TORIC MORPHISMS BETWEEN p-COMPACT GROUPS
JESPER M. MOLLER
Abstract.It is well-known that any morphism between two p-compact groups *
*will lift, non-
uniquely, to an admissible morphism between the maximal tori. We identify*
* here a class of p-
compact group morphisms, the p-toric morphisms, which can be perceived as*
* generalized rational
isomorphisms, enjoying the stronger property of lifting uniquely to a mor*
*phism between the max-
imal torus normalizers. We investigate the class of p-toric morphisms and*
* apply our observations
to determine the mapping spaces map(BSU(3); BF4), map(BG 2; BF4), and map*
*(BSU(3); BG 2)
where the classifying spaces have been completed at the prime p = 3.
1. Introduction
The classification up to homotopy of maps between classifying spaces of compa*
*ct Lie groups
is a traditional project of algebraic topology [13, 21]. One line of developmen*
*t started with the
investigations 25 years ago by Hubbuck [10, 11] and Adams-Mahmud [1]. They note*
*d the close
relationship between maps between classifying spaces and admissible homomorphis*
*ms between
maximal tori. The regular admissible homomorphisms, in particular, turned out t*
*o have especially
nice properties. It is the purpose of this paper to study regular admissible m*
*orphisms, here
called toric admissible morphisms, in light of the more recent theory by Dwyer-*
*Wilkerson [6] of
p-compact groups. As case studies, we classify homotopy homomorphisms SU(3) -!F*
*4, G2 -!F4,
and SU(3) -!G 2at the prime p = 3.
In order to describe the content in more detail, let X1 and X2 be p-compact g*
*roups, for the sake
of this introduction assumed to be connected, with maximal tori T (X1) -!X1 and*
* T (X2) -!X2,
respectively. For any morphism f :X1 ! X2there is a lift T (f): T (X1) ! T (X2)*
*, unique up the
action of the Weyl group of X2, such that the diagram
T(f)
T (X1)____//T (X2)
i1|| i2||
fflffl| fflffl|
X1 ___f___//_X2
commutes up to conjugacy. The morphism T (f) is admissible in the sense that fo*
*r any element
w1 of the Weyl group of X1 there exists and element w2 of the Weyl group of X2 *
*such that
T (f)w1 = w2T (f). In general, w2 is not uniquely determined by w1, but if it *
*is, we say (2.1)
that f is p-toric. (If f is p-toric, the centralizer CX2(fi1T (X1)) of the maxi*
*mal torus of X1 in X2
will be isomorphic to the maximal torus of X2.) In that case, the correspondenc*
*e w1 -!w2 is a
homomorphism of Weyl groups and, by Theorem 3.5, there is a unique lift N(f): N*
*(X1) ! N(X2)
to a map between the maximal torus normalizers such that the diagram
N(f)
N(X1) _____//N(X2)
| |
| |
fflffl| fflffl|
X1 ___f___//_X2
commutes up to conjugacy.
As a first example, we consider the case where the domain X1 = SU(3), the cod*
*omain X2 =
F4, and the prime p = 3. The compact Lie group F4 contains a unique copy of SU*
*(3; 3) =
___________
Date: March 6, 1998.
1991 Mathematics Subject Classification. 55R35, 55S37.
Key words and phrases. Weyl group, Quillen category, Bousfield-Kan spectral s*
*equence.
1
2 J. M. MOLLER
SU(3) xZ(SU(3))SU(3) as a subgroup of maximal rank. Any morphism SU(3) -!SU (3;*
* 3) is of the
form u v
(u;v):SU(3) -! SU(3) x SU(3) --x--!SU (3) x SU(3) -!SU (3; 3)
where u and v are 3-adic units or zero. Composing with the inclusion e: SU(3; 3*
*) ! F4we obtain
the morphism e (u;v):SU(3) ! F4. Observe that e (u;v)= e (-u;-v)since the incl*
*usion e is
invariant under the action of the Weyl group WF4(SU (3; 3) which is of order tw*
*o generated by the
self-map -1 xZ(SU(3)) -1 of SU(3; 3). These maps e (u;v), u; v 2 Z*3[ {0}, wit*
*h the relation
e (u;v)= e (-u;-v), turn out to describe Rep(SU (3); F4) = [B SU(3)^3; (B F4)^3*
*] completely.
Theorem 1.1. The map
e O -: WF4(SU (3; 3))\ Rep(SU (3); SU(3; 3)) ! Rep(SU (3); F4)
is a bijection when p = 3.
See (4.16, 5.7, 6.7) for information about the centralizers of these maps. T*
*he proof of The-
orem 1.1 is divided into three cases: Monomorphisms SU (3) -! F4 (4.13), p-tor*
*ic monomor-
phisms PU(3) -!F4 (5.4), and, the technically most demanding case, non-p-toric *
*monomorphisms
PU(3) -!F4 (6.1).
As a second example, we consider the case where X1 = G2 and X2 = F4and p = 3 *
*and reprove
a result from Jackowski-McClure-Oliver [14]. To state the theorem, we recall th*
*at the compact Lie
group G2 contains a unique copy of SU(3) as a subgroup of maximal rank. Thus we*
* may restrict
morphisms defined on G2 to this subgroup SU(3) G2.
Theorem 1.2. [14, 3.4] The restriction map
Rep(G 2; F4) -!Rep(SU (3); F4)
is a bijection when p = 3.
When working with this paper, I made use of a MAGMA program written by K. And*
*ersen
for computing admissible homomorphisms. I also wish to thank Chuck McGibbon for*
* a clarifying
remark.
2.Toric morphisms
In this section I introduce the concept of a p-toric morphism, relate it to o*
*ther, more familiar,
types of morphisms between p-compact groups, and provide examples of morphisms *
*that are p-toric
and others that are not.
Let X1 and X2 be p-compact groups (or extended p-compact tori) with maximal t*
*ori T1 =
T (X1) ! X1, T2 = T (X2) ! X2 and Weyl groups W1 = W (X1) and W2 = W (X2), resp*
*ectively.
Write Rep(X1; X2) for the set [BX1; BX2] of conjugacy classes of loop space mor*
*phisms.
Definition 2.1. 1.A loop space morphism T1 ! X2 is p-toric (or regular [1, 2.22*
*], [14, 1.3])
if its centralizer CX2(T1) is a p-compact toral group.
2.A loop space morphism X1 ! X2 is p-toric if its composition with T1 ! X1 is*
* p-toric.
Note that the centralizer CX2(T1) in (2.1.1) is known to be a p-compact group*
* [6, x6] [7, 2.5].
We shall now consider some alternative criteria for a morphism to be p-toric.
For any loop space morphism f :X1 ! X2 between p-compact groups or extended p*
*-compact
tori there exists a loop space morphism T (f): T1 ! T2between the maximal tori *
*such that
T(f)
(2.2) T1_____//T2
i1|| |i2|
fflffl|fflffl|
X1__f__//X2
commutes up to conjugacy. As W2T (f) = f1|T1 2 W2\ Rep(T1; T2) = Rep(T1; X2) is*
* an invariant
of f, T (f) in Rep(T1; T2) is uniquely determined up to the action of the Weyl *
*group W2. In
particular, the stabilizer subgroup
W2T(f)= {w2 2 W2 | w2. T (f) = T (f)}
TORIC MORPHISMS 3
at T (f) for the action of W2 on Rep(T1; T2) is determined up to conjugacy by f.
In case X1 and X2 are extended p-compact tori, there is a short exact sequenc*
*e of loop spaces
T2 -!CX2(T (f)T1) -!W2T(f)
from which we see that
f :X1 ! X2 is p-toric, W2T(f)= ss0(CX2(T1)) is a p-group:
In case X1 and X2 are p-compact groups, CX2(T1) -!X2 is a monomorphism of max*
*imal rank
[7, x4], so
f :X1 ! X2 is p-toric, CX2(T1)0 -!X2 is a maximal torus
where subscript 0 indicates identity component. (If X2 is connected, the centra*
*lizer of T1 is also
connected [20, 3.11] [7, 7.8] so in this case f is p-toric if and only if CX2(T*
* (f)T1) -! X2 is a
maximal torus for X2 if and only if the stabilizer W2T(f)is trivial [17, 3.2].)
We consider also an enlarged version of diagram (2.2) in the form of the diag*
*ram
T(f)
(2.3) T1_____//|T2
| | 0
| |i2
| fflffl|
i1|| N2
| |
| |j2
fflffl|fflffl|
X1 __f__//X2
where j2:N2 ! X2is the normalizer [6, 9.8] of the maximal torus. Using that CN2*
*(T1) ! CX2(T1)
is a maximal torus normalizer, we get [17]
(2.4) f is p-toric, T1 i1-!X1 f-!X2 is p-toric
(2.5) , CN2(T1) ! CX2(T1) is an isomorphism
0
(2.6) ) T1 T(f)---!T2 i2-!N2 is p-toric
(2.7) ,The isotropy subgroup W2T(f)is a finite p-group.
If p > 2, also the converse of the third implication holds because for odd p a *
*p-compact group is
a p-compact toral group if and only if its Weyl group is a finite p-group [15].
In some cases, see e.g. [17] or (3.5) below, it is possible to lift f to a lo*
*op space morphism N(f)
between the maximal torus normalizers such that
N(f)
(2.8) N1_____//N2
j1|| |j2|
fflffl|fflffl|
X1__f__//X2
commutes up to conjugacy. In this situation
(2.9) f is p-toric) N(f) is p-toric
and for p > 2 also the converse holds. (Use (2.5, 2.6) to see this.)
In the following examples and elsewhere
oTRep (X1; X2) Rep(X1; X2) denotes the set of conjugacy classes of p-toric *
*morphisms
oMono (X1; X2) Rep(X1; X2) denotes the set of conjugacy classes of monomorp*
*hisms
oTMono (X1; X2) = Mono(X1; X2) \ TRep(X1; X2)
o"Q(X1; X2) Rep(X1; X2) is the set of rational isomorphisms
o"Q(X1) = "Q(X1; X1) is the monoid of rational automorphisms of X1
oOut(X1) is the group of conjugacy classes of automorphisms of X1.
Above, a loop space morphism between extended p-compact tori is a monomorphism *
*if its discrete
approximation [7, 3.12] is a monomorphism.
4 J. M. MOLLER
Example 2.10. If X1 and X2 have the same rank,
Mono (X1; X2) TRep(X1; X2) "Q(X1; X2)
because any monomorphism (rational isomorphism) restricts to an isomorphism (ep*
*imorphism)
between maximal tori.
If X1 and X2 are locally isomorphic, simple p-compact groups
TRep(X1; X2) = Rep(X1; X2) - {0} = "Q(X1; X2)
because f is p-toric or a rational isomorphism if and only if T (f) is non-triv*
*ial if and only if f is
non-trivial.
Example 2.11. For any p-compact group X and any integer m > 0, TRep(X; Xm )*
* =
(TRep (X; X))m . If X is simple,
TRep(X; Xm ) = (Rep(X; X) - {0})m = "Q(X)m p||W|=Out(X)m ;
where the last identity holds under the assumption that p divides the order of *
*the Weyl group.
Proposition 2.12.Assume that X1 is connected and that z :Z1 ! X1is a central mo*
*nomor-
phism. Then there are bijections
oRep(X1=Z1; X2) -!{f 2 Rep(X1; X2) | f O z is trivial}
oTRep (X1=Z1; X2) -!{f 2 TRep(X1; X2) | f O z is trivial}
induced by the epimorphism X1 -!X1=Z1.
In fact, map(B(X1=Z1); BX2) is homotopy equivalent to a union of connected co*
*mponents of
map(BX1; BX2).
Proof.The epimorphism of X1 to X1=Z1 induces a homotopy equivalence b*
*etween
map(B(X1=Z1); BX2) and a collection of components of map(BX1; BX2). This shows *
*the injec-
tion of sets of representations and when applied with X1 replaced by T1, it sho*
*ws that a morphism_
X1 -!X2 is p-toric if and only if its composition with the epimorphism X1 -!X1=*
*Z1 is p-toric. |__|
Proposition 2.13.Assume that X1 is simply connected, X2 is connected, and that *
*z :Z2 ! X2
is a central monomorphism. Then there are bijections
oRep(X1; X2) -!Rep(X1; X2=Z2)
oTRep (X1; X2) -!TRep (X1; X2=Z2)
induced by the epimorphism X2 -!X2=Z2.
Proof.Obstruction theory (remember that BX1 is 3-connected) shows that Rep(X1; *
*X2) =
Rep(X1; X2=Z2) and the existence of a short exact sequence of p-compact groups
K -!CX2(X1) -!CX2=Z2(X1)
where BK is one component of the homotopy fixed point set BZhX12; in particular*
* K is a p-compact __
toral group. It follows that CX2(X1) is a p-compact toral group if and only if *
*CX2=Z2(X1) is. |__|
Example 2.14. For any simply connected, simple p-compact group X and any centra*
*l monomor-
phism Z -!Xm ,
TRep(X; Xm =Z) = TRep(X; Xm ) = "Q(X)m p||W|=Out(X)m
where the last identity holds if p divides the order of the Weyl group.
Example 2.15. Let p be an odd prime and let SU(p; p) denote the quotient of SU(*
*p) x SU(p) be
the central subgroup generated by (iE; i-1E) where i 6= 1 is a pth root of unit*
*y. Then (2.13)
Rep(SU (p); SU(p; p)) = Rep(SU (p); SU(p) x Rep(SU (p); SU(p))
TRep (SU (p); SU(p; p)) = Out(SU (p)) x Out(SU (p))
and Rep(SU (p); SU(p)) = Z*p[ {0}. Relative to this identification
(2.16) Mono(SU (p); SU(p; p)) = {(u; v) 2 (Z*p[ {0})2 | u + v 2 Z*p}
TORIC MORPHISMS 5
for the morphism (u;v)defined as the composition
ux v
SU(p) -! SU(p) x SU(p) ----! SU (p) x SU(p) -!SU (p; p)
is a monomorphism if and only if u + v 2 Z*p. The monoid Rep(SU (p; p); SU(p; *
*p)) is (2.13)
isomorphic to a submonoid of Rep(SU (p) x SU(p); SU(p) x SU(p)) and, in particu*
*lar,
Out(SU (p; p)) = {(u; v) 2 Z*px Z*p| u v mod p} o
where o is the automorphism that swaps the two SU(p)-factors.
The set of monomorphisms (2.16) consists of two orbits, represented by (1;1)*
*and (1;0), un-
der the action of the automorphism group Out(SU (p; p)). It follows that the ce*
*ntralizers of the
monomorphism (u;v)are
(
(2.17) CSU(p;p)( (u;v)SU(p)) ~= Z(SU (p))if u 6= 0 and v 6= 0
SU(p) if u = 0 or v = 0
i.e. that (u;v)is centric precisely when it is p-toric. (To prove that (1;1)i*
*s centric one uses the
fact that Z(SU (p)) -! Z(SU (p) x SU(p)) -! Z(SU (p; p)) is an isomorphism of c*
*enters.) In the
non-toric case, observe that the projection morphism SU(p) x SU(p) -!SU (p; p) *
*restricts to (1;0)
on the first factor and to (0;1)on the second factor. This gives a factorizati*
*on
SU (p) -!CSU(p;p)( (1;0)SU(p)) -!SU (p; p)
of (0;1)through the centralizer of (1;0)where the first map is an isomorphism*
*. We conclude that if
f :SU(p) ! SU(p; p)is a non-toric monomorphism, so is the evaluation monomorphi*
*sm SU(p) =
CSU(p;p)(f SU(p)) -! SU(p; p). The Weyl group, WSU(p;p)( (u;v)SU(p)), of any m*
*onomorphism
(u;v)is trivial [18, 5.6].
Finally, we note that by (2.12),
Rep(PU (p); SU(p; p)) = {(u; v) 2 (Z*p[ {0})2 | u + v 2 pZp}
TRep(PU (p); SU(p; p)) = {(u; v) 2 (Z*p)2 | u + v 2 pZp}
so that Rep (PU (p); SU(p; p)) = {0} [ Mono(PU (p); SU(p; p)) and Mono (PU (p*
*); SU(p; p)) =
TRep(PU (p); SU(p; p)).
Lemma 2.18. Let f :X ! Y1 be any morphism and g :Y1 ! Y2a monomorphism between *
*p-
compact groups. Then
g O f :X ! Y2is p-toric) f :X ! Y1is p-toric:
Proof.Let T be a maximal torus of X1. Since composition with Bg, CY1(fiT ) ! CY*
*2(gfiT ), is a
monomorphism, CY2(gfiT ) is a p-compact toral group if CY1(fiT ) is a p-compact*
* toral_group [20,
3.5.(1)]. *
*|__|
The converse of (2.18) is not true in general; take for instance Y1 to be the*
* maximal torus of Y2.
3. Lifting p-toric morphisms
In this section I show that all p-toric morphisms between two p-compact group*
*s lift uniquely to
p-toric morphisms between the maximal torus normalizers.
Recall that X1 and X2 are p-compact groups or extended p-compact tori and tha*
*t j1:N1 ! X1
and j2:N2 ! X2 are normalizers of the respective maximal tori, i1:T1 ! X1and i2*
*:T2 ! X2.
By the very definition of a p-toric morphism, the maps j1 and j2 induce maps
(3.1) TRep(X1; X2) -!TRep (N1; X2) - TRep (N1; N2)
of sets of p-toric representations. Our first objective is to prove that the ar*
*row to the right is a
bijection. This will enable us to define a map from TRep(X1; X2) to TRep(N1; N*
*2). Note the
favorable input provided by the information [17, 3.2] that
(3.2) TRep (T1; X2) - TRep (T1; N2)
is a bijection and
(3.3) CX2(T1) - CN2(T1)
6 J. M. MOLLER
an isomorphism for any p-toric morphism T1 -!N2.
For any set S Rep(X1; X2), write map(BX1; BX2)S for the space of all maps BX*
*1 -!BX2
homotopic to a member of S.
Lemma 3.4. The map, induced by j2,
map(BN1; BX2)TRep(N1;X2)-map (BN1; BN2)TRep(N1;N2)
is a homotopy equivalence.
Proof.The map of the lemma is the map on homotopy fixed point spaces
hW1
map(BN1; BG2)TRep(N1;G2)= map(BT1; BG2)TRep(T1;G2) ; G2 = N2; X2;
induced by the map
map(BT1; BX2)TRep(T1;X2)-map(BT1; BN2)TRep(T1;N2)
which is known to be a homotopy equivalence (3.2, 3.3). *
* |___|
This lemma immediately leads to the main result of this section.
Theorem 3.5. (Cf. [1, 2.22]) Let X1 and X2 be p-compact groups and f :X1 ! X2 a*
* p-toric
morphism. Then there exists a morphism N(f): N1 ! N2 between extended p-compact*
* tori such
that
N(f)
N1_____//N2
j1|| |j2|
fflffl|fflffl|
X1__f__//X2
commutes up to conjugacy. Moreover,
oN(f) is unique up to conjugacy
oN(f) is p-toric
oCX2(fj1N1) - CN2(N(f)N1) is an isomorphism of loop spaces
Proof.The map
(3.6) N :TRep (X1; X2) ! TRep(N1; N2)
is defined as the composition of the map TRep (X1; X2) -! TRep (X1; N2) with th*
*e inverse of
the bijection TRep(N1; X2) - TRep(N1; N2) from (3.1). That N(f) is p-toric is *
*(2.9) and_the
isomorphism of centralizers is (3.4). *
* |__|
Example 3.7. If X is simple and N -! X the normalizer of the maximal torus, th*
*e map
TRep(X; Xm ) -!TRep (N; Nm ) is injective if "Q(X) -!Rep(N; N) is injective; e.*
*g. if X = PU(p),
X = G2 and p = 3, or X = DI(2) and p = 3.
The above theorem is intended as a tool to facilitate the computation of TRep*
* (X1; X2) in
concrete cases. We now address injectivity of (3.6).
Remark 3.8. According to the homology decomposition theorem of Jackowski-McClur*
*e [12] and
Dwyer-Wilkerson [5], the exists an Fp-equivalence
hocolimAopBCX1() ! BX1
where the homotopy colimit is taken over some full subcategory A of the Quillen*
* category A(X1).
Let us assume that
oAny object :V ! X1 of A admits a factorization : V ! T1 through the maxim*
*al torus
and
oN :TRep (CX1(); X2) ! TRep(CN1(); N2)is injective for all objects :V ! X1 *
*of A
and let now f and f0 be two p-toric morphisms with N(f) = ' = N(f0) for some ' 2
TRep(CN1(); N2). Then the two possible compositions
e() __f_//_
CX1() _____//X1___//_X2
f0
TORIC MORPHISMS 7
are again p-toric morphisms for CX2(fe()CT1()) = CX2(fi1T1) is a p-compact toru*
*s and similarly
for the other morphism f0. Since also,
N(f O e()) = ' O e() = N(f0O e())
we have f O e() ' f0O e() for all objects of A by hypothesis. (Here, e(): CX (*
*) ! X stands
for the evaluation monomorphism.) The obstructions to constructing a homotopy b*
*etween Bf and
Bf0 lie in
limiAssi(map (BCX1(); BX2)B(fOe())); i 1
which is an abelian group for i > 1 but just a set if i = 1 and the fundamental*
* groups are
non-abelian.
It is possible that (3.8) can be generalized to a more general situation usin*
*g the preferred lifts
of [16].
While (3.8) applies to the case where X1 is center-free, the following lemma *
*can be helpful if
X1 has a non-trivial center.
Consider the following situation
Z APPP
AAPPPPz2
z||z AAAPPPPP
fflffl|1A PP''P
X Y1__g_//_Y2
of p-compact groups and loop space morphisms. Let Rep(X; Y1)z!z1 = {f 2 Rep(X;*
* Y1) | f O
z = z1} denote the set of conjugacy classes of morphisms under Z and map(BX; BY*
* )z!z1 the
corresponding mapping space.
Lemma 3.9. Assume that z :Z ! X is a central monomorphism into the connected p-*
*compact
group X and that composition with Bg is an isomorphism g_:CY1(z1Z) ! CY2(z2Z)of*
* centralizers.
Then composition with Bg,
Bg O -: map(BX; BY1)z!z1! map(BX; BY2)z!z2
is a homotopy equivalence.
Proof.The spaces BCYi(zi) = map(BZ; BYi)Bzi; i = 1; 2, are X=Z-spaces and comp*
*osition with
Bg, Bg_:BCY1(z1) ! BCY2(z2), is an X=Z-map inducing a map
(3.10)
map(BX; BY1)z!z1= BCY1(z1)h(X=Z)! BCY2(z2)h(X=Z)= map(BX; BY2)z!z2
of homotopy fixed point spaces. If g_is a homotopy equivalence, so is (3.10). *
* |___|
Here is a typical application of (3.9). In the diagram
V HSSS
HHSSSSS z
z1|| _HHHHHSS2SSSSS
fflffl|z2##H SSSS))S
X1 CX2(V )e(V_)//_X2
V is an elementary abelian p-group, z1 a central monomorphism, z2 a monomorphis*
*m and _z2
the canonical factorization of z2 through its centralizer. Since the evaluatio*
*n monomorphism
CX2(V ) -!X2 clearly satisfies the hypothesis of (3.9) we see that
(3.11) map(BX1; BCX2(V ))z1!_z2! map(BX1; BX2)z1!z2
is a homotopy equivalence.
Definition 3.12.Let R be a subset of Rep(X1; X2). We say that R is T -determine*
*d if the impli-
cation
f|T (X1) = g|T (X1) ) f = g
holds for all f 2 R and all g 2 Rep(X1; X2).
8 J. M. MOLLER
Example 3.13. If the order of W (X1) is prime to p, then
(3.14) Rep(X1; X2) = W (X2)\ Adm(T (X1); T (X2))
where Adm (T (X1); T (X2)) consists of all OE 2 Rep(T (X1); T (X2)) with the pr*
*operty that for all
w1 2 W (X1), OEw1 = w2OE for some w2 2 W (X2). Thus Rep(X1; X2) is T -determine*
*d in this case.
The bijection (3.14) follows by exploiting the H*Fp-equivalence BN(X1) -!BX1.
Remark 3.15. Let S1 -! G1 -! ss0(G1) and S2 -! G2 -! ss0(G2) be two extensions *
*of finite
groups, ss0(G1) and ss0(G2), by p-compact tori, S1 and S2. Let Hom (G1; G2) = *
*[BG1; *; BG2]
denote the set of based and Rep(G1; G2) = [BG1; BG2] = ss0(G2)\ Hom(G1; G2) the*
* set of free
homotopy classes of maps of BG1 into BG2.
The two functors ss1 and ss2 define a map
(3.16) Hom (G1; G2) ! Hom (ss0(G1);ss0(G2))(S1; S2)
into the set Hom (ss0(G1);ss0(G2))(S1; S2) of pairs (O; OE) 2 Hom (ss0(G1); ss0*
*(G2)) x Hom(S1; S2) such
that OE is O-equivariant. The fibre over (O; OE) is either empty or in bijectio*
*n with the set
(3.17) ss0(map (BS1; BS2)ss0(G1)BOE) = H2(ss0(G1); ss2(BS2)) = H1O(ss0(G1);*
* S2)
where ss0(G1) acts on S2, the discrete approximation to S2, through O.
If we put w2 . (O; OE) = (w2Ow-12; w2OE) for all w2 2 ss0(G2) and *
* all (O; OE) 2
Hom (ss0(G1);ss0(G2))(S1; S2) then (3.16) becomes ss0(G2)-equivariant, so it de*
*scends to a map
(3.18) Rep(G1; G2) ! ss0(G2)\ Hom(ss0(G1);ss0(G2))(S1; S2)
of ss0(G2)-orbit sets. The fibre over the orbit ss0(G2)(O; OE) is either empty *
*or in bijection with the
orbit set
ss0(G2)(O;OE)\H1O(ss0(G1); S2)
for the action of the stabilizer group ss0(G2)(O;OE), consisting of all w2 2 ss*
*0(G2) such that w2O = Ow2
and w2OE = OE, on the fibre (3.17).
Proposition 3.19.Let (O; OE) be an element of Hom(ss0(G1);ss0(G2))(S1; S2) and *
*suppose that the
stabilizer subgroup ss0(G2)(O;OE)acts transitively on the cohomology group H1O(*
*ss0(G1); S2). Then at
most one element of Rep(G1; G2) is mapped to the orbit ss0(G2)(O; OE) under the*
* map (3.18).
The rest of the paper consists of an analysis of the special case where X1 = *
*SU (3) or G 2,
X2 = F4, and the prime p = 3.
4. Embeddings of SU(3) in F4
In this section we apply the concepts of the previous sections to investigate*
* monomorphisms
from SU(3) to F4at the prime p = 3. First, a few facts about the Quillen catego*
*ry A(F4) of F4.
Lemma 4.1. [23, 8.2.2] Let E1 be an elementary abelian group of order 31. The s*
*et Mono(E1; F4)
of conjugacy classes of monomorphisms of E1 into F4has three elements e11; e12;*
* e13. The centralizers
of these three elements are connected 3-compact groups with Weyl groups of orde*
*r 36, 48, and 48,
respectively. The centralizer CF4(e11) of e11is isomorphic to SU(3; 3). The a*
*utomorphism group
Aut(E1) acts trivially on Mono(E1; F4).
Lemma 4.2. (Cf. [23, 8.2.4], [22, 7.5]) Let E2 be an elementary abelian group o*
*f order 32. The
set Mono(E2; F4)= Aut(E2) of isomorphism classes of conjugacy classes of monomo*
*rphisms of E2
into F4 has 5 elements, e21; e22; e23; e24; e25, with Quillen automorphism grou*
*ps of order 8, 4, 12, 12,
48, and with centralizer Weyl groups of order 4, 6, 6, 8, 3, respectively. The *
*centralizer, CF4(e25),
of e25is a 3-compact toral group of maximal rank with component group ss0(CF4(e*
*25)) of order 3.
There are no maps in the Quillen category from e12or e13to e25.
Proofs of (4.1) and (4.2).With computer assistance it is easy to determine, usi*
*ng [19, 2.5] and
[17, 3.2], that Mono(E1; F4) is a trivial Aut(E1)-set containing three elements*
* whose centralizers
are connected 3-compact groups with Weyl groups of order 36, 48, 48, respective*
*ly. See [14, 3.3]
for the precise structure of CF4(a). Since each centralizer of E1 is connected,*
* any monomorphism_
E2 -!F4 will factor through the maximal torus. *
* |__|
TORIC MORPHISMS 9
The Quillen automorphism group referred to in (4.2) consists of all automorph*
*ism of E2 that
leaves e2i2 Mono(E2; F4) invariant.
We now show that for any monomorphism of SU(3) or SU(3; 3) to F4the triangles
(4.3) xE1 AA vE1 AA
zxxxx AAe11AA zvvvvv AAe11AA
__xxx AA --vvv AA
SU (3)//_________//_F4 SU (3; 3)//________//_F4
where z :E1 ! SU(3)and z :E1 ! SU(3; 3)are centers, will commute up to conjugac*
*y. This
observation is the key to the classification of monomorphisms of SU(3) ae F4.
Lemma 4.4. 1.Mono(SU (3); F4)z!e11= Mono(SU (3); F4).
2.Mono (SU (3; 3); F4)z!e11= Mono(SU (3; 3); F4).
The proof of this lemma uses admissible homomorphisms which we now discuss.
The Weyl group W1 = W (SU (3)) of SU(3) is [18, 2.6] o Aut(0(Z33)) where*
* 0(Z33) is the
free Z3-module with basis (1; -1; 0); (0; 1; -1) 2 Z33and
oe = 01 -1-1 ; o = -01 -10
The Weyl group W (F4) = W (F4) < GL(4; Z3) of F4is the group (of order 1152 =*
* 384 . 3)
(4.5) W (F4) = W (B4)E [ W (B4)H1[ W (B4)H2
where W (B4) is the reflection group (of order 384 = 24 . 4!) of all signed per*
*mutation matrices,
and H1 and H2 are the matrices
0 1 0 1
-1 1 1 1 1 1 1 1
B 1 -1 1 1 C 1B 1 1 -1 -1 C
H1 = 1_2B@1 1 -1 1 CA; H2 = _2B@1 -1 1 -1 CA
1 1 1 -1 1 -1 -1 1
satisfying H21= E = H22; H2H1 = -H2; H1H2 = diag(-1; 1; 1; 1)H1.
We say that a linear map A: 0(Z33) ! Z43is admissible if AW (SU (3)) W (F4)A*
*. The linear
map A(u; v): 0(Z33) !,Z43for instance, with matrix
0 1 0 1 0 1
-u v -1 0 0 1
B u v - uC B 1 -1 C B 0 1 C
(4.6) A(u; v) = B@ 0 v + uCA = u B@ 0 1CA+ v B@ 0 1 CA; u; v 2 Z3;
-2v v 0 0 -2 1
is admissible since it is O-equivariant where O: W (SU (3)) ! W (F4)is the grou*
*p homomorphism
given by
0 1 0 1
-1 -1 1 -1 -1 -1 1 1
B 1 -1 -1 -1 C 1B -1 1 -1 1 C
(4.7) O(oe) = 1_2B@-1 1 -1 -1 CA; O(o) = _2B@ 1 -1 -1 1 CA
1 1 1 -1 1 1 1 1
The next lemma classifies the admissible homomorphisms. Note that A(u; v) and -*
*A(u; v) lie in
the same orbit under the action of W (F4) as -E 2 W (F4).
Lemma 4.8. 1.Let A: 0(Z33) ! Z43be a linear map. Then A is admissible with *
*respect to
W (SU (3)) and W (F4) if and only if A 2 W (F4)A(u; v) for some 3-adic inte*
*gers u; v 2 Z3.
2.A(u; v) is split injective if and only if u + v is a 3-adic unit.
3.The map
<(-1; -1)>\(Z3)2-!W (F4)\ HomZ3(0(Z33); Z43)
(u; v)-! W (F4)A(u; v)
is injective.
10 J. M. MOLLER
Proof.1. It is possible to show, using a computer, that, up to inner automorphi*
*sms, any admissible
homomorphism 0(Z33) -!Z43must be O-equivariant. Given this, one simply solves t*
*he system of
linear equations Aw = O(w)A for A where w runs through a generating set for W (*
*SU (3)).
2. The matrix A(u; v) is equivalent to the matrix
0 1
u - 2v 0
BB 0 2v - uCC
@ 3u 0 A
-u v
which is split injective if and only if u - 2v or, equivalently, (u - 2v) + 3v *
*= u + v is a 3-adic unit.
3. The claim is that for any w in W (F4) the set of solutions to the homogene*
*ous system of linear
equations
wA(u1; v1) - A(u2; v2) = 0
in the four unknowns (u1; v1; u2; v2) is contained in the diagonal (u1; v1) = (*
*u2; v2) or in the_
anti-diagonal (u1; v1) = -(u2; v2). This is easily verified on a computer. *
* |__|
Our interest in the admissible homomorphisms lies in the fact that the induce*
*d homomorphism
ss1(T (f)) is admissible for any lift T (f): T (SU (3)) ! Tt(F4)o the maximal t*
*ori of any morphism
f :SU(3) ! F4. Thus we must have ss1(T (f)) 2 W (F4)A(u; v) for some 3-adic int*
*egers u and v.
However, as we shall shortly see, not all the homomorphisms A(u; v) are induced*
* in this way from
morphisms SU(3) -!F4.
The proof of (4.4) follows immediately from (4.8.1).
Proof of Lemma 4.4.1. Let f :SU(3) ! F4be any monomorphism. Then ss1(T (f)) is *
*admissible,
so we may assume that ss1(T (f)) = A(u; v) for some 3-adic integers u; v 2 Z3. *
* The restriction
fz :E1 ! F4of f to the center z :E1 ! SU(3)of SU(3) is given by
0 1
B u + vC
(4.9) A(u; v) -11 = B@uu++vvCA
0
where we have reduced modulo 3. Since fz is a monomorphism, u + v 6 0 mod 3 and*
* then the
stabilizer in W (F4) of (u + v; u + v; u + v; 0) 2 (Z=3)4 has order 36. Thus fz*
* ' e112 Mono(E1; F4).
2. Let f :SU(3; 3) ! F4be any monomorphism and choose some monomorphi*
*sm __
g :SU(3) ! SU(3; 3)such that gz = z, e.g. g = (1;0). Then fz = fgz = e11. *
* |__|
Let e: SU(3; 3) = CF4(e11) !dF4enote the inclusion of the centralizer of e11i*
*nto F4; this map
is described in detail in [14, 3.3].
Corollary 4.10.The maps
Mono (SU (3); SU(3; 3))z!zeO---!Mono(SU (3); F4)
Out(SU (3; 3))z!zeO---!Mono(SU (3; 3); F4)
are bijections.
Proof.By (3.9) and (4.4),
Mono(SU (3); SU(3; 3)z!z = Mono(SU (3); F4)z-!e11= Mono(SU (3); F4)
and similarly for morphisms from SU(3; 3). *
* |___|
Lemma 4.11. Let (u;v):SU(3) ! SU(3; 3)be the morphism (2.15) indexed by u; v 2*
* Z*3[ {0}.
Then W (F4)ss1(T (e (u;v))) = W (F4)A(u; v).
Proof.The monomorphism e: SU(3; 3) ! F4is [14, 3.3] realizable on the level of *
*compact Lie
groups as an inclusion SU(3) ,! F4such that the restriction 0(Z3)x0(Z3) ! 2(Z4)*
* to the inte-
gral lattices of the composite morphism SU(3)xSU (3) i SU(3; 3) ,! F4takes (x1;*
* x2; x3; y1; y2; y3)
TORIC MORPHISMS 11
to (x1+ y3; x2+ y3; x3+ y3; y1- y2). Thus
0 1 0 1 0 1
1 0 0 -1 u 0 u -v
BB -1 1 0 -1 CCBB 0 u CC BB-u u - vCC
@ 0 -1 0 -1 A @ v 0 A = @ 0 -u - vA = -A(u; v)
0 0 2 -1 0 v 2v -v
represents ss1(T (e (u;v))). *
* |___|
Lemma 4.12. Let u and v be 3-adic integers and A(u; v) the corresponding admiss*
*ible homomor-
phism.
1.There exists a morphism f :SU(3) ! F4such that W (F4)ss1(T (f)) = W (F4)A(u*
*; v) if and
only if both u and v are in the set Z*3[ {0}.
2.There exists a monomorphism f :SU(3) ! F4such that W (F4)ss1(T (f)) = W (F4*
*)A(u; v) if
and only if u; v 2 Z*3[ {0} and u + v 2 Z*3.
Proof.We have already seen (4.11) that A(u; v) is realizable for all u; v 2 Z*3*
*[ {0}.
Suppose, conversely, that ss1(T (f)) = A(u; v) for some 3-adic integers, u an*
*d v, and some
morphism f :SU(3) ! F4. If f is a monomorphism, then f = e (u;v)for some u; v 2*
* Z*3[ {0}
with u + v 2 Z*3by (4.10). If f is not a monomorphism, A(u; v) is not split inj*
*ective [19,_5.2]_[20,
3.6.1], so u + v is not a 3-adic unit (4.8.2). *
* |__|
Theorem 4.13. 1.Mono (SU (3); F4) is T -determined.
2.The map
<(-1; -1)>\{(u; v) 2 (Z*3[ {0})2|u +-v!2MZ*3}ono(SU (3); F4)
(u; v) -!e (u;v)
is a bijection.
Proof.1. The restriction map Mono(SU (3); F4) -! Mono(T (SU (3)); F4) can be id*
*entified to the
map
{(u; v) 2 (Z*3[ {0})2 | u + v 1-mod!3}W (F4)\ Hom(0(Z33); Z43)
(u; v)-!W (F4)A(u; v)
which is injective by (4.8.3). *
* __
2. This is immediate from (2.16) and (4.10). *
* |__|
Here is an alternative formulation of (4.10): Consider the commutative diagra*
*ms
Mono(SU (3); SU(3;W3))z!z_//< -1 x -1>\ Mono(SU (3); SU(3; 3))
WWWWWWWW~=WWW |
eO-WWWWWWWW++W eO-fflffl||
Mono(SU (3); F4)
Out(SU (3; 3))z!z_________//_W< -1 x -1>\ Out(SU (3; 3)
WWWWWWWW~=WWW |
eO-WWWWWWWW++W eO-fflffl||
Mono (SU (3; 3); F4)
where the slanted arrows are bijections. The vertical arrows exist because e( -*
*1 x -1) = e by
[14, 3.3]. Noting (2.15) that
Mono (SU (3); SU(3; 3))z!z= {(u; v) 2 (Z*3[ {0})2 | u + v 1 mod 3}
Out(SU (3; 3))z!z= {(u; v) 2 (Z*3)2 | u 1 v mod 3} o
12 J. M. MOLLER
we see that the vertical arrow in each of the diagrams is a bijection, too, and*
* hence that the vertical
arrow of the upper (lower) diagram is a bijection of right Out(SU (3))- (Out(SU*
* (3; 3))-) sets. Thus
the action
(4.14) Mono(SU (3; 3); F4) x Out(SU (3; 3)) ! Mono(SU (3; 3); F4)
is transitive and the stabilizer subgroup at the centric monomorphism e, i.e. t*
*he Weyl group [18,
5.6]
(4.15) WF4(e SU(3; 3)) = < -1 x -1>
is cyclic of order two.
The next lemma lists the centralizers of all monomorphisms SU(3) ae F4. We le*
*t -1 denote
the automorphism -1 xZ(SU(3)) -1 of T (SU (3) xZ(SU(3))SU(3) [17, 4.3].
Lemma 4.16. Let (u; v) 2 (Z*3[ {0})2 and u + v 2 Z*3. If uv 6= 0, then
CF4(e (u;v)SU(3)) = Z(SU (3))
CF4(e (u;v)T (SU (3))) = T (F4)
If uv = 0, then
CF4(e (u;v)SU(3)) = Z(SU (3) xZ(SU(3))SU(3)
CF4(e (u;v)T (SU (3))) = T (SU (3) xZ(SU(3))SU(3)
In all cases, CF4( -1) = -1.
Proof.It only remains to determine the map CF4( -1) induced by -1 since the ce*
*ntralizers
themselves are given by (2.17, 3.9). Let us, for example, consider the case whe*
*re (u; v) = (0; 1).
Consider the morphism : (SU (3) x T (SU (3))) x T (SU (3)) -! SU (3) x T (SU (3*
*)) -! SU (3; 3)
constructed from the multiplication on the maximal torus and the projection map*
*. Since
e((1 x 1) x -1) = e( -1 x -1)((1 x 1) x -1) = e(( -1 x -1) x 1)
it follows from (4.17) that CF4( -1) = -1 on CF4(e (0;1)T (SU (3))). The other*
*_cases are similar.
|__|
Lemma 4.17. If the diagram of p-compact groups
1xf2 f1x1 0 0
X1x X2 oo___X1x X02 ____//_X1x X2
|| 0||
fflffl| fflffl|
Y __________h__________//_Y 0
commutes up to conjugacy, so does the induced diagram
ad()
X1____________//CY (X2)
f1|| |Ch(f2)|
fflffl|ad(0) fflffl|
X01__________//CY 0(X02)
where the horizontal arrows are adjoints of and 0.
Corollary 4.18.Let N be a (topological) group with subgroups g1:G1 ! N and g2:*
*G2 ! N .
Suppose that n 2 N is an element such that conjugation with n, c(n)(m) = nmn-1,*
* m 2 N,
takes G1 into G2. Then conjugation with n-1 takes the centralizer CN (G2) into *
*CN (G1) and the
diagram
BCN (G1)O____//_map(BG1;OBN)Bg1OO
Bc(n-1)|| |____Bc(n)|
| |
BCN (G2) ____//_map(BG2; BN)Bg2
commutes up to homotopy.
TORIC MORPHISMS 13
Proof.We have (c(n) x 1) = c(n)(1 x c(n-1)) where is group multiplication and *
*where the_
induced map Bc(n): BN ! BN is homotopic to the identity. *
* |__|
5. Toric representations of PU (3) in F4
In this section I classify the p-toric morphisms from PU(3) to F4 viewed as 3*
*-compact groups.
The first step is the determination of the admissible homomorphisms.
Let X be a connected p-compact group with maximal torus i: T ! X. We want to *
*describe
the integral lattice of the central quotients of X. Suppose that Z is a subgrou*
*p of the discrete
approximation T = (ss1(T ) Q)=ss1(T ) such that the composition Z ! T ! X is a*
* central
monomorphism. Then we may form the p-compact group X=Z [6, 8.3] with induced ma*
*ximal torus
i=Z :T=Z ! X=Z[20, 4.6] that fits into the commutative diagram
0 _____//_-1(0)______//-1(Z)_______//_Z____//_0
|| | |
|| | |
|| fflffl| fflffl|
0 _____//_ss1(T_)___//ss1(T )__Q___//_T____//_0
| |~ |
| |= |
fflffl| fflffl| fflffl|
0 ____//_ss1(T=Z)_//_ss1(T=Z) _Q__//_T=Z___//_0
with exact rows. From this we get an isomorphism
0 ____//_ss1(T_)_//ss1(T=Z)__//Z____//0
|| | ||
|| | |
|| |fflffl fflffl|
0 ____//_-1(0)___//-1(Z)_____//Z____//0
of extensions of WT(X) = WT=Z(X=Z)-modules.
In particular, let 0(Z33) 0(Q33) be the free Z3-submodule with basis e1 = (1*
*; -1; 0) and
e2 = (0; 1; -1); this is the integral lattice for SU(3). Put f = 1_3(e1- e2) an*
*d let P 0(Z33) be the
free Z3-submodule of Q33with basis {e1; f}. Then there is an exact sequence
0 -!0(Z33) -!P 0(Z33) -!Z=3 -!0
of Z3[3]-modules and P 0(Z33) corresponds to the maximal torus for PU(3).
Note that there is an extension, B(u; v), of A(u; v),
0(Z33)____//_P 0(Z33)
ss
A(u;v)||sssss
fflffl|B(u;v)yysss
L4
if and only if u + v is divisible by 3 and in that case the extension is unique*
* and given by
0 1_ 1
-1 B -u -3(u1+_v)C
B(u; v) = A(u; v) 10 -13 = B@ u0 -3(2u1-_v)CA
3(u + v)
-2v -v
where u and v are 3-adic integers and u + v 2 3Z3. Moreover, the inclusion is*
* W (SU (3)) =
W (PU (3))-equivariant and B(u; v) is O-equivariant where O is the group homomo*
*rphism from
(4.7).
Lemma 5.1. 1.A Z3-linear map B :P 0(Z33) ! Z43is admissible with respect to *
*W (PU (3))
and W (F4) is and only if B 2 W (F4)B(u; v) where u and v are 3-adic intege*
*rs whose sum
is divisible by 3.
2.B(u; v) is split-injective when u and v are 3-adic units.
14 J. M. MOLLER
3.The map
<(-1; -1)>\{(u; v) 2 (Z3)2|u + v!2W3Z3}(F4)\ HomZ3(P 0(Z33); Z43)
(u; v)! W (F4)B(u; v)
is injective.
Proof.1. B is admissible if and only if B O is, i.e. if and only if B is an ext*
*ension of A(u; v) (4.8.1)
for some 3-adic integers, u and v.
2. If u and v are units then
-u-1 0 u-1 0
2u-1 0 2u-1 -v-1
is a left inverse of B(u; v).
3. If B(u1; v1) 2 W (F4)B(u2; v2) then also A(u1; v1) 2 W (F4)A(u2; v2) and t*
*hen (4.8.3)_(u1; v1)
and (u2; v2) are equal up to sign. *
* |__|
__(*
*u;v)
When u; v 2 Z*3[{0} with sum u+v 2 3Z3 there is a unique conjugacy class, *
* , that makes
the diagram
(u;v) e
SU(3)____//_SU(3;_3)_//_F499
| tttt
| t__(u;v)tt
fflffl|tt
PU (3)
__(u;v)
commutes up to conjugation. By construction, W (F4)ss1(T (eO (T (PU (3)))))*
* = W (F4)B(u; v).
Lemma 5.2. Let u and v be 3-adic integers with sum u + v 2 3Z3 and let B(u; v):*
* P 0(Z33) ! Z43
be the corresponding admissible homomorphism.
1.There exists a morphism f :PU (3) ! F4such that W (F4)ss1(T (f)) = W (F4)B(*
*u; v) if and
only if u = 0 = v or u; v 2 Z*3.
2.There exists a monomorphism f :PU (3) ! F4such that W (F4)ss1(T (f)) = W (F*
*4)B(u; v) if
and only if u; v 2 Z*3.
Proof.We have already seen that W (F4)B(u; v) is realizable by a morphism f :PU*
* (3) ! F4if
u = 0 = v or u; v 2 Z*3; if both u and v are non-zero then f is a monomorphism *
*by (5.1.2).
Conversely, if W (F4)B(u; v) is realizable, so is W (F4)A(u; v) and then (4.12)*
* u;_v 2 Z*3[ {0},
u + v 62 Z*3. *
* |__|
Alternatively, (5.2) says that any non-trivial morphism PU(3) -!F4 is a monom*
*orphism.
Proposition 5.3.(Cf. [1, 2.27.(ii)]) Suppose that u and v are 3-adic units wit*
*h u + v 2 3Z3.
Then
T (PU (3)) B(u;v)----!T (F4) i2-!F4
is toric if and only if (u; v) 62 Z*3(2; 1) [ Z*3(1; -1).
Proof.Explicit (computer aided) computations of W (F4)B(u;v)= W (F4)A(u;v). *
* |___|
The two generic non-3-toric morphisms
0 1 0 1
-2 -1 -1 0
B 2 1 C B 1 1 C
B(2; 1) = B@ 0 -1 CA and B(1; -1) = B@ 0 0 CA
-2 -1 2 1
are related by the equation "B(2; 1) = 2B(1; -1) where
0 1
0 0 -1 1
B 0 0 -1 -1 C
" = B@-1 -1 0 0 CA
-1 1 0 0
is the admissible automorphism of Z43corresponding to the exotic automorphism o*
*f F4. (In general,
W (F4)("A(u; v)) = W (F4)(A(2v; -u)), cf. [1, 2.11].)
TORIC MORPHISMS 15
Theorem 5.4. 1.TRep (PU (3); F4) is T -determined.
2.The map
<(-1; -1)>\({(u; v) 2 (Z*3)2 | u + v 2 3Z3}) \ (Z*3(2;!1)T[RZ*3(1;e-1)))p(P*
*U (3); F4)
__(u;v)
(u; v) ! e O
is a bijection.
Consider the set Rep(N(PU (3)); N(F4)) of conjugacy classes of maps from the *
*maximal torus
normalizer N(P U(3)) of PU (3) to the maximal torus normalizer N(F4) of F4. As *
*we have seen
(3.18), there is a map
Rep(N(PU (3)); N(F4)) ! W (F4)\ Hom(W(PU(3));W(F4))(T (PU (3)); T (F4))
induced by the functors ss1 and ss2. It is easy to calculate directly that the*
* cohomology group
H2(; ss1(T (F4))) is trivial. Then also
(5.5) H2O(W (PU (3)); ss1(T (F4))) = 0
for is a Sylow 3-subgroup of the Weyl group of PU(3) and we get
Lemma 5.6. There is at most one element of Rep(N(PU (3)); N(F4)) corresponding *
*to the orbit
W (F4)(O; B(u; v)), (u; v) 2 (Z*3)2, u + v 2 3Z3.
Proof of Theorem 5.4.Let f1; f2 2 T Rep(PU (3); F4) be two toric representation*
*s and suppose
that their restrictions to the maximal torus of PU(3) agree. Under the map
TRep (PU (3); F4) -!TRep (N(PU (3)); N(F4)) -!W (F4)\ Hom(W(PU(3));W(F4))(T (PU*
* (3)); T (F4))
f1 and f2 go to the same element of the target and it follows (5.6) that the li*
*fts (3.5) N(f1) and
N(f2) are conjugate, i.e. that f1 and f2 have conjugate restrictions to the max*
*imal torus normalizer
N(PU (3)). In fact, N(f1) = B(u; v) o O = N(f2) for some (u; v) 2 (Z*3)2\ (Z*3(*
*2; 1) [ Z*3(1; -1)).
We may approximate BPU (3) by a homotopy colimit over a category I = I(SL(2; *
*F3); S3) (a
full subcategory of the Quillen category that may be described as formed from t*
*he inclusion of a
Sylow 3-subgroup S3 into the special linear group SL(2; F3)) with just two obje*
*cts, : E1 ! PU(3)
and :E2 ! PU(3), where E1 and E2 are elementary abelian groups of order 3 and *
*32, respec-
tively [12, 6.8, 7,7], see [19] for the notation used here. Since f1 and f2 agr*
*ee on the centralizers,
CPU(3)(E1) = N3(PU (3)) and CPU(3)(E2) = E2, it only remains to compute the rel*
*evant Wo-
jtkowiak obstruction groups [24]. For this we need information about the centra*
*lizer CF4(fiE2)
and CF4(fiN3(PU (3))).
We must have f1|E2 = e25= f2|E2 for only e252 Mono(E2; F4) can contain in its*
* automorphism
group the automorphism group SL(2; F3) of (E2; ). Thus CF4(fiE2) is a p-compact*
* toral group
of maximal rank with E1 as its component group (4.2).
The centralizer CF4(fiN3(PU (3))) is (3.4) the p-compact toral group
CT(F4)oW(F4)(T (PU (3)) o ) = T(F4)= t(F4)= E2
where t(F4) T(F4) denotes the maximal elmentary abelian subgroup of the discre*
*te approxima-
tion T(F4) to T (F4) and
The obstructions to a homotopy between the two maps Bf1; Bf2:BPU (3) ! BF 4li*
*e in the
abelian groups lim1Iss_1and lim2Iss_2where ss_1and ss_2are the abelian I-groups
______________________________________________SL(2;F3*
*)=S3______________________________________________
Z=2_5E25_______________________________________________*
*_______//_E1SL(2;F3)ii________________________________________________@
____________________________________________________*
*____________SL(2;F3)=S3_4_____________________________________________@
Z=2__:0:_______________________________________________*
*____________________//_Z3SL(2;F3)hh___________________________________@
given by the homotopy groups of the above centralizers. The group SL(2; F3) ha*
*s no normal
subgroups of index two, so it necessarily acts trivially on E1. It now follows *
*from [18, 3.1] that
both obstruction groups are trivial and we conclude that f1 and f2 are conjugat*
*e. This shows that
TRep(PU (3); F4) is T -determined.
Let now f :PU (3) ! F4be any toric monomorphism. Then there is (5.1.3, 5.3) *
*a unique,
up to sign, pair of units (u; v) 2 (Z*3)2, u + v 2 3Z3, (u; v) 62 Z*3(2; 1) [ Z*
**3(1; -1), such that
16 J. M. MOLLER
__(u;v)
W (F4)ss1(T (f)) = W (F4)B(u; v) and then f = since the p-toric monomorphi*
*sms_are T -
determined. |*
*__|
Lemma 5.7. Let (u; v) 2 (Z*3)2, u + v 2 3Z3, (u; v) 62 Z*3(2; 1) [ Z*3(1; -1). *
*Then
CF4(e (u;v)SU(3))= T(F4)O(W(SU(3)))
CF4(e (u;v)T (SU (3)))= T (F4)
and CF4( -1) = -1 in both cases.
Proof.Since e (u;v)is toric, the centralizer in F4of e (u;v)T (SU (3)) equals t*
*he maximal torus of
F4. Proceed as in (4.16) to show that CF4( -1) = -1.
__(u;v)
The centralizer BCF4(e PU (3)) is the homotopy colimit of the I-space
__________________________SL(2;F3)=S3_________________*
*_________
Z=2_B(0)--__________________________________________//_B(*
*1)SL(2;F3)qq___________________________________________
where B(0) = BT (F4)and B(0) = BCF4(e25). We need to be more specific ab*
*out the group
actions that occur here.
The 3-normalizer N3(PU (3)) = CN(PU(3))(T (PU (3))) is the centralizer in*
* N(PU (3)) of
T(PU (3))= E1. Since conjugation by (0; o) restricts to the non-trivial au*
*tomorphism of
T(PU (3))we see that the induced action on N3(PU (3)) = T (PU (3)) o i*
*s given by con-
jugation with (0; o) 2 N(PU (3)) = T(PU (3)) o W (PU (3)).
Since B(u; v) o O: N3(PU (3)) ! N(F4)is O-equivariant whith the Weyl groups a*
*cting by con-
jugation, we see (4.17) that Z=2-acts on T(F4)= CN (F4)(N3 (PU (3))) as *
*conjugation with
(0; O(o)). With this information it is now easy to see, using [18, 3.1], that
lim0Iss1 = (T (F4))= T(F4)O(W(SU(3)))
is the only non-trivial contribution from the I-groups ss_1and ss_2to the Bousf*
*ield-Kan spectral
sequence. This means that the morphisms
__(u;v) __(u;v) __(u;v)
CF4(e PU (3)) -!CF4(N(e )(N(PU (3)))) - CN(F4)(N(e )(N(PU (3))*
*))
are isomorphisms. Consider the corresponding group homo*
*morphism
: T(F4)O(W(SU(3)))x N(SU (3)) ! N(F4)which is the inclusion on the first *
* factor and
equals N(e (u;v)) on the second factor. Since -1 o 1 is inner on N*
*(F4), we have
(1 x ( -1 o 1)) = ( -1 o 1)(1 x ( -1 o 1)) = ( -1 x (1 o 1)) up to inner au*
*to-
morphism. This shows (4.17) that CF4( -1) = -1 is the non-trivial automorp*
*hism_of
CF4(e (u;v)SU(3)) = E1. *
*|__|
6.Non-toric morphisms of PU (3) to F4
The non-toric morphisms of PU (3) to F4 require special treatment. It is the*
* object of this
section to show that also the non-toric morphisms are T -determined, i.e. to co*
*mplete the proof of
the following theorem.
Theorem 6.1. 1.Mono (PU (3); F4) is T -determined.
2.The map
<(-1; -1)>\{(u; v) 2 (Z*3)2|u + v-2!3Z3}Mono(PU (3); F4)
__(u;v)
(u; v)-! e
is a bijection.
Since the toric morphisms were dealt with in (5.4) only the non-toric ones ne*
*ed be considered
in order to finish the proof of (6.1).
The first lemma, which is of a general nature, assures the existence of a kin*
*d of preferred lifts
in certain situations.
Let G be a p-compact toral group sitting in short exact sequence S -i1!G -!ss*
*0(G) where S is
a p-compact torus and ss0(G) cyclic p-group. Let j :N ! X be the maximal torus *
*normalizer of a
TORIC MORPHISMS 17
p-compact group, X, and let i2:T ! N be the inclusion of the identity component*
*. Suppose that
we are given a morphisms, B and f, such that the diagram
S __B_//_T
i1|| ji2||
fflffl|fflffl|
G __f_//_X
commutes up to conjugacy and B is admissible in the sense that for any 2 ss0(G*
*) there exists
some w in the Weyl group for X such that B = wB.
Lemma 6.2. Assuming that the component group ss0(G) is cyclic there is a unique*
* representation
OE 2 Rep(G; N) such that the diagram
"T??
""
"" i2||
B """ fflffl|
""" N>>
"" OE""
""" """ j||
"" "" fflffl|
S __i1_//G_f_//_X
commutes up to conjugacy and such that the morphism
Cj:CN (OEG) ! CX (fG);
induced by j, is a maximal torus normalizer for the centralizer CX (fG) of G in*
* X.
Proof.The ss0(G)-map induced by j
BCN (i2BS)hss0(G)______________//PBCX (ji2BS)hss0(G)
PPPP nnnnn
PPPP nnnn
PP((P vvnnn
Bss0(G)
between the ss0(G)-spaces BCN (i2BS) = map (BS; BN)i2B and BCX (ji2BS)*
* =
map(BS; BX)ji2Bis a maximal torus normalizer. There is an induced map
(6.3)
map(BG; BN)i1!i2B= BCN (i2BS)hss0(G)! BCX (ji2BS)hss0(G)= map(BG; BX)i1!ji2B
of homotopy fixed point spaces.
According to [16, 4.6], the section Bf 2 BCX (ji2BS)hss0(G)admits, since ss0(*
*G) is assumed to be
cyclic, a unique lift BOE 2 BCN (i2BS)hss0(G)such that the restriction of (6.3)*
* to the corresponding
components,
BCN (OEG) = map(BG; BN)BOE! map(BG; BX)Bf = BCX (fG)
is a maximal torus normalizer for the p-compact group CX (fG). *
* |___|
After these general and preparatory remarks, we now return to the discussion *
*of non-toric
morphisms from PU(3) to F4.
Let f :PU (3) ! F4be a morphism of 3-compact groups such that f|T (P*
*U (3)) =
W (F4)B(2; 1) 2 [BT (PU (3)); BF 4]. By (6.2), there is a unique OE(2; 1) 2 Rep*
*(N3(PU (3)); N(F4)),
extending B(2; 1), such that CN(F4)(OE(2; 1)N3(PU (3))) is a maximal to*
*rus normalizer for
CF4(fN3(PU (3))). We shall now determine this map OE(2; 1).
Let N3 = T1o and N2 = T2o W2 be the discrete approximations to the the 3*
*-normalizer
N3(PU (3)) and the maximal torus normalizer N(F4), respectively. Also, let B(2;*
* 1): T1! T2be
a discrete approximation to B(2; 1). The stabilizer subgroup W (F4)B(2;1)at B(2*
*; 1) for the action
18 J. M. MOLLER
of W (F4) on Hom (T1; T2) is isomorphic to the permutation group 3 and generate*
*d by the two
Weyl group elements
0 1 0 1
0 0 0 1 0 -1 0 0
B -1 0 0 0 C B -1 0 0 0 C
w1 = B@ 0 0 1 0 CA and w2 = B@ 0 0 1 0 CA
0 -1 0 0 0 0 0 1
of order 3 and 2, respectively.
Lemma 6.4. The discrete approximation OE(2; 1): T1o ! T2o Wt(F4)o OE(2; 1)*
* is conjugate
to B(2; 1) o O.
Proof.For general reasons, the discrete approximation OE(2; 1) to OE(2; 1) has *
*the form OE(2; 1)(t; 1) =
(B (2; 1)(t); 1) and OE(2; 1)(0; oe) = (a; (oe)) where : ! W (F4)is a grou*
*p homomorphism,
B(2; 1) is -equivariant and a 2 Z1(<(oe)>; T(F4)) is a 1-cocycle.
Since the homomorphism B(2; 1) is O-equivariant we know that (oe) is an eleme*
*nt of order 3
B(2;1) *
* 2
in the coset O(oe)W2 . This leaves the three possibilities O(oe); O(oe)w1, a*
*nd O(oe)w1 for (oe).
Since w2 conjugates O(oe) into O(oe)w21we can ignore the third possibility. We *
*now rule out the
second possibility.
Assume for the moment that (oe) = O(oe)w1. Explicit computation s*
*hows that
H0(; T(F4)) is a 3-discrete torus of rank 2 and that H0(; T(F*
*4)) is cyclic of
order 3 generated by the cohomology class of the 1-cocycle
0 1
0
B 0 C
a = B@1 CA2 t(F4) T(F4)
0
which is fixed by W (F4)B(2;1). It follows that the centralizer
CT(F4)oW(F4)(OE(2; 1)N3=)CT(F4)oW(F4)(B (2; 1)T (PU (3))) \ CT(F4)oW(F4)(a;*
* O(oe)w1)
= (T (F4) o W (F4)B(2;1)) \ CT(F4)oW(F4)(a; O(oe)w1)
= CT(F4)oW(F4)B(2;1)(a; O(oe)w1)
= T(F4)o W (F4)B(2;1)
is the (discrete) maximal torus normalizer for SU(3) and hence (6.2) that CF4(f*
*N3(PU (3))) is iso-
morphic to the N-determined 3-compact group SU(3) [19]. Thus OE(2; 1): N3(PU (3*
*))e!xF4tends
to a morphism N3(PU (3)) x SU(3) -!F4 which is a non-toric monomorphism on the *
*second factor
and we get a factorization
N3(PU (3)) -!CF4(SU (3)) = SU(3) -!F4
of OE(2; 1) through another non-toric monomorphism of SU(3) to F4. The restrict*
*ion of this map
to the maximal tori
T (PU (3)) -!T (SU (3)) -!T (F4)
provides a factorization, up to left action by W (F4), of B(2; 1) as the compos*
*ition of an isomorphism
followed by A(u; 0) or A(0; u), u 2 Z*3, and hence we have that the set
W (F4) . A(2; 1) . GL(0(Q33)) Hom Q3(0(Q33); Q43)
contains A(1; 0) or A(0; 1). It is easy to verify, using a computer, that this *
*is not the case, so we
have arrived at a contradiction.
Thus (oe) = O(oe)w1 can not occur and we are left with (oe) = O(oe) as the on*
*ly possibility. As
H1(; T(F4)) = 0 (5.5), OE(2; 1) = B(2; 1) o O is, up to conjugation, the*
* only extension_of the
pair (B (2; 1); O) to a homomorphism T1o ! T(F4) o W (F4). *
* |__|
A similar statement holds for the non-toric morphism B(1; -1) which differs f*
*rom B(2; 1) by an
automorphism of F4.
TORIC MORPHISMS 19
Proof of Theorem 6.1.It suffices to show that f1 ' f2 whenever f1; f2: PU(3) ! *
*F4are monomor-
phisms such that f1|T (PU (3)) = W (F4)B(2; 1) = f2|T (PU (3)). We already know*
* (6.4) that the
two morphisms become conjugate when restricted to N3(PU (3)). Therefore, the si*
*tuation is now
exactly as in the proof of Theorem 5.4: In order to compute the relevant Wojtko*
*wiak obstruction
groups [24] we need information about the centralizer CF4(fiE2) and CF4(fiN3(PU*
* (3))).
Again, we must have f1|E2 = e25= f2|E2 and CF4(fiE2) is a p-compact toral gro*
*up of maximal
rank with Z=3 as its component group (4.2).
Also, we know (6.2, 6.4) that the centralizer in T(F4)oW (F4) of OE(2; 1) is *
*the (discrete) maximal
torus normalizer for CF4(fiN3(PU (3))). Since
CT(F4)oW(F4)(OE(2; 1)N3=)CT(F4)oW(F4)(B (2; 1)T1) \ CT(F4)oW(F4)(O(oe))
= (T (F4) o W (F4)B(2;1)) \ (T (F4)O(oe)o CW(F4)(O(o*
*e)))
= T(F4)O(oe)o CW(F4)B(2;1)(O(oe))
= t(F4)O(oe)o
is a finite group (of order 27 and with center of order 3) it follows that also*
* CF4(fiN3(PU (3))) is
this finite, but non-abelian, 3-group.
The obstructions to a homotopy between the two maps Bf1; Bf2:BPU (3) ! BFl4ie*
* in the set
lim1Iss_1and in the abelian group lim2Iss_2where ss_1and ss_2are the I-groups
____________________________________________________*
*______SL(2;F3)=S3_____________________________________________
Z=2__9ss9______________________________________________*
*_____________________//_E1SL(2;F3)ii__________________________________@
____________________________________________________*
*____________SL(2;F3)=S3_4_____________________________________________@
Z=2__:0:_______________________________________________*
*____________________//_Z3SL(2;F3)hh___________________________________@
given by the homotopy groups of the above centralizers, e.g. ss = t(F4)O(oe)o <*
*w1>. The group
lim2Iss_2is trivial for general reasons [18, 3.1]. That also lim1Iss_1= * follo*
*ws from (6.5) below since
both the central I-subgroup _________________________________________________*
*__
_________________________________________________*
*__
Z=2__:0:____________________________________________*
*_______________________//_Z=3SL(2;F3)ll_______________________________@
as well as the quotient I-group
_________________________________________________*
*______________________________________________________________________
Z=2___ss99__________________________________________*
*________________________//_0SL(2;F3)dd________________________________@
where SL(2; F3) necessarily acts trivially, have vanishing lim1by [19, 3.1] and*
* (6.6). |___|
The following observations were used to compute the non-abelian lim1.
Let I be a small category. Define an I-group to be a functor from the categor*
*y I to the category
of groups. Let A ! E ! G be a central extension of I-groups meaning that A; E,*
* and G are
I-groups, the arrows are natural transformations, and that the evaluation at ea*
*ch object of I yields
a central extension of groups.
Lemma 6.5. Any central extension of I-groups A ! E ! G induces an exact sequence
* -!lim0IA -!lim0IE -!lim0IG -!lim1IA -!lim1IE -!lim1IG -!lim2IA
of sets. Moreover, the fibres of the map lim1IE -! lim1IG are precisely the orb*
*its for an induced
action of the abelian group lim1IA on the set lim1IE.
Corollary 6.6.Let I be a finite group acting on a finite group ss. If the ss is*
* a p-group and p does
not divide the order of I, then H1(I; ss) = *.
Proof.This follows, using the preceding lemma, by induction over the order of s*
*s since any_non-
trivial p-group has a non-trivial center. *
* |__|
Proof of Theorem 1.1.Modulo the action of the Weyl group WF4(SU (3; 3)) of orde*
*r 2 (4.15), the
sets
Rep(SU (3); SU(3; 3)) = {0} [ Mono(SU (3); SU(3; 3)) [ Mono(PU (3); SU(*
*3; 3))
and
Rep (SU (3); F4) = {0} [ Mono(SU (3); F4) [ Mono(PU (3); F4)
are (4.13, 6.1) in correspondence. *
* |___|
20 J. M. MOLLER
Lemma 6.7. Let (u; v) 2 Z*3(2; 1) [ Z*3(1; -1). Then
CF4(e (u;v)SU(3))= T(F4)O(W(SU(3)))
CF4(e (u;v)T (SU (3)))= T (SU (3)) xZ(SU(3))SU(3)
and CF4( -1) = -1 in both cases.
Proof.We shall apply the Bousfield-Kan spectral sequence [2] to map(BPU (3); BF*
* 4)e__(u;v)where
BPU (3) is viewed as the homotopy colimit of the I-space
__________________________SL(2;F3)=S3_________________*
*_________
(6.8) Z=2_B(0)--__________________________________________//_B(*
*1)SL(2;F3)qq___________________________________________
__(u;v)
where B(0) = BCF4(e N3(PU (3)))and B(1) = BCF4(e25). It represents no loss *
*of generality
to assume that (u; v) = (2; 1).
As we saw in the proof of (5.7), Z=2-acts on N3(PU (3)) = T(PU (3)) o as*
* conjugation with
(0; o) 2 N(PU (3)) = T(PU (3)) o W (PU (3)). But this is again the restriction *
*to
'(2; 1)(N3 (PU (3))) T(F4) o W (F4)
of conjugation by (0; O(o)). Thus (4.18) the Z=2-action on
__(2;1)
CF4(e N3(PU (3))) = CT(F4)oW(F4)('(2; 1)N3 ) = T(F4) o
is through conjugation with (0; O(o)).
Note also that the multiplication map : CT(F4)oW(F4)('(2; 1)N3 ) x N3! T(F4) *
*o Ws(F4)at-
isfies
( -1 x 1) = -1( -1 x 1) = (1 x N3( -1))
__(2;1)
up to inner automorphism. This means that the induced action on CF4(e N3(P*
*U (3))) is
CF4(N3( -1)) = -1 o 1.
Recall from [3] that there is an essentially unique monomorphism : *
*DI2! F4
inducing a monomorphism t(): t(DI2) ! t(F4)and a group monomorphism
O: GL(2; F3) = W (DI2) ! W (F4)extending (4.7). Now, t() is isomorphic to e25a*
*nd from
the commutative diagram
-1
t(DI2)_______w__________//t(DI2)
t()|| t()||
fflffl| fflffl|
T (F4) o W (F4)_________//T(F ) o W (F )
(0;O(w)) 4 4
we see (4.18) that w 2 GL (2; F3) acts on CN (F4)(t(DI2)) = T(F4) o W (F4)t(DI2*
*)as conjugation
with the element (0; O(w)) of the semi-direct product. The restriction to SL(2;*
* F3) of this action
gives the action on CN(F4)(t(DI2)) = CF4(t(DI2)) in (6.8).
The conclusion of this is that
lim0Iss_1= (T (F4)o )= T(F4)O(W(SU(3)))
is the only non-trivial contribution from the groups lim-iIss_j, i+j 0, of the*
* Bousfield-Kan spectral
sequence. Consequently, CF4(e (2;1)SU(3)) is isomorphic to this group of order *
*3. The action of
CF4( -1), which is the restriction of the action of CF4(N3( -1)), is given by *
*-1.
The centralizer
CT(F4)oW(F4)(e (2;1)T(SU (3))) = T(F4) o W (F4)A(2;1)
is the (discrete) maximal torus normalizer for CF4(e (2;1)T(SU (3))) and the ce*
*ntralizer
CT(F4)oW(F4)(e (0;1)T(SU (3))) = T(F4) o W (F4)A(0;1)
is the (discrete) maximal torus normalizer for CF4(e (0;1)T(SU (3))) = SU(3) xZ*
*(SU(3))T (SU (3))
[17, 3.4.3]. Since the two stabilizer subgroups W (F4)A(2;1)and W (F4)A(0;1)are*
* conjugate in W (F4),
the two maximal torus normalizers are isomorphic and hence the two centralizers*
* are isomorphic,
too, by N-determinism [15] [19].
TORIC MORPHISMS 21
The group homomorphism : (T (F4) o W (F4)A(2;1)-!T(F4) o W (F4) which is the *
*inclusion
on the first factor and equals A(2; 1) on the second factor satisfies
((1 o 1) x -1) = ( -1 o 1)((1 o 1) x -1) = (( -1 o 1) x 1
up to inner automorphisms. This shows (4.17) that CF4( -1) = -1. *
* |___|
7.Morphisms from G2 to F4 at the prime p = 3
Using the Jackowski-McClure decomposition of BG 2and the Bousfield-Kan spectr*
*al sequence
we classify morphisms G2 -!F4 viewed as 3-compact groups and compute their cent*
*ralizers.
The Weyl group of G2, W (G 2) < GL(0(Z33)) is the product of the Weyl group W*
* (SU (3)) =
oof SU(3) and the central group <-1 > of order 2. The group morphism O fr*
*om (4.7) ex-
tends to a group homomorphism O: W (G 2) ! W (F4)simply by putting O(-1) = -1. *
* Let
I = I(W (G 2); W (SU (3))) denote the category
_____________________________________________________*
*___________W(G2)=W(SU(3))_____________________________________________@
<-1>__0::_______________________________________________*
*____________________//_1W(G2)dd_______________________________________@
of the central inclusion of W (SU (3)) into W (G 2). Then BG 2is [18, x4] H*F3-*
*equivalent to the
homotopy colimit of an Iop-space
____________________________________________________W(S*
*U(3))op\W(G2)op__________________________
(7.1) < -1_-B(0)-__________________________________________>--___*
*________________________________________B(1)oo_W(G2)opqq______________@
where B(0) = BSU (3) and B(1) = BT (SU (3)).
Theorem 7.2. The restriction map
Rep(G 2; F4) -!Rep(SU (3); F4)
is bijective. The centralizer CF4(e (u;v)G2), u; v 2 Z*3[ {0}, is isomorphic to*
* SU(2) if uv = 0 and
trivial otherwise.
Proof.We must show that any morphism SU(3) -!F 4extends uniquely to G2. Since t*
*his is true
for the trivial morphism by [17, 6.7], we only need here to consider non-trivia*
*l morphisms.
Let (u; v) 2 (Z*3[ {0})2, (u; v) 6= (0; 0). Since e (u;v):SU(3) ! F4is invari*
*ant under -1, this
map e (u;v)and its restriction to the maximal torus form a homotopy coherent se*
*t of maps out of
the Iop-space (7.1). Thus it suffices to show that lim-iIss_j(u; v) = 0 for i +*
* j -1 where ss_j(u; v)
is the I-group
_______________________W(G2)=W(SU(3))_________________*
*______
Z=2_ssj(0)++_______________________________________//_ssj*
*(1)W(G2)ss________________________________________
where the group ssj(0) = ssj(u; v)(0) = ssj(BCF4(e (u;v)SU(3))) and the gro*
*up ssj(1) =
ssj(u; v)(1) = ssj(BCF4(e (u;v)T (SU (3)))). Since the abelian I-groups ss_j(u*
*; v) are in fact Z3[I]-
modules and W (SU (3)) is normal in W (G 2), it follows from [18, 3.1] that lim*
*0Iss_j(u; v) =
ssj(u; v)(0)Z=2= ssj(BCF4(e (u;v)SU(3)))Z=2is the subgroup that is invariant un*
*der the action
of -1 and that the higher limits are automatically trivial. By (4.16, 5.7, 6.7*
*), ssj(u; v)(0)Z=2is
trivial except when either u = 0 or v = 0 when it equals the invariants ssj(BSU*
* (3))**.
We now examine the case (u; v) = (0; 1) more closely. According to Dynkin [8*
*, 9] the Lie
group F4 contains a copy of (a central quotient of) SU(2) x G2. The restrictio*
*n to G2 of this
inclusion SU(2) x G2 -!F4 equals, up to an automorphism of F4, the map e (0;1)f*
*or otherwise
the restriction to the other factor, the inclusion of SU(2) into F4, would fact*
*or through the trivial
3-compact group. The homotopy class of the restriction
BSU (2) x BSU (3) -!BSU (2) x BG 2-! BF 4
to SU(2) x SU(3) is determined by its adjoint in ss0(map (BSU (2); map(BSU (3);*
* BF 4)B(e (0;1))) =
ss0(map (BSU (2); BSU (3))) = Rep(SU (2); SU(3)) so. Since SU (3) contains (7.*
*3) an essentially
unique copy of SU(2), we conclude that the diagram of 3-compact groups
S(2;3)x1
SU(2) x SU(3)__________//SU(3) x SU(3)
| |
| |
fflffl| fflffl|
SU (2) x G2________________//F4
22 J. M. MOLLER
commutes up to conjugacy. After taking adjoint maps we end up with
BSU (2)______//map(BG 2; BF 4)B(e (0;1))
BS(2;3)|| ||
fflffl|' fflffl|
BSU (3)____//_map(BSU (3); BF 4)B(e (0;1))
which commutes up to homotopy and where the lower horizontal arrow represents (*
*4.16) a homo-
topy equivalence homotopy equivariant under the action ****. By the above co*
*mputations with
the Bousfield-Kan spectral sequence,
*
* -1>
ss*(map (BG 2; BF 4); B(e (0;1))) = ss*(map (BSU (3); BF 4); B(e (0;1))*
*)**** and Mono(SU (2); SU(3)).
Proof.This follows from (3.14) that identifies both Rep(SU (2); SU(2)) and Rep(*
*SU_(2); SU(3)) to
Z3=<-1 >. |*
*__|
Since -1S(2; 3) = S(2; 3) -1 = S(2; 3), the image of ss*(BSU (2)) in ss*(BSU*
* (3)) is invariant
under the action of the group ****.
Lemma 7.4. There is an isommorphism, induced by S(2; 3),
-1>
ss*(BSU (2)) -!ss*(BSU (3))****-invariant subgroup of the homot*
*opy of BSU (3).
Proof.There is a short exact sequence of homotopy groups
0 -!ss*(SU (2)) -!ss*(SU (3)) -!ss*(S5) -!0
of F3-complete spaces induced by the fibration of SU(3) onto S5 with fibre SU(2*
*). This fibration
splits since ss4(S3) Z3 = 0. The homomorphism -1, complex conjugation of matr*
*ices, restricts
to the identity on the fibre and induces the degree -1-map on the base. Since t*
*his map induces_
multiplication by -1 on the homotopy groups of S5, the claim follows. *
* |__|
8.Morphisms from SU(3) to G2 at the prime p = 3
The classification of morphisms SU 3) -!G 2of 3-compact groups proceeds very *
*much like the
classification of morphisms SU(3) -!F4.
Lemma 8.1. The set Mono(E1; G2) contains two elements, e11, e12, with centraliz*
*er Weyl groups
of order 2, 6, and Quillen automorphism groups of order 2, 2, respectively. The*
* centralizer CG2(e12)
is isomorphic to SU(3).
The set Mono(E2; G2)= Aut(E2) contains a unique element, e22= t(G 2), with Qu*
*illen automor-
phism group W (G 2) of order 12.
Let O1:W (SU (3)) ! W (Gb2)e the inclusion and O2:W (SU (3)) ! W (Gt2)he inje*
*ction given
by O2(oe) = oe and O2(o) = -o. Then the identity map A1:0(Z33) ! 0(Z33)is O1-eq*
*uivariant
and the Z3-linear map A2:0(Z33) ! 0(Z33)with matrix
A2 = 12 -2-1
is O2-equivariant.
TORIC MORPHISMS 23
Lemma 8.2. A Z3-linear map 0(Z33) -! 0(Z33) is admissible with respect to W (SU*
* (3)) and
W (G 2) is and only it belongs to W (G 2)(uA1) or W (G 2)(uA2) for some scalar *
*u 2 Z3.
Proof.Computerized calculations show that any admissible homomorphism must, up *
*to inner au-
tomorphisms, be either O1- or O2-equivariant. Next, one solves the two systems *
*of linear_equations
Aw = Oi(w)A, w 2 W (SU (3)), i = 1; 2. *
* |__|
Proposition 8.3.Any non-trivial morphism f :SU(3) ! G2is a monomorphism.
Proof.Let f :SU(3) ! G2be any non-trivial morphism and T (f): T (SU (3)) ! Ta(G*
*l2)ift of
f to the maximal tori. Then W (G 2)ss1(T (f)) equals W (G 2)(uA1) or W (G 2)(u*
*A2) for some 3-
adic integer, u. In fact, since the order of W (SU (3)) is divisible by 3, u mu*
*st be a unit [3]. In
the first case, W (G 2)ss1(T (f)) = W (G 2)(uA1), f is a monomorphism. And if W*
* (G 2)ss1(T (f)) =
W_(G 2)(uA2), the kernel of T(f) equals the center of SU(3) and f factors throu*
*gh a monomorphism
f: PU(3) ! G2. However, such a monomorphism can not exist since the Quillen cat*
*egory of PU(3)
contains an object E2 ae PU(3) with Quillen automorphism group SL(2; F3) of ord*
*er 24 exceeding_
the order of the Quillen automorphism group of e222 Mono(E2; G2). *
* |__|
Consider now the diagram
E1ISSS
IISSSSSe1
z || IzIIIIS2SSSSS
fflffl|I$$ SSSS))S
SU (3) SU (3)_e__//_G2
where the SU(3) to the right stands for CG2(e12) and z stands for center. Here,*
* e -1 = e since
CG2( -1) = -1.
Lemma 8.4. For any monomorphism f :SU(3) ! G2, fz = e12.
Proof.Since ss1(T (f)) = uA1, u 2 Z*3, the reduction mod 3, t(f): t(SU (3)) ! t*
*(G,2)takes the
center, (1; -1), of SU(3) to the element u(1; -1) 2 t(G 2) whose stabilizer sub*
*group_is W (SU (3)).
|__|
It follows (3.9) that
Mono (SU (3); SU(3))z!z = Mono(SU (3); G2)z!e12= Mono(SU (3); G2)
or, alternatively, that the map
<-1 >\Z*3-!Mono(SU (3); G2)
u -! e u
is a bijection. Also, any monomorphism f :SU(3) ! G2is centric [4] in the sense*
* that the map
given by composition with Bf,
map(B SU(3); B SU(3))B1 -!map (B SU(3); B G2)Bf
is a homotopy equivalence. Clearly, f is toric as well (2.10).
Theorem 8.5. 1.Rep(SU (3); G2) = {0} [ Mono(SU (3); G2) is T -determined.
2.The action
Mono(SU (3); G2) x Out(SU (3)) -!Mono (SU (3); G2)
is transitive and the stabilizer at f 2 Mono(SU (3); G2) equals WG2(f SU(3)*
*) = < -1>.
Proof.This is clear from the explicit description of the set Rep(SU (3); G2). *
*For instance, the
restriction map
Mono (SU (3); G2) -!Mono (T (SU (3)); G2)
can be identified to the map
{u 2 Z*3| u 1 mod 3} -!W (G 2)(uA1)
which clearly is injective. *
* |___|
24 J. M. MOLLER
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