '. Elementary calculations of characters provide *
*the
following
Corollary 4.13. 5-invariant representations of L11 restrict to L0 as follows:
ffk~ j507! j71(1)` ~j507! j70 o ~j507! j72(0)
ffk~ j517! j72(1)` ~j517! j73(2)o ~j517! j71(0)
ffk~ j527! j73(1)` ~j527! j72(0)o ~j527! j73(0)
1 L1
Moreover, resL1L0fi2k~j5i' res1L0(` o) ~j5i.
14 KRZYSZTOF ZIEMIA'NSKI
Representations of M0 and M11. Isomorphisms
M x O(2)1 3 (g, h)7! (g, h, deth) 2 M11 G \ (O(4) x O(2) x O(1))
M x { 1}2 3 (g, h1, h2)7! (g, h1, h2, h1h2) 2 M0 G \ (O(4) x O(1)3)
provide an identifications IR(M11) ' IR(M x O(2)1 ) ' IR(M) x IR(O(2)1 ) and
IR(M0) ' IR(M x { 1}2) ' IR(M) x IR({ 1}2). Furthermore, the restriction
from M11 ' M x O(2)1 to M0 ' M x { 1}2 is product-wise.
Representations of K0. All irreducibleNrepresentations of K ' { 1}oC22are sub-
representations of indK{ 1}4~ a2C22~a for ~a 2 IR({ 1}) = {`, '}. By Mackey's
criterion representations
fl1 := indK{ 1}4` ~` ~` ~'
fl3 := indK{ 1}4` ~' ~' ~'
are irreducible. Furthermore, For ~, 2 IR(C2) = {`, '} define
2 ~, C2
fl~,0:= resC2K(~ ~ ), fl4 := res2K(~ ~ ) det.
There are decompositions
2 M ~,
ind{{1}oC2`1}4~` ~`'~` fl0
~, 2IR(C2)
2 M ~,
ind{{1}oC2'1}4~' ~''~' fl4 .
~, 2IR(C2)
For any a 2 C22\ {(0, 0)} let ia := ` ~` ~' ~' 2 IR({ 1}4) (where a is the
difference between coordinates with the same isomorphism4class of representatio*
*n).
Following [11, Section 8.2] we see that ind{{1} n1}4ia splits onto the sum o*
*f non-
isomorphic one-dimensional representations i+aand i-a. Moreover, for ffl 2 {+, *
*-}
representations flffl2,a:= indK{ 1}4nifflaare irreducible. Obviously indK{ *
*1}4ia '
fl+2,a fl-2,a. As a consequence we obtain
Corollary 4.14.
IR(K) = {fl1, fl3} [ {fl~,0, fl~,4}~, 2IR(C2)[ {flffl2,a}ffl2{+,-}a2C2*
*2\{(0,0)}.
Since the action of WO(4)(K) ~=GL 2(F2) ' 3 on K is natural, we have
Corollary 4.15. Here follow the orbits of an action of WO(4)(K) on IR(K):
{fl`,`0}, {fl`,'0, fl',`0, fl','0}, {fl`,`4}, {fl`,'4, fl',`4, fl','4}*
*, {fl1}, {fl3}
{fl+2,(0,1), fl+2,(1,0), fl+2,(1,1)}, {fl-2,(0,1), fl-2,(1,0), fl*
*-2,(1,1)}
Finally we show how WG (K0)-invariant representations of K0 restrict to L0. An
isomorphism
K x { 1}2 3 (g, h1, h2) 7! (g, h1, h2, h1h2) 2 K0 G \ (O(4) x O(1)3)
identifies IR(K0) and IR(K) x IR({ 1}2) (and obviously an action of WG (K0) '
3 x 3 is product-wise).LDenote oe := ` ~' ' ~` ' ~' 2 Rep({ 1}2). Finally,
let j7i[j]:= o(~1,...,~7), where the sum is taken over all sequences such tha*
*t ex-
actly j of the representations ~1, ~2, ~3, ~4 are isomorphic to ' and exactly i*
* of the
representations ~1, . .,.~7 are isomorphic to '. By comparing characters we obt*
*ain
HOMOTOPY REPRESENTATIONS OF SO(7) AND Spin(7) AT PRIME 2 15
Corollary 4.16. WG (K0)-invariant representations of K0 restrict to L0 as follo*
*ws:
fl~,0~`7! j70 fl~,0~oe7! j72[0]fl~,4~`7! j73[0]fl~,4~oe7! j71[0]
fl1~ `7! j71[1] fl1~ oe7! j73[1]fl3~ `7! j73[3] fl3~ oe7! j72[1]
flffl2,a~`7! j72[2]flffl2,a~oe7! j73[2]
for any ~, 2 IR({ 1}), a 2 C22\ {0, 0}, ffl 2 {+, -}.
5. Cohomology of Z^2[R2(L)]-modules
By an R[C]-module, where R is a commutative ring R and C is a small category,
we mean a contravariant functor M : C ! R - Mod and Hn(C; M) stands for a
higher limit limnCM. Let A be a ring of p-adic integers and let be a finite
group. For any A[ ]-module M let FM be an atomic Op( )-module with value M
concentrated on =1. Following [7, 5.3] define n( ; M) := Hn(Op( ); FM ). An
importance of groups * comes from the following
Theorem 5.1. ([7, 5.4]) Let L be a compact Lie group and M an atomic A[Rp(L)]-
module concentrated on an object L=Q. Then
H*(Rp(L); M) ~= *(Aut Rp(L)(L=Q); M(L=Q)) = *(WL(Q); M(L=Q)).
Jackowski, McClure and Oliver [7] provided the following inductive method of
calculation of groups *( , M).
Proposition 5.2. Let M be an A[ ]-module. Then
(1) [7, 6.1.(i)] If p divides | |, then 0( ; M) = 0. Otherwise 0( ; M) = M
and i( ; M) = 0 for i > 0,
(2) [7, 6.2.(ii)] Let p be a Sylow p-subgroup of and let ~ be the equival*
*ence
relation among p-Sylow subgroups generated by nontrivial intersection. S*
*et
:= {g 2 : g-1 pg ~ p}. Then 1( ; M) ~=M =M .
(3) [7, 5.2.(ii)] Define A[ ]-modules F0M( =P ) := MP and F00M( =P ) := F0M=*
*FM .
Then Hi(Rp( ); F0M) = 0 for i > 0 and H0(Rp( ); F0M) = M . As a con-
sequence,
i( ; M) = Hi(Rp( ); FM ) ' Hi-1(Rp( ); F00M) ' Hi-1(Rp( ) \ { =1}; F00M)
for i > 1 (Rp( ) \ { =1} stands for a full subcategory with =1 omitted).
If an A[Rp(L)]-module M is not atomic, then there is a spectral sequence
converging to H*(Rp(L); M). Choose a strictly decreasing map of posets ht :
Ob(Rp(L)) ! Z. Here follows reformulation of [6, 1.3].
Theorem 5.3. There is a spectral sequence E*,**:= E(Rp(L), M)*,**with the first
term M
Es,t1:= s+t(WL(P ); M(L=P ))
ht(L=P)=s
converging to H*(Rp(L); M).
A differential ds,t1: Es,t1! Es+1,t1is a sum
M s,t
ds,t1= d1 (P, Q),
ht(L=P)=s, ht(L=Q)=s+1
16 KRZYSZTOF ZIEMIA'NSKI
where ds,t1(P, Q) : s+t(WL(P ); M(L=P )) ! s+t+1(WL(Q); M(L=Q)) is a differ-
ential from a long exact sequence associated to a short exact sequence
0 -! M|{Q} -! M|{P,Q}-! M|{Q} -! 0,
The module M|X , for X Ob(Rp(L)) is
(
(5.4) M|X (L=P ) = M(L=P ) if L=P 2 X
0 if L=P 62 X
and M|X (L=P ! L=Q) = M(L=P ! L=Q) if L=P, L=Q 2 X (of course, this
definition is valid only for subsets X which are convex, i.e. for any sequence *
*L=P !
L=Q ! L=R such that L=P, L=R 2 X also L=Q 2 X).
Proposition 5.5. If ht(L=P )+1 = ht(L=1) = n, then dn-1,t1(P, 1) is a compositi*
*on
n-1+t(W (P ); M( =P )) -! n-1+t(W (P ); M( =1)P )
-! Hn-1+t(Op( ); F00M( =1)) ' n+t( ; M( =1))
Proof.A homomorphism M ! F0M( =1)which is an identity on =1 induces a
commutative diagram
ds,t1(P,1)
n-1+t(W (P ); M( =P ))______wn+t( ; M( =1))
||||||||| |
| ||||||||||||||||
| ||||||||||||| |
| ||||||||||||||||
|u ||||||||
n-1+t(W (P ); M( =1)P )____w n+t( ; M( =1))
The conclusion follows.
For any group let dimp( ) be the greatest integer n such that n( ; M) 6= 0
for some A[ ]-module M. By [7, p. 229] dim p( ) is less or equal to the rank of
p-Sylow-subgroup of . An application of a spectral sequence 5.3 to an A[Rp( )]-
module F00Mshows that
(5.6) dimp( ) 1 + max dim p(W (P )).
=P2Rp( )
The remaining part of the section contains calculations of groups *( ; M) for
some groups and modules M for p = 2.
Cohomology of R2( 3). Note that J := R2( 3) has two objects, namely 3=1
and 3=C2. By 5.2 and 5.6 we have
(
2=M 3 for n = 1
(5.7) n( 3; M) = M
0 for n 6= 1.
Define a height function of J by ht( 3=1) = 1, ht( 3=C2) = 0. For any A[J ]-
module M, the spectral sequence 5.3 degenerates to an exact sequence
0 -! H0(J ; M) -! M( 3= 2)
-d!M( 2 3 1
3=1) =M( 3=1) -! H (J ; M) -! 0,
where (by 5.5) d is the composition
(5.8) M( 3= 2) M(1-2)----!M( 3=1) 2 -i M( 3=1) 2=M( 3=1) 3.
HOMOTOPY REPRESENTATIONS OF SO(7) AND Spin(7) AT PRIME 2 17
Cohomology of R2( 5). By (3.14) there are, up to conjugacy, four stubborn
subgroups of 5, namely 1, C2, 1 o C22and C2 o C2. Its Weyl groups (in 5) are
isomorphic to 5, 3, 3 and 1 respectively.
Proposition 5.9. Let M be an A[ 5]-module. Then
(
2x 2x1=(M 2x 3 + M 4x1) for n = 2
n( 5; M) = M
0 for n 6=.2
Proof.By 5.2 and 5.6 we have n( 5; M) = 0 for n 6= 2 and 2( 5; M) ~=
H1(R2( 5); F00M). Introduce a height function on R2( 5) by ht( 5=(C2o C2)) = 0,
ht( 5=(1 o C22)) = ht( 5=C2) = 1 and ht( 5=1) = 2. The spectral sequence (5.3)
calculating H*(R; F00M) degenerates to an exact sequence
d0,01(1,C2) d0,01(1,1oC22)
0 -! H0(R; F00M) -! MC2oC2---------------!
MC2xC2=MC2x 3 MC2oC2=M 4 -! H1(R; F00M) -! 0.
Full subcategories with object sets respectively { 5=(C2 o C2), 5=(1 o C22)} a*
*nd
{ 5=(C2o C2), 5=C2} are both isomorphic to R2( 3). Thus by 5.5 d0,01(1, C2) is*
* a
composition
MC2oC2,! MC2xC2 i MC2xC2=MC2x 3
and d0,01(1, 1 o C22) is a natural projection. Thus 2( 5; M) ~= H1(R; F00M) '
M 2x 2x1=(M 2x 3 + M 4x1).
The groups *( 7; -). *-groups of 7 are more difficult to calculate.
Denote R := R2( 7), and let a = (12), b = (34), c = (56), s = (13)(24). By
3.14 there are 7 conjugacy classes of stubborn subgroups of 7, namely 1, C2, C*
*32,
K := 1 o C22(the Klein group), K x C2, D8 ' C2 o C2, D8 x C2. Its automorphism
groups are respectively 1 (for D8 x C2), 3 (for D8, C22and K x C2), 5 (for C2*
*),
3 x 3 (for K) and 7 (for 1).
Proposition 5.10. If M is an A[ 7]-module, then i( 7; M) = 0 for i 6= 2, 3.
Proof.For n = 0, 1 the conclusion is clear. Since dim2(Aut R( 7=P )) 2 for a*
*ll
orbits 7=P 2 R except 7=1, then for each n > 3
n( 7; M) ~=Hn-1(R; F00M) ~=Hn-1(R \ { 7=1}; F00M) = 0.
Proposition 5.11. i( 7; M(7, 1)) = 0 for each i 0.
Proof.It is enough to prove that for each 7=P 2 R and each n > 1 holds
nP:= n(Aut R( 7=P ); FM00(7,1)( 7=P )) = 0.
For P = L0 = 1 we have FM00(7,1)( 7=1) = 0 and for P = N1, N0, K1, L3 we have
dim2(P ) < 2 and therefore nP= 0 for n 2. Moreover
F00M(7,1)( 7=C2)= M(7, 1)C2 ~=A{x1 + x2} M(5, 1){xi}7i=3
F00M(7,1)( 7=K)~=A ~A{x1 + x2 + x3 + x4} A ~M(3, 1){xi}7i=5
Then, for i > 1, we have nL1= 0 by 5.9 and iK0= 0 by 5.7 and [7, 6.1.v].
Proposition 5.12. 3( 7; M(7, 2)) = 0.
18 KRZYSZTOF ZIEMIA'NSKI
Proof.Introduce a height function on R := R2( 7) by
8
>>>0for P = D8 x C2
<1 for P = D , C2, K x C
ht( 7=P ) = > 8 2 2
>>:2for P = C2, K
3 for P = 1
Let E*,**:= E(R2( 7), F00M(7,2))*,**. Since dim 2(W 7(P )) ht( 7=P ) for al*
*l 2-
stubborn P it is sufficient to prove that
E2,01= 2( 5; M(7, 2)C2) 2( 23; M(7, 2)K )
is killed by the differential d1,01. We have
M(7, 2)C2~=A{x12} M(5, 1){x1i+ x2i}7i=3 M(5, 2){xij}3 i
and the conclusion follows.
The groups *( 3 o C2; -). There are (up to conjugacy) three 2-stubborn sub-
groups of 3oC2, namely 1, 2 and 2oC2. Its Weyl groups are respectively 3oC2,
3 and 1. The full subcategory of R := R2( 3 o C2) containing 2 and 2 o C2 is
isomorphic to J = R2( 3).
Proposition 5.13. Let M be an A[ 3 o C2]-module. Then
(
2x 2=M 2x 3 + M 2oC2 for n = 2
n( 3 o C2; M) = M
0 for n 6=.2
Proof.Since the rank of 3oC2 is even, and the relation ~ from 5.2.(2) is trans*
*itive,
then n( 3 o C2; M) = 0 for n = 0, 1. For n > 1 we have
n( 3 o C2; M) ~=Hn-1(R2( 3 o C2); F00M) ~=Hn-1(J ; resRJFM00).
By 5.8 we have
(
2x 2=M 2x 3 + M 2oC2 for n = 2
Hn-1(J ; resRJFM00) ~= M
0 for n 6=.2
6. Homotopy representations of G and "G
Let R := R2(G) ~=R2(G"). In this section we prove that for every R-invariant
representation of N (resp. N") the map B'^2extends to a homotopy representation
of G (resp. G").
Denote A := Z^2. Let ' be an R-invariant representation of N and let := (')
be a module introduced in Section 2. By Theorem 2.5.(a) we need to prove that
H3(R; ) vanishes, and that cohomology of R with coefficient in any A[R]-module
vanish above dimension 4.
Proposition 6.1. For each G=P 2 R we have dim2(WG (P )) 3.
Proof.Each group WG (P ) is isomorphic to one of the following groups: 1, 3,
3x 3, 5, 3oC2, 3oC2x 3, 7 (cf. 4.6). We have dim2(1) = 0, dim2( 3) = 1
(by 5.7), dim2( 5) = 2 (by 5.9), dim2( 3oC2) 2 (by 5.13) and dim2( 7) 3
(by 5.10). Moreover, dim2( 3 x 3) = 2 and dim2( 3 o C2 x 3) = dim2( 3 o
C2) + 1 3.
As a consequence we obtain
Corollary 6.2. cdim2(R) 3.
In order to calculate H3(R; ) we use a spectral sequence (5.3). Define a hei*
*ght
function on R by putting
8
>>>0for P = N1
<1 for P = N , L , K , J
(6.3) ht(G=P ) = > 0 3 1 1
>>:2for P = L3, K0, J0, M1
3 for P = L0, M0
20 KRZYSZTOF ZIEMIA'NSKI
Let E*,**:= E(R; )*,**.
Proposition 6.4. H3(R; ) = coker(d2,01: E2,01! E3,01).
Proof.By 6.1 we have i(WG (P )) = 0 for i > ht(P ). Hence Es,t1vanishes if t >*
* 0
or if s > 3 (since 3 is a maximal value of the height function). Therefore E3,0*
*1is
the only rank 3 entry at the first table which possibly does not vanish and d2,*
*01the
only possibly non-trivial differential hitting E3,01.
Proposition 6.5. A homomorphism
d2,01(M1, M0) : 2( 3 o C2; (G=M1)) -! 3( 3 o C2 x 3; (G=M0))
(cf. 5.3) is an epimorphism.
1
Proof.Let {Xi}i2Ibe orbits of an action of 3oC2x 3 on IR(M0, resN1M0'). Each
Xihas the form YixZi, where Yiis an 3oC2-orbit in IR(M) and Ziis an 3-orbit
in IR({ 1}2) (see p.14). By [7, 6.1.v]
M
3( 3 o C2 x 3; (G=M0)) ~= 3( 3 o C2 x 3; A[Xi])
i2I M
' 2( 3 o C2; A[Yi]) 1( 3; A[Zi])
i2I
Each Zi is equal either to {` ~`}, or to {` ~', ' ~`, ' ~'}. In the first1case *
*we have
1( 3; A[Zi]) =.0In the second case, there is an orbit X0iin IR(M11, resN1M11')*
* '
IR( 14) x IR(O(2)1 ) such that X0i= Yix {ff2k+1} for some k 2 A. Furthermore,
the restriction of d2,01to 2( 3 o C2; A[X0i]) is a composition
2( 3 o C2; A[X0i]) -'! 2( 3 o C2; A[Yi0]) A{ff2k+1} 1xd--!
2( 3 o C2; A[Yi]) 1( 3; A{` ~', ' ~`, ' ~'}) 3( 3 o C2 x 3; (G=M0)).
The differential d : A[{ff2k+1}] ! 1( 3; A[{` ~', ' ~`, ' ~'}]) is an epimorph*
*ism by
5.8. Hence d2,01is an epimorphism.
Proposition 6.6. We have
F0M(7,3)(L1)' M(5, 1){x12k}k 3 M(5, 2){x1kl+ x2kl}k,l 3
M(5, 2){xklm}k,l,m 3
F0M(7,3)(K0)' (A ~A){x123+ x124+ x134+ x234}
(M(3, 1) ~M(3, 1)){x12k+ x34k, x13+ x24k, x14+ x23k}k 5
(A ~M(3, 1)){x1kl+ x2kl+ x3kl+ x4kl}k,l 5(A ~A){x567}
Proposition 6.7. A homomorphism
d2,01(L1, L0) d2,01(K0, L0) : 2( 5; (G=L1)) 2( 23; (G=K0)) -! 3( 7; (G=L0*
*))
is an epimorphism.
Proof.Let I : R2( 7) ! R2(O(7)) ~=R be a full inclusion from Theorem 3.22
and let F := F0 (G=L0). Let ~E*,**:= E(R2( 7), F)*,**. There is a homomorphism
of A[R2( 7)]-modules I* ! F0 (G=L0)which is an isomorphism on L0 and it
HOMOTOPY REPRESENTATIONS OF SO(7) AND Spin(7) AT PRIME 2 21
induces a transformation of spectral sequences E*,**! ~E*,**. In particular, th*
*ere is
a commutative diagram
2( 5; (G=L1)) 2( 23; (G=K0))___w 2( 5; (L0)C2) 2( 23; (G=L0)K )
| |
| |
|d2,0 |~d2,0
| 1 | 1
| |
|u |uu
3( 7; (G=L0)) _______________________ 3( 7; (G=L0))
(where K 4 7 is a Klein group). Since
1
resN1L0' ' (j70) l0 (j71) l1 (j72) l2 (j73) l3,
(cf. 4.12) then (G=L0) is a direct sum of some of the modules M(7, i), i =1
0, . .,.3, where M(7, i) appears as a summand of (G=L0) if and only if resN1L0'
contains a subrepresentation1isomorphic to j7i. From 5.11 and 5.12 follows that
3( 7; (G=L0)) = 0 if resN1L0' does not contain a subrepresentation isomorphic
to j73, so assume otherwise. Let xijk, 1 i < j < k 7 be the generators of t*
*he sub-
module of1 (G=L0) corresponding to j73. By 4.13 there exists a subrepresentation
of resN1L11' which is isomorphic either to o ~j52or to fi2k~j52for some k 2 A, *
*and
another subrepresentation isomorphic to ff2k0+1~j52, where k02 A. Then (G=L1)
contains the direct sum of two A[ 5]-submodules isomorphic to M(5, 2): the one
generated by irreducible subrepresentations of o ~j52(or fi2k~j52) maps onto the
summand
M(5, 2){xklm}3 k 0.
Theorem 6.10. Any R-invariant representation of N"11extends to a homotopy
representation of "G.
Proof.It follows from 6.1, 6.9 and 2.5.
References
[1]J. L. Alperin and P. Fong, Weights for Symmetric and General Linear Group*
*s, J. Alg.
131 (1990), 2-22
[2]A. K. Bousfield, The localization of spaces with respect to homology, Top*
*ology 14 (1975),
133-150
[3]A. K. Bousfield and D. M. Kan, Homotopy Limits, Completions and Localizat*
*ions, Lec-
ture Notes in Mathematics Vol. 304. Springer, New York (1972)
[4]W. G. Dwyer and C. W. Wilkerson, A new finite loop space at the prime 2, *
*J. AMS 6
(1993), 37-63
[5]W. G. Dwyer and C. W. Wilkerson, Homotopy fixed point methods for Lie gro*
*ups and
finite loop spaces, Ann. Math. 139 (1994), 395-442
[6]J. Grodal, Higher limits via subgroup complexes Ann. Math. 155 (2002), 40*
*5-457
[7]S. Jackowski, J. McClure and B. Oliver, Homotopy classification of self-m*
*aps of BG via
G-actions, Ann. Math. 135 (1992), 189-270
[8]S. Jackowski, J. McClure and B. Oliver, Self-homotopy equivalences of cla*
*ssifying spaces
of compact connected Lie groups, Fund. Math. 147(1995), 99-126
[9]B. Oliver, p-Stubborn subgroups of classical compact Lie groups, J. Pure *
*Appl. Algebra
92 (1994), 55-78
[10]B. Oliver, Higher limits via Steinberg representations, Comm. Alg. 22 (1*
*994), 1381-1393
[11]J.-P. Serre Repr'esentations Lin'eaires des Groupes Finis, Hermann, Pari*
*s, 1967
[12]D. Winter Representations of locally finite groups, Bull. Amer. Math. So*
*c. 74 (1968)
145-148
[13]Z. Wojtkowiak, On maps from hocolimF to Z, Algebraic Topology, Barcelona*
*, 1986,
Lecture Notes in Math. 1298, Springer-Verlag, 1987, 227-236
[14]K. Ziemiaski, A faithful unitary representation of the 2-compact group D*
*I(4) Ph.D. thesis,
Uniwersytet Warszawski, 2005