TORIC MORPHISMS BETWEEN p-COMPACT GROUPS

JESPER M. MØLLER

Abstract. It is well-known that any morphism between two p-compact groups will lift, nonuniquely, to an admissible morphism between the maximal tori. We identify here a class of pcompact group morphisms, the p-toric morphisms, which can be perceived as generalized rational isomorphisms, enjoying the stronger property of lifting uniquely to a morphism between the maximal torus normalizers. We investigate the class of p-toric morphisms and apply our observations to determine the mapping spaces map(BSU(3), BF_{4}), map(BG_{2}, BF_{4}), 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 compact Lie groups is a traditional project of algebraic topology [13, 21]. One line of development started with the investigations 25 years ago by Hubbuck [10, 11] and Adams-Mahmud [1]. They noted the close relationship between maps between classifying spaces and admissible homomorphisms between maximal tori. The regular admissible homomorphisms, in particular, turned out to have especially nice properties. It is the purpose of this paper to study regular admissible morphisms, 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}, G_{2} ⟶ F_{4}, and SU(3) ⟶ G_{2} at the prime p = 3.

In order to describe the content in more detail, let X_{1} and X_{2} be p-compact groups, for the sake of this introduction assumed to be connected, with maximal tori T (X_{1}) ⟶ X_{1} and T (X_{2}) ⟶ X_{2}, respectively. For any morphism f: X_{1} → X_{2} there is a lift T (f): T (X_{1}) → T (X_{2}), unique up the action of the Weyl group of X_{2}, such that the diagram

\xymatrix{T (X_{1}) \ar[r]^{T (f)} \ar[d]_{i_{1}} & T (X_{2}) \ar[d]^{i_{2}} \\ X_{1} \ar[r]_{f} & X_{2}}

commutes up to conjugacy. The morphism T (f) is admissible in the sense that for any element w_{1} of the Weyl group of X_{1} there exists and element w_{2} of the Weyl group of X_{2} such that T (f)w_{1} = w_{2}T (f). In general, w_{2} is not uniquely determined by w_{1}, but if it is, we say (2.1) that f is p-toric. (If f is p-toric, the centralizer C_{X_{2}}(fi_{1}T (X_{1})) of the maximal torus of X_{1} in X_{2} will be isomorphic to the maximal torus of X_{2}.) In that case, the correspondence w_{1} ⟶ w_{2} is a homomorphism of Weyl groups and, by Theorem 3.5, there is a unique lift N(f): N(X_{1}) → N(X_{2}) to a map between the maximal torus normalizers such that the diagram

\xymatrix{N(X_{1}) \ar[r]^{N(f)} \ar[d] & N(X_{2}) \ar[d] \\ X_{1} \ar[r]_{f} & X_{2}}

commutes up to conjugacy.

As a first example, we consider the case where the domain X_{1} = SU(3), the codomain X_{2} = F_{4}, and the prime p = 3. The compact Lie group F_{4} contains a unique copy of SU(3, 3) = SU(3) ×_{Z(SU(3))} SU(3) as a subgroup of maximal rank. Any morphism SU(3) ⟶ SU(3, 3) is of the form

Date: March 6, 1998.

1991 Mathematics Subject Classification. 55R35, 55S37.

Key words and phrases. Weyl group, Quillen category, Bousfield-Kan spectral sequence.

ψ^{(u,v)} : SU(3) ⟶^{Δ} SU(3) × SU(3) ⟶^{ψ^{u}×ψ^{v}} SU(3) × SU(3) ⟶ SU(3, 3)

where u and v are 3-adic units or zero. Composing with the inclusion e: SU(3, 3) → F_{4} we obtain the morphism eψ^{(u,v)} : SU(3) → F_{4}. Observe that eψ^{(u,v)} = eψ^{(−u,−v)} since the inclusion e is invariant under the action of the Weyl group W_{F_{4}}(SU(3, 3) which is of order two generated by the self-map ψ^{−1} ×_{Z(SU(3))} ψ^{−1} of SU(3, 3). These maps eψ^{(u,v)}, u, v ∈ ℤ_{3}^{∗} ∪ {0}, with the relation eψ^{(u,v)} = eψ^{(−u,−v)}, turn out to describe Rep(SU(3), F_{4}) = [B SU(3)_{3}^{∧}, (B F_{4})_{3}^{∧}] completely.

Theorem 1.1. The map

e ∘ −: W_{F_{4}}(SU(3, 3))\ Rep(SU(3), SU(3, 3)) → Rep(SU(3), F_{4})

is a bijection when p = 3.

See (4.16, 5.7, 6.7) for information about the centralizers of these maps. The proof of Theorem 1.1 is divided into three cases: Monomorphisms SU(3) ⟶ F_{4} (4.13), p-toric monomorphisms PU(3) ⟶ F_{4} (5.4), and, the technically most demanding case, non-p-toric monomorphisms PU(3) ⟶ F_{4} (6.1).

As a second example, we consider the case where X_{1} = G_{2} and X_{2} = F_{4} and p = 3 and reprove a result from Jackowski-McClure-Oliver [14]. To state the theorem, we recall that the compact Lie group G_{2} contains a unique copy of SU(3) as a subgroup of maximal rank. Thus we may restrict morphisms defined on G_{2} to this subgroup SU(3) ⊂ G_{2}.

Theorem 1.2. [14, 3.4] The restriction map

Rep(G_{2}, F_{4}) ⟶ Rep(SU(3), F_{4})

is a bijection when p = 3.

When working with this paper, I made use of a MAGMA program written by K. Andersen 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 other, more familiar, types of morphisms between p-compact groups, and provide examples of morphisms that are p-toric and others that are not.

Let X_{1} and X_{2} be p-compact groups (or extended p-compact tori) with maximal tori T_{1} = T (X_{1}) → X_{1}, T_{2} = T (X_{2}) → X_{2} and Weyl groups W_{1} = W (X_{1}) and W_{2} = W (X_{2}), respectively. Write Rep(X_{1}, X_{2}) for the set [BX_{1}, BX_{2}] of conjugacy classes of loop space morphisms.

Definition 2.1. 1. A loop space morphism T_{1} → X_{2} is p-toric (or regular [1, 2.22], [14, 1.3])

if its centralizer C_{X_{2}}(T_{1}) is a p-compact toral group.

2. A loop space morphism X_{1} → X_{2} is p-toric if its composition with T_{1} → X_{1} is p-toric.

Note that the centralizer C_{X_{2}}(T_{1}) in (2.1.1) is known to be a p-compact group [6, §6] [7, 2.5]. We shall now consider some alternative criteria for a morphism to be p-toric.

For any loop space morphism f: X_{1} → X_{2} between p-compact groups or extended p-compact tori there exists a loop space morphism T (f): T_{1} → T_{2} between the maximal tori such that

\xymatrix{T_{1} \ar[r]^{T (f)} \ar[d]_{i_{1}} & T_{2} \ar[d]^{i_{2}} \\ X_{1} \ar[r]_{f} & X_{2}} (2.2)

commutes up to conjugacy. As W_{2}T (f) = f_{1}|T_{1} ∈ W_{2}\ Rep(T_{1}, T_{2}) = Rep(T_{1}, X_{2}) is an invariant of f, T (f) in Rep(T_{1}, T_{2}) is uniquely determined up to the action of the Weyl group W_{2}. In particular, the stabilizer subgroup

W_{2}^{T (f)} = {w_{2} ∈ W_{2} | w_{2} · T (f) = T (f)}

at T (f) for the action of W_{2} on Rep(T_{1}, T_{2}) is determined up to conjugacy by f.

In case X_{1} and X_{2} are extended p-compact tori, there is a short exact sequence of loop spaces

T_{2} ⟶ C_{X_{2}}(T (f)T_{1}) ⟶ W_{2}^{T (f)}

from which we see that

f: X_{1} → X_{2} is p-toric ⇔ W_{2}^{T (f)} = π_{0}(C_{X_{2}}(T_{1})) is a p-group.

In case X_{1} and X_{2} are p-compact groups, C_{X_{2}}(T_{1}) ⟶ X_{2} is a monomorphism of maximal rank [7, §4], so

f: X_{1} → X_{2} is p-toric ⇔ C_{X_{2}}(T_{1})_{0} ⟶ X_{2} is a maximal torus

where subscript 0 indicates identity component. (If X_{2} is connected, the centralizer of T_{1} is also connected [20, 3.11] [7, 7.8] so in this case f is p-toric if and only if C_{X_{2}}(T (f)T_{1}) ⟶ X_{2} is a maximal torus for X_{2} if and only if the stabilizer W_{2}^{T (f)} is trivial [17, 3.2].)

We consider also an enlarged version of diagram (2.2) in the form of the diagram

\xymatrix{T_{1} \ar[r]^{T (f)} \ar[dd]_{i_{1}} & T_{2} \ar[d]^{i′_{2}} \\ & N_{2} \ar[d]^{j_{2}} \\ X_{1} \ar[r]_{f} & X_{2}} (2.3)

where j_{2} : N_{2} → X_{2} is the normalizer [6, 9.8] of the maximal torus. Using that C_{N_{2}}(T_{1}) → C_{X_{2}}(T_{1}) is a maximal torus normalizer, we get [17]

(2.4)  f is p-toric ⇔ T_{1} ⟶^{i_{1}} X_{1} ⟶^{f} X_{2} is p-toric

(2.5) ⇔ C_{N_{2}}(T_{1}) → C_{X_{2}}(T_{1}) is an isomorphism

⇒ T_{1} ⟶^{T (f)} T_{2} ⟶^{i′_{2}} N_{2} is p-toric (2.6)

⇔The isotropy subgroup W_{2}^{T (f)} is a finite p-group.

(2.7)

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 loop space morphism N(f) between the maximal torus normalizers such that

\xymatrix{N_{1} \ar[r]^{N(f)} \ar[d]_{j_{1}} & N_{2} \ar[d]^{j_{2}} \\ X_{1} \ar[r]_{f} & X_{2}} (2.8)

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

• TRep(X_{1}, X_{2}) ⊂ Rep(X_{1}, X_{2}) denotes the set of conjugacy classes of p-toric morphisms • Mono(X_{1}, X_{2}) ⊂ Rep(X_{1}, X_{2}) denotes the set of conjugacy classes of monomorphisms

• TMono(X_{1}, X_{2}) = Mono(X_{1}, X_{2}) ∩ TRep(X_{1}, X_{2})

• ε_{ℚ}(X_{1}, X_{2}) ⊂ Rep(X_{1}, X_{2}) is the set of rational isomorphisms

• ε_{ℚ}(X_{1}) = ε_{ℚ}(X_{1}, X_{1}) is the monoid of rational automorphisms of X_{1}

• Out(X_{1}) is the group of conjugacy classes of automorphisms of X_{1}. Above, a loop space morphism between extended p-compact tori is a monomorphism if its discrete approximation [7, 3.12] is a monomorphism.

Example 2.10. If X_{1} and X_{2} have the same rank,

Mono(X_{1}, X_{2}) ⊂ TRep(X_{1}, X_{2}) ⊃ ε_{ℚ}(X_{1}, X_{2})

because any monomorphism (rational isomorphism) restricts to an isomorphism (epimorphism) between maximal tori.

If X_{1} and X_{2} are locally isomorphic, simple p-compact groups

TRep(X_{1}, X_{2}) = Rep(X_{1}, X_{2}) − {0} = ε_{ℚ}(X_{1}, X_{2})

because f is p-toric or a rational isomorphism if and only if T (f) is non-trivial if and only if f is non-trivial.

Example 2.11. For any p-compact group X and any integer m > 0, TRep(X, X^{m}) = (TRep(X, X))^{m}. If X is simple,

TRep(X, X^{m}) = (Rep(X, X) − {0})^{m} = ε_{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 X_{1} is connected and that z: Z_{1} → X_{1} is a central monomorphism. Then there are bijections

• Rep(X_{1}/Z_{1}, X_{2}) ⟶ {f ∈ Rep(X_{1}, X_{2}) | f ∘ z is trivial}

• TRep(X_{1}/Z_{1}, X_{2}) ⟶ {f ∈ TRep(X_{1}, X_{2}) | f ∘ z is trivial} induced by the epimorphism X_{1} ⟶ X_{1}/Z_{1}.

In fact, map(B(X_{1}/Z_{1}), BX_{2}) is homotopy equivalent to a union of connected components of map(BX_{1}, BX_{2}).

Proof. The epimorphism of X_{1} to X_{1}/Z_{1} induces a homotopy equivalence between map(B(X_{1}/Z_{1}), BX_{2}) and a collection of components of map(BX_{1}, BX_{2}). This shows the injection of sets of representations and when applied with X_{1} replaced by T_{1}, it shows that a morphism X_{1} ⟶ X_{2} is p-toric if and only if its composition with the epimorphism X_{1} ⟶ X_{1}/Z_{1} is p-toric.

Proposition 2.13. Assume that X_{1} is simply connected, X_{2} is connected, and that z: Z_{2} → X_{2} is a central monomorphism. Then there are bijections

• Rep(X_{1}, X_{2}) ⟶ Rep(X_{1}, X_{2}/Z_{2})

• TRep(X_{1}, X_{2}) ⟶ TRep(X_{1}, X_{2}/Z_{2}) induced by the epimorphism X_{2} ⟶ X_{2}/Z_{2}.

Proof. Obstruction theory (remember that BX_{1} is 3-connected) shows that Rep(X_{1}, X_{2}) = Rep(X_{1}, X_{2}/Z_{2}) and the existence of a short exact sequence of p-compact groups

K ⟶ C_{X_{2}}(X_{1}) ⟶ C_{X_{2}/Z_{2}}(X_{1})

where BK is one component of the homotopy fixed point set BZ_{2}^{hX_{1}}; in particular K is a p-compact toral group. It follows that C_{X_{2}}(X_{1}) is a p-compact toral group if and only if C_{X_{2}/Z_{2}}(X_{1}) is.

Example 2.14. For any simply connected, simple p-compact group X and any central monomorphism Z ⟶ X^{m},

TRep(X, X^{m}/Z) = TRep(X, X^{m}) = ε_{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) × SU(p) be the central subgroup generated by (ζE, ζ^{−1}E) where ζ ≠ 1 is a pth root of unity. Then (2.13)

Rep(SU(p), SU(p, p)) = Rep(SU(p), SU(p) × Rep(SU(p), SU(p))

TRep(SU(p), SU(p, p)) = Out(SU(p)) × 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) ∈ (ℤ_{p}^{∗} ∪ {0})^{2} | u + v ∈ ℤ_{p}^{∗}} for the morphism ψ^{(u,v)} defined as the composition

SU(p) ⟶^{Δ} SU(p) × SU(p) ⟶^{ψ^{u}×ψ^{v}} SU(p) × SU(p) ⟶ SU(p, p)

is a monomorphism if and only if u + v ∈ ℤ_{p}^{∗}. The monoid Rep(SU(p, p), SU(p, p)) is (2.13) isomorphic to a submonoid of Rep(SU(p) × SU(p), SU(p) × SU(p)) and, in particular,

Out(SU(p, p)) = {(u, v) ∈ ℤ_{p}^{∗} × ℤ_{p}^{∗} | u ≡ v mod p} ⋊ ⟨τ⟩

where τ is the automorphism that swaps the two SU(p)-factors.

The set of monomorphisms (2.16) consists of two orbits, represented by ψ^{(1,1)} and ψ^{(1,0)}, under the action of the automorphism group Out(SU(p, p)). It follows that the centralizers of the monomorphism ψ^{(u,v)} are

(2.17) C_{SU(p,p)}(ψ^{(u,v)} SU(p)) ≅ \begin{cases}Z(SU(p)) & if u ≠ 0 and v ≠ 0 \\ SU(p) & if u = 0 or v = 0 \end{cases}

i.e. that ψ^{(u,v)} is centric precisely when it is p-toric. (To prove that ψ^{(1,1)} is centric one uses the fact that Z(SU(p)) ⟶^{Δ} Z(SU(p) × SU(p)) ⟶ Z(SU(p, p)) is an isomorphism of centers.) In the non-toric case, observe that the projection morphism SU(p) × SU(p) ⟶ SU(p, p) restricts to ψ^{(1,0)} on the first factor and to ψ^{(0,1)} on the second factor. This gives a factorization

SU(p) ⟶ C_{SU(p,p)}(ψ^{(1,0)} SU(p)) ⟶ SU(p, p)

of ψ^{(0,1)} through the centralizer of ψ^{(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 monomorphism SU(p) = C_{SU(p,p)}(f SU(p)) ⟶ SU(p, p). The Weyl group, W_{SU(p,p)}(ψ^{(u,v)} SU(p)), of any monomorphism ψ^{(u,v)} is trivial [18, 5.6].

Finally, we note that by (2.12),

Rep(PU(p), SU(p, p)) = {(u, v) ∈ (ℤ_{p}^{∗} ∪ {0})^{2} | u + v ∈ pℤ_{p}}
TRep(PU(p), SU(p, p)) = {(u, v) ∈ (ℤ_{p}^{∗})^{2} | u + v ∈ pℤ_{p}}

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 → Y_{1} be any morphism and g: Y_{1} → Y_{2} a monomorphism between pcompact groups. Then

g ∘ f: X → Y_{2} is p-toric ⇒ f: X → Y_{1} is p-toric.

Proof. Let T be a maximal torus of X_{1}. Since composition with Bg, C_{Y_{1}}(fiT) → C_{Y_{2}}(gfiT), is a monomorphism, C_{Y_{2}}(gfiT) is a p-compact toral group if C_{Y_{1}}(fiT) is a p-compact toral group [20, 3.5.(1)].

The converse of (2.18) is not true in general; take for instance Y_{1} to be the maximal torus of Y_{2}.

3. Lifting p-toric morphisms

In this section I show that all p-toric morphisms between two p-compact groups lift uniquely to p-toric morphisms between the maximal torus normalizers.

Recall that X_{1} and X_{2} are p-compact groups or extended p-compact tori and that j_{1} : N_{1} → X_{1} and j_{2} : N_{2} → X_{2} are normalizers of the respective maximal tori, i_{1} : T_{1} → X_{1} and i_{2} : T_{2} → X_{2}.

By the very definition of a p-toric morphism, the maps j_{1} and j_{2} induce maps

(3.1) TRep(X_{1}, X_{2}) ⟶ TRep(N_{1}, X_{2}) ⟵ TRep(N_{1}, N_{2})

of sets of p-toric representations. Our first objective is to prove that the arrow to the right is a bijection. This will enable us to define a map from TRep(X_{1}, X_{2}) to TRep(N_{1}, N_{2}). Note the favorable input provided by the information [17, 3.2] that

(3.2) TRep(T_{1}, X_{2}) ⟵ TRep(T_{1}, N_{2})

is a bijection and

C_{X_{2}}(T_{1}) ⟵ C_{N_{2}}(T_{1}) (3.3) an isomorphism for any p-toric morphism T_{1} ⟶ N_{2}.

For any set S ⊂ Rep(X_{1}, X_{2}), write map(BX_{1}, BX_{2})_{S} for the space of all maps BX_{1} ⟶ BX_{2} homotopic to a member of S.

Lemma 3.4. The map, induced by j_{2},

map(BN_{1}, BX_{2})_{TRep(N_{1},X_{2})} ⟵ map(BN_{1}, BN_{2})_{TRep(N_{1},N_{2})}

is a homotopy equivalence.

Proof. The map of the lemma is the map on homotopy fixed point spaces

map(BN_{1}, BG_{2})_{TRep(N_{1},G_{2})} = (map(BT_{1}, BG_{2})_{TRep(T_{1},G_{2})})^{hW_{1}} , G_{2} = N_{2}, X_{2},

induced by the map

map(BT_{1}, BX_{2})_{TRep(T_{1},X_{2})} ⟵ map(BT_{1}, BN_{2})_{TRep(T_{1},N_{2})}

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 X_{1} and X_{2} be p-compact groups and f: X_{1} → X_{2} a p-toric morphism. Then there exists a morphism N(f): N_{1} → N_{2} between extended p-compact tori such that

\xymatrix{N_{1} \ar[r]^{N(f)} \ar[d]_{j_{1}} & N_{2} \ar[d]^{j_{2}} \\ X_{1} \ar[r]_{f} & X_{2}}

commutes up to conjugacy. Moreover, • N(f) is unique up to conjugacy

• N(f) is p-toric

• C_{X_{2}}(fj_{1}N_{1}) ⟵ C_{N_{2}}(N(f)N_{1}) is an isomorphism of loop spaces

Proof. The map

(3.6) N: TRep(X_{1}, X_{2}) → TRep(N_{1}, N_{2})

is defined as the composition of the map TRep(X_{1}, X_{2}) ⟶ TRep(X_{1}, N_{2}) with the inverse of the bijection TRep(N_{1}, X_{2}) ⟵ TRep(N_{1}, N_{2}) 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, the map TRep(X, X^{m}) ⟶ TRep(N, N^{m}) is injective if ε_{ℚ}(X) ⟶ Rep(N, N) is injective; e.g. if X = PU(p), X = G_{2} and p = 3, or X = DI(2) and p = 3.

The above theorem is intended as a tool to facilitate the computation of TRep(X_{1}, X_{2}) in concrete cases. We now address injectivity of (3.6).

Remark 3.8. According to the homology decomposition theorem of Jackowski-McClure [12] and Dwyer-Wilkerson [5], the exists an 𝔽_{p}-equivalence

hocolim_{𝔸^{op}} BC_{X_{1}}(ν) → BX_{1}

where the homotopy colimit is taken over some full subcategory 𝔸 of the Quillen category 𝔸(X_{1}). Let us assume that

• Any object ν: V → X_{1} of 𝔸 admits a factorization μ: V → T_{1} through the maximal torus and

• N: TRep(C_{X_{1}}(ν), X_{2}) → TRep(C_{N_{1}}(μ), N_{2}) is injective for all objects ν: V → X_{1} of 𝔸 and let now f and f′ be two p-toric morphisms with N(f) = φ = N(f′) for some φ ∈ TRep(C_{N_{1}}(μ), N_{2}). Then the two possible compositions

\xymatrix{C_{X_{1}}(ν) \ar[r]^{e(ν)} & X_{1} \ar[r]^{f} \ar[r]_{f′} & X_{2}}

are again p-toric morphisms for C_{X_{2}}(fe(ν)C_{T_{1}}(μ)) = C_{X_{2}}(fi_{1}T_{1}) is a p-compact torus and similarly for the other morphism f′. Since also,

N(f ∘ e(ν)) = φ ∘ e(μ) = N(f′ ∘ e(ν))

we have f ∘ e(ν) ≃ f′ ∘ e(ν) for all objects ν of 𝔸 by hypothesis. (Here, e(ν): C_{X}(ν) → X stands for the evaluation monomorphism.) The obstructions to constructing a homotopy between Bf and Bf′ lie in

lim_{𝔸}^{i}π_{i}(map(BC_{X_{1}}(ν), BX_{2})_{B(f∘e(ν))}), 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 using the preferred lifts of [16].

While (3.8) applies to the case where X_{1} is center-free, the following lemma can be helpful if X_{1} has a non-trivial center.

Consider the following situation

\xymatrix{Z \ar[d]_{z} \ar[dr]_{z_{1}} \ar[drr]^{z_{2}} \\ X & Y_{1} \ar[r]_{g} & Y_{2}}

of p-compact groups and loop space morphisms. Let Rep(X, Y_{1})_{z→z_{1}} = {f ∈ Rep(X, Y_{1}) | f ∘ z = z_{1}} denote the set of conjugacy classes of morphisms under Z and map(BX, BY)_{z→z_{1}} 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 \underline{g}: C_{Y_{1}}(z_{1}Z) → C_{Y_{2}}(z_{2}Z) of centralizers. Then composition with Bg,

Bg ∘ −: map(BX, BY_{1})_{z→z_{1}} → map(BX, BY_{2})_{z→z_{2}}

is a homotopy equivalence.

Proof. The spaces BC_{Y_{i}}(z_{i}) = map(BZ, BY_{i})_{Bz_{i}}, i = 1, 2, are X/Z-spaces and composition with Bg, B\underline{g}: BC_{Y_{1}}(z_{1}) → BC_{Y_{2}}(z_{2}), is an X/Z-map inducing a map

(3.10) map(BX, BY_{1})_{z→z_{1}} = BC_{Y_{1}}(z_{1})^{h(X/Z)} → BC_{Y_{2}}(z_{2})^{h(X/Z)} = map(BX, BY_{2})_{z→z_{2}}

of homotopy fixed point spaces. If \underline{g} is a homotopy equivalence, so is (3.10).

Here is a typical application of (3.9). In the diagram

\xymatrix{V \ar[d]_{z_{1}} \ar[dr]_{_{\overline{z}_{2}}} \ar[drr]^{z_{2}} \\ X_{1} & C_{X_{2}}(V) \ar[r]_{e(V)} & X_{2}}

V is an elementary abelian p-group, z_{1} a central monomorphism, z_{2} a monomorphism and \overline{z}_{2} the canonical factorization of z_{2} through its centralizer. Since the evaluation monomorphism C_{X_{2}}(V) ⟶ X_{2} clearly satisfies the hypothesis of (3.9) we see that

map(BX_{1}, BC_{X_{2}}(V))_{z_{1}→\overline{z}_{2}} → map(BX_{1}, BX_{2})_{z_{1}→z_{2}} (3.11)

is a homotopy equivalence.

Definition 3.12. Let R be a subset of Rep(X_{1}, X_{2}). We say that R is T -determined if the implication

f|T (X_{1}) = g|T (X_{1}) ⇒ f = g

holds for all f ∈ R and all g ∈ Rep(X_{1}, X_{2}).

Example 3.13. If the order of W (X_{1}) is prime to p, then

Rep(X_{1}, X_{2}) = W (X_{2})\ Adm(T (X_{1}), T (X_{2})) (3.14)

where Adm(T (X_{1}), T (X_{2})) consists of all ϕ ∈ Rep(T (X_{1}), T (X_{2})) with the property that for all w_{1} ∈ W (X_{1}), ϕw_{1} = w_{2}ϕ for some w_{2} ∈ W (X_{2}). Thus Rep(X_{1}, X_{2}) is T -determined in this case. The bijection (3.14) follows by exploiting the H^{∗}𝔽_{p}-equivalence BN(X_{1}) ⟶ BX_{1}.

Remark 3.15. Let S_{1} ⟶ G_{1} ⟶ π_{0}(G_{1}) and S_{2} ⟶ G_{2} ⟶ π_{0}(G_{2}) be two extensions of finite groups, π_{0}(G_{1}) and π_{0}(G_{2}), by p-compact tori, S_{1} and S_{2}. Let Hom(G_{1}, G_{2}) = [BG_{1}, ∗; BG_{2}] denote the set of based and Rep(G_{1}, G_{2}) = [BG_{1}, BG_{2}] = π_{0}(G_{2})\ Hom(G_{1}, G_{2}) the set of free homotopy classes of maps of BG_{1} into BG_{2}.

The two functors π_{1} and π_{2} define a map

Hom(G_{1}, G_{2}) → Hom_{(π_{0}(G_{1}),π_{0}(G_{2}))}(S_{1}, S_{2}) (3.16)

into the set Hom_{(π_{0}(G_{1}),π_{0}(G_{2}))}(S_{1}, S_{2}) of pairs (χ, ϕ) ∈ Hom(π_{0}(G_{1}), π_{0}(G_{2})) × Hom(S_{1}, S_{2}) such that ϕ is χ-equivariant. The fibre over (χ, ϕ) is either empty or in bijection with the set

π_{0}(map(BS_{1}, BS_{2})_{Bϕ}^{π_{0}(G_{1})}) = H^{2}(π_{0}(G_{1}); π_{2}(BS_{2})) = H_{χ}^{1}(π_{0}(G_{1}); Š_{2}) (3.17)

where π_{0}(G_{1}) acts on Š_{2}, the discrete approximation to S_{2}, through χ.

If we put w_{2} · (χ, ϕ) = (w_{2}χw_{2}^{−1}, w_{2}ϕ) for all w_{2} ∈ π_{0}(G_{2}) and all (χ, ϕ) ∈ Hom_{(π_{0}(G_{1}),π_{0}(G_{2}))}(S_{1}, S_{2}) then (3.16) becomes π_{0}(G_{2})-equivariant, so it descends to a map

Rep(G_{1}, G_{2}) → π_{0}(G_{2})\ Hom_{(π_{0}(G_{1}),π_{0}(G_{2}))}(S_{1}, S_{2}) (3.18)

of π_{0}(G_{2})-orbit sets. The fibre over the orbit π_{0}(G_{2})(χ, ϕ) is either empty or in bijection with the orbit set

π_{0}(G_{2})^{(χ,ϕ)}\H_{χ}^{1}(π_{0}(G_{1}), Š_{2})

for the action of the stabilizer group π_{0}(G_{2})^{(χ,ϕ)}, consisting of all w_{2} ∈ π_{0}(G_{2}) such that w_{2}χ = χw_{2} and w_{2}ϕ = ϕ, on the fibre (3.17).

Proposition 3.19. Let (χ, ϕ) be an element of Hom_{(π_{0}(G_{1}),π_{0}(G_{2}))}(S_{1}, S_{2}) and suppose that the stabilizer subgroup π_{0}(G_{2})^{(χ,ϕ)} acts transitively on the cohomology group H_{χ}^{1}(π_{0}(G_{1}), Š_{2}). Then at most one element of Rep(G_{1}, G_{2}) is mapped to the orbit π_{0}(G_{2})(χ, ϕ) under the map (3.18).

The rest of the paper consists of an analysis of the special case where X_{1} = SU(3) or G_{2}, X_{2} = F_{4}, and the prime p = 3.

4. Embeddings of SU(3) in F_{4}

In this section we apply the concepts of the previous sections to investigate monomorphisms from SU(3) to F_{4} at the prime p = 3. First, a few facts about the Quillen category 𝔸(F_{4}) of F_{4}.

Lemma 4.1. [23, 8.2.2] Let E^{1} be an elementary abelian group of order 3^{1}. The set Mono(E^{1}, F_{4}) of conjugacy classes of monomorphisms of E^{1} into F_{4} has three elements e_{1}^{1}, e_{2}^{1}, e_{3}^{1}. The centralizers of these three elements are connected 3-compact groups with Weyl groups of order 36, 48, and 48, respectively. The centralizer C_{F_{4}}(e_{1}^{1}) of e_{1}^{1} is isomorphic to SU(3, 3). The automorphism group Aut(E^{1}) acts trivially on Mono(E^{1}, F_{4}).

Lemma 4.2. (Cf. [23, 8.2.4], [22, 7.5]) Let E^{2} be an elementary abelian group of order 3^{2}. The set Mono(E^{2}, F_{4})/ Aut(E^{2}) of isomorphism classes of conjugacy classes of monomorphisms of E^{2} into F_{4} has 5 elements, e_{1}^{2}, e_{2}^{2}, e_{3}^{2}, e_{4}^{2}, e_{5}^{2}, with Quillen automorphism groups of order 8, 4, 12, 12, 48, and with centralizer Weyl groups of order 4, 6, 6, 8, 3, respectively. The centralizer, C_{F_{4}}(e_{5}^{2}), of e_{5}^{2} is a 3-compact toral group of maximal rank with component group π_{0}(C_{F_{4}}(e_{5}^{2})) of order 3. There are no maps in the Quillen category from e_{2}^{1} or e_{3}^{1} to e_{5}^{2}.

Proofs of (4.1) and (4.2). With computer assistance it is easy to determine, using [19, 2.5] and [17, 3.2], that Mono(E^{1}, F_{4}) is a trivial Aut(E^{1})-set containing three elements whose centralizers are connected 3-compact groups with Weyl groups of order 36, 48, 48, respectively. See [14, 3.3] for the precise structure of C_{F_{4}}(a). Since each centralizer of E^{1} is connected, any monomorphism E^{2} ⟶ F_{4} will factor through the maximal torus.

The Quillen automorphism group referred to in (4.2) consists of all automorphism of E^{2} that leaves e_{i}^{2} ∈ Mono(E^{2}, F_{4}) invariant.

We now show that for any monomorphism of SU(3) or SU(3, 3) to F_{4} the triangles

[figure: E^{1}; z; e_{1}^{1}; E^{1}; z; e_{1}^{1}; SU(3); _{F_{4}}; SU(3, 3); _{F_{4}}] (4.3)

where z: E^{1} → SU(3) and z: E^{1} → SU(3, 3) are centers, will commute up to conjugacy. This observation is the key to the classification of monomorphisms of SU(3) ↣ F_{4}.

Lemma 4.4. 1. Mono(SU(3), F_{4})_{z→e_{1}^{1}} = Mono(SU(3), F_{4}).

2. Mono(SU(3, 3), F_{4})_{z→e_{1}^{1}} = Mono(SU(3, 3), F_{4}).

The proof of this lemma uses admissible homomorphisms which we now discuss.

The Weyl group W_{1} = W (SU(3)) of SU(3) is [18, 2.6] ⟨σ, τ⟩ ⊆ Aut(Σ_{0}(ℤ_{3}^{3})) where Σ_{0}(ℤ_{3}^{3}) is the free ℤ_{3}-module with basis (1, −1, 0), (0, 1, −1) ∈ ℤ_{3}^{3} and

σ = ( \begin{matrix}0 & −1 \\ 1 & −1 \end{matrix} ) , τ = ( \begin{matrix}0 & −1 \\ −1 & 0 \end{matrix} )

The Weyl group W (F_{4}) = W (F_{4}) < GL(4, ℤ_{3}) of F_{4} is the group (of order 1152 = 384 · 3)

W (F_{4}) = W (B_{4})E ∪ W (B_{4})H_{1} ∪ W (B_{4})H_{2} (4.5)

where W (B_{4}) is the reflection group (of order 384 = 2^{4} · 4!) of all signed permutation matrices, and H_{1} and H_{2} are the matrices

H_{1} = \frac{1}{2} ( \begin{matrix}−1 & 1 & 1 & 1 \\ 1 & −1 & 1 & 1 \\ 1 & 1 & −1 & 1 \\ 1 & 1 & 1 & −1 \end{matrix} ) , H_{2} = \frac{1}{2} ( \begin{matrix}1 & 1 & 1 & 1 \\ 1 & 1 & −1 & −1 \\ 1 & −1 & 1 & −1 \\ 1 & −1 & −1 & 1 \end{matrix} )

satisfying H_{1}^{2} = E = H_{2}^{2}, H_{2}H_{1} = −H_{2}, H_{1}H_{2} = diag(−1, 1, 1, 1)H_{1}.

We say that a linear map A: Σ_{0}(ℤ_{3}^{3}) → ℤ_{3}^{4} is admissible if AW (SU(3)) ⊆ W (F_{4})A. The linear map A(u, v): Σ_{0}(ℤ_{3}^{3}) → ℤ_{3}^{4}, for instance, with matrix

A(u, v) = ( \begin{matrix}−u & v \\ u & v − u \\ 0 & v + u \\ −2v & v \end{matrix} ) = u ( \begin{matrix}−1 & 0 \\ 1 & −1 \\ 0 & 1 \\ 0 & 0 \end{matrix} ) + v ( \begin{matrix}0 & 1 \\ 0 & 1 \\ 0 & 1 \\ −2 & 1 \end{matrix} ) , u, v ∈ ℤ_{3}, (4.6)

is admissible since it is χ-equivariant where χ: W (SU(3)) → W (F_{4}) is the group homomorphism given by

χ(σ) = \frac{1}{2} ( \begin{matrix}−1 & −1 & 1 & −1 \\ 1 & −1 & −1 & −1 \\ −1 & 1 & −1 & −1 \\ 1 & 1 & 1 & −1 \end{matrix} ) , χ(τ) = \frac{1}{2} ( \begin{matrix}−1 & −1 & 1 & 1 \\ −1 & 1 & −1 & 1 \\ 1 & −1 & −1 & 1 \\ 1 & 1 & 1 & 1 \end{matrix} ) (4.7)

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 (F_{4}) as −E ∈ W (F_{4}).

Lemma 4.8. 1. Let A: Σ_{0}(ℤ_{3}^{3}) → ℤ_{3}^{4} be a linear map. Then A is admissible with respect to

W (SU(3)) and W (F_{4}) if and only if A ∈ W (F_{4})A(u, v) for some 3-adic integers u, v ∈ ℤ_{3}.

2. A(u, v) is split injective if and only if u + v is a 3-adic unit.

3. The map

⟨(−1, −1)⟩\(ℤ_{3})^{2} ⟶ W (F_{4})\ Hom_{ℤ_{3}}(Σ_{0}(ℤ_{3}^{3}), ℤ_{3}^{4})
±(u, v) ⟶ W (F_{4})A(u, v)

is injective.

Proof. 1. It is possible to show, using a computer, that, up to inner automorphisms, any admissible homomorphism Σ_{0}(ℤ_{3}^{3}) ⟶ ℤ_{3}^{4} must be χ-equivariant. Given this, one simply solves the system of linear equations Aw = χ(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

( \begin{matrix}u − 2v & 0 \\ 0 & 2v − u \\ 3u & 0 \\ −u & v \end{matrix} )

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 (F_{4}) the set of solutions to the homogeneous system of linear equations

wA(u_{1}, v_{1}) − A(u_{2}, v_{2}) = 0

in the four unknowns (u_{1}, v_{1}, u_{2}, v_{2}) is contained in the diagonal (u_{1}, v_{1}) = (u_{2}, v_{2}) or in the anti-diagonal (u_{1}, v_{1}) = −(u_{2}, v_{2}). This is easily verified on a computer.

Our interest in the admissible homomorphisms lies in the fact that the induced homomorphism π_{1}(T (f)) is admissible for any lift T (f): T (SU(3)) → T (F_{4}) to the maximal tori of any morphism f: SU(3) → F_{4}. Thus we must have π_{1}(T (f)) ∈ W (F_{4})A(u, v) for some 3-adic integers 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) ⟶ F_{4}.

The proof of (4.4) follows immediately from (4.8.1).

Proof of Lemma 4.4. 1. Let f: SU(3) → F_{4} be any monomorphism. Then π_{1}(T (f)) is admissible, so we may assume that π_{1}(T (f)) = A(u, v) for some 3-adic integers u, v ∈ ℤ_{3}. The restriction fz: E^{1} → F_{4} of f to the center z: E^{1} → SU(3) of SU(3) is given by

A(u, v) ( \begin{matrix}−1 \\ 1 \end{matrix} ) = ( \begin{matrix}u + v \\ u + v \\ u + v \\ 0 \end{matrix} ) (4.9)

where we have reduced modulo 3. Since fz is a monomorphism, u + v ≢ 0 mod 3 and then the stabilizer in W (F_{4}) of (u + v, u + v, u + v, 0) ∈ (ℤ/3)^{4} has order 36. Thus fz ≃ e_{1}^{1} ∈ Mono(E^{1}, F_{4}).

2. Let f: SU(3, 3) → F_{4} be any monomorphism and choose some monomorphism g: SU(3) → SU(3, 3) such that gz = z, e.g. g = ψ^{(1,0)}. Then fz = fgz = e_{1}^{1}.

Let e: SU(3, 3) = C_{F_{4}}(e_{1}^{1}) → F_{4} denote the inclusion of the centralizer of e_{1}^{1} into F_{4}; this map is described in detail in [14, 3.3].

Corollary 4.10. The maps

Mono(SU(3), SU(3, 3))_{z→z} ⟶^{e∘−} Mono(SU(3), F_{4})
Out(SU(3, 3))_{z→z} ⟶^{e∘−} Mono(SU(3, 3), F_{4})

are bijections.

Proof. By (3.9) and (4.4),

Mono(SU(3), SU(3, 3)_{z→z} = Mono(SU(3), F_{4})_{z⟶e_{1}^{1}} = Mono(SU(3), F_{4})

and similarly for morphisms from SU(3, 3).

Lemma 4.11. Let ψ^{(u,v)} : SU(3) → SU(3, 3) be the morphism (2.15) indexed by u, v ∈ ℤ_{3}^{∗} ∪ {0}. Then W (F_{4})π_{1}(T (eψ^{(u,v)})) = W (F_{4})A(u, v).

Proof. The monomorphism e: SU(3, 3) → F_{4} is [14, 3.3] realizable on the level of compact Lie groups as an inclusion SU(3) ↪ F_{4} such that the restriction Σ_{0}(ℤ^{3})×Σ_{0}(ℤ^{3}) → Σ_{2}(ℤ^{4}) to the integral lattices of the composite morphism SU(3)×SU(3) ↠ SU(3, 3) ↪ F_{4} takes (x_{1}, x_{2}, x_{3}; y_{1}, y_{2}, y_{3}) to (x_{1} + y_{3}, x_{2} + y_{3}, x_{3} + y_{3}, y_{1} − y_{2}). Thus

( \begin{matrix}1 & 0 & 0 & −1 \\ −1 & 1 & 0 & −1 \\ 0 & −1 & 0 & −1 \\ 0 & 0 & 2 & −1 \end{matrix} ) ( \begin{matrix}u & 0 \\ 0 & u \\ v & 0 \\ 0 & v \end{matrix} ) = ( \begin{matrix}u & −v \\ −u & u − v \\ 0 & −u − v \\ 2v & −v \end{matrix} ) = −A(u, v)

represents π_{1}(T (eψ^{(u,v)})).

Lemma 4.12. Let u and v be 3-adic integers and A(u, v) the corresponding admissible homomorphism.

1. There exists a morphism f: SU(3) → F_{4} such that W (F_{4})π_{1}(T (f)) = W (F_{4})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) → F_{4} such that W (F_{4})π_{1}(T (f)) = W (F_{4})A(u, v) if and only if u, v ∈ ℤ_{3}^{∗} ∪ {0} and u + v ∈ ℤ_{3}^{∗}.

Proof. We have already seen (4.11) that A(u, v) is realizable for all u, v ∈ ℤ_{3}^{∗} ∪ {0}.

Suppose, conversely, that π_{1}(T (f)) = A(u, v) for some 3-adic integers, u and v, and some morphism f: SU(3) → F_{4}. If f is a monomorphism, then f = eψ^{(u,v)} for some u, v ∈ ℤ_{3}^{∗} ∪ {0} with u + v ∈ ℤ_{3}^{∗} by (4.10). If f is not a monomorphism, A(u, v) is not split injective [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), F_{4}) is T -determined.

2. The map

⟨(−1, −1)⟩\{(u, v) ∈ (ℤ_{3}^{∗} ∪ {0})^{2}|u + v ∈ ℤ_{3}^{∗}} ⟶ Mono(SU(3), F_{4})
±(u, v) ⟶ eψ^{(u,v)}

is a bijection.

Proof. 1. The restriction map Mono(SU(3), F_{4}) ⟶ Mono(T (SU(3)), F_{4}) can be identified to the map

{(u, v) ∈ (ℤ_{3}^{∗} ∪ {0})^{2} | u + v ≡ 1 mod 3} ⟶ W (F_{4})\ Hom(Σ_{0}(ℤ_{3}^{3}), ℤ_{3}^{4})
(u, v) ⟶ W (F_{4})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 diagrams

\xymatrix{Mono(SU(3), SU(3, 3))_{z→z} \ar[dr]_{e∘−}^{≅} \ar[r] & ⟨ψ^{−1} × ψ^{−1}⟩\ Mono(SU(3), SU(3, 3)) \ar[d]^{e∘−} \\  & Mono(SU(3), F_{4}) \\ Out(SU(3, 3))_{z→z} \ar[r] \ar[dr]_{e∘−}^{≅} & ⟨ψ^{−1} × ψ^{−1}⟩\ Out(SU(3, 3) \ar[d]^{e∘−} \\  & Mono(SU(3, 3), F_{4})}

where the slanted arrows are bijections. The vertical arrows exist because e(ψ^{−1} × ψ^{−1}) = e by [14, 3.3]. Noting (2.15) that

Mono(SU(3), SU(3, 3))_{z→z} = {(u, v) ∈ (ℤ_{3}^{∗} ∪ {0})^{2} | u + v ≡ 1 mod 3}
Out(SU(3, 3))_{z→z} = {(u, v) ∈ (ℤ_{3}^{∗})^{2} | u ≡ 1 ≡ v mod 3} ⋊ ⟨τ⟩

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), F_{4}) × Out(SU(3, 3)) → Mono(SU(3, 3), F_{4})

is transitive and the stabilizer subgroup at the centric monomorphism e, i.e. the Weyl group [18, 5.6]

W_{F_{4}}(e SU(3, 3)) = ⟨ψ^{−1} × ψ^{−1}⟩ (4.15)

is cyclic of order two.

The next lemma lists the centralizers of all monomorphisms SU(3) ↣ F_{4}. We let ψ^{−1} denote the automorphism ψ^{−1} ×_{Z(SU(3))} ψ^{−1} of T (SU(3) ×_{Z(SU(3))} SU(3) [17, 4.3].

Lemma 4.16. Let (u, v) ∈ (ℤ_{3}^{∗} ∪ {0})^{2} and u + v ∈ ℤ_{3}^{∗}. If uv ≠ 0, then

C_{F_{4}}(eψ^{(u,v)} SU(3)) = Z(SU(3))
C_{F_{4}}(eψ^{(u,v)}T (SU(3))) = T (F_{4})

If uv = 0, then

C_{F_{4}}(eψ^{(u,v)} SU(3)) = Z(SU(3) ×_{Z(SU(3))} SU(3)
C_{F_{4}}(eψ^{(u,v)}T (SU(3))) = T (SU(3) ×_{Z(SU(3))} SU(3)

In all cases, C_{F_{4}}(ψ^{−1}) = ψ^{−1}.

Proof. It only remains to determine the map C_{F_{4}}(ψ^{−1}) induced by ψ^{−1} since the centralizers themselves are given by (2.17, 3.9). Let us, for example, consider the case where (u, v) = (0, 1). Consider the morphism μ: (SU(3) × T (SU(3))) × T (SU(3)) ⟶ SU(3) × T (SU(3)) ⟶ SU(3, 3) constructed from the multiplication on the maximal torus and the projection map. Since

eμ((1 × 1) × ψ^{−1}) = e(ψ^{−1} × ψ^{−1})μ((1 × 1) × ψ^{−1}) = eμ((ψ^{−1} × ψ^{−1}) × 1)

it follows from (4.17) that C_{F_{4}}(ψ^{−1}) = ψ^{−1} on C_{F_{4}}(eψ^{(0,1)}T (SU(3))). The other cases are similar.

Lemma 4.17. If the diagram of p-compact groups

\xymatrix{X_{1} × X_{2} \ar[d]_{μ} & X_{1} × X′_{2} \ar[l]_{1×f_{2}} \ar[r]^{f_{1}×1} & X′_{1} × X′_{2} \ar[d]^{μ′} \\ Y \ar[rr]_{h} &  & Y′}

commutes up to conjugacy, so does the induced diagram

\xymatrix{X_{1} \ar[d]_{f_{1}} \ar[r]^{ad(μ)} & C_{Y} (X_{2}) \ar[d]^{C_{h}(f_{2})} \\ X′_{1} \ar[r]^{ad(μ′)} & C_{Y′}(X′_{2})}

where the horizontal arrows are adjoints of μ and μ′.

Corollary 4.18. Let N be a (topological) group with subgroups g_{1} : G_{1} → N and g_{2} : G_{2} → N. Suppose that n ∈ N is an element such that conjugation with n, c(n)(m) = nmn^{−1}, m ∈ N, takes G_{1} into G_{2}. Then conjugation with n^{−1} takes the centralizer C_{N}(G_{2}) into C_{N}(G_{1}) and the diagram

\xymatrix{BC_{N}(G_{1}) \ar[r] & map(BG_{1}, BN)_{Bg_{1}} \\ BC_{N}(G_{2}) \ar[u]^{Bc(n^{−1})} \ar[r] & map(BG_{2}, BN)_{Bg_{2}} \ar[u]_{\overline{Bc(n)}}}

commutes up to homotopy.

Proof. We have μ(c(n) × 1) = c(n)μ(1 × c(n^{−1})) where μ is group multiplication and where the induced map Bc(n): BN → BN is homotopic to the identity.

5. Toric representations of PU(3) in F_{4}

In this section I classify the p-toric morphisms from PU(3) to F_{4} 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 subgroup of the discrete approximation Ť = (π_{1}(T) ⊗ ℚ)/π_{1}(T) such that the composition Z → Ť → X is a central monomorphism. Then we may form the p-compact group X/Z [6, 8.3] with induced maximal torus i/Z: T/Z → X/Z [20, 4.6] that fits into the commutative diagram

[figure: 0; κ^{−1}(0); κ^{−1}(Z); Z; 0; 0; π_{1}(T); π_{1}(T) ⊗ ℚ; κ; ≅; Ť; 0; 0; π_{1}(T/Z); π_{1}(T/Z) ⊗ ℚ; Ť/Z; 0]

with exact rows. From this we get an isomorphism

[figure: 0; π_{1}(T); π_{1}(T/Z); Z; 0; 0; κ^{−1}(0); κ^{−1}(Z); Z; 0]

of extensions of W_{T} (X) = W_{T/Z}(X/Z)-modules.

In particular, let Σ_{0}(ℤ_{3}^{3}) ⊆ Σ_{0}(ℚ_{3}^{3}) be the free ℤ_{3}-submodule with basis e_{1} = (1, −1, 0) and e_{2} = (0, 1, −1); this is the integral lattice for SU(3). Put f = \frac{1}{3}(e_{1} − e_{2}) and let P Σ_{0}(ℤ_{3}^{3}) be the free ℤ_{3}-submodule of ℚ_{3}^{3} with basis {e_{1}, f}. Then there is an exact sequence

0 ⟶ Σ_{0}(ℤ_{3}^{3}) ⟶^{ι} P Σ_{0}(ℤ_{3}^{3}) ⟶ ℤ/3 ⟶ 0

of ℤ_{3}[Σ_{3}]-modules and P Σ_{0}(ℤ_{3}^{3}) corresponds to the maximal torus for PU(3).

Note that there is an extension, B(u, v), of A(u, v),

\xymatrix{Σ_{0}(ℤ_{3}^{3}) \ar[r]^{ι} \ar[d]_{A(u,v)} & P Σ_{0}(ℤ_{3}^{3}) \ar[dl]^{B(u,v)} \\ L_{4}}

if and only if u + v is divisible by 3 and in that case the extension is unique and given by

B(u, v) = A(u, v) ( \begin{matrix}1 & 1 \\ 0 & −3 \end{matrix} )^{−1} = ( \begin{matrix}−u & −\frac{1}{3}(u + v) \\ u & \frac{1}{3}(2u − v) \\ 0 & −\frac{1}{3}(u + v) \\ −2v & −v \end{matrix} )

where u and v are 3-adic integers and u + v ∈ 3ℤ_{3}. Moreover, the inclusion ι is W (SU(3)) = W (PU(3))-equivariant and B(u, v) is χ-equivariant where χ is the group homomorphism from (4.7).

Lemma 5.1. 1. A ℤ_{3}-linear map B: P Σ_{0}(ℤ_{3}^{3}) → ℤ_{3}^{4} is admissible with respect to W (PU(3)) and W (F_{4}) is and only if B ∈ W (F_{4})B(u, v) where u and v are 3-adic integers whose sum is divisible by 3.

2. B(u, v) is split-injective when u and v are 3-adic units.

3. The map

⟨(−1, −1)⟩\{(u, v) ∈ (ℤ_{3})^{2}|u + v ∈ 3ℤ_{3}} → W (F_{4})\ Hom_{ℤ_{3}}(P Σ_{0}(ℤ_{3}^{3}), ℤ_{3}^{4})
±(u, v) → W (F_{4})B(u, v)

is injective.

Proof. 1. B is admissible if and only if B ∘ι is, i.e. if and only if B is an extension of A(u, v) (4.8.1) for some 3-adic integers, u and v.

2. If u and v are units then

( \begin{matrix}−u^{−1} & 0 & u^{−1} & 0 \\ 2u^{−1} & 0 & 2u^{−1} & −v^{−1} \end{matrix} )

is a left inverse of B(u, v).

3. If B(u_{1}, v_{1}) ∈ W (F_{4})B(u_{2}, v_{2}) then also A(u_{1}, v_{1}) ∈ W (F_{4})A(u_{2}, v_{2}) and then (4.8.3) (u_{1}, v_{1}) and (u_{2}, v_{2}) are equal up to sign.

When u, v ∈ ℤ_{3}^{∗} ∪\{0\} with sum u+v ∈ 3ℤ_{3} there is a unique conjugacy class, \overline{ψ}^{(u,v)}, that makes the diagram

\xymatrix{SU(3) \ar[d] \ar[r]^{ψ^{(u,v)}} & SU(3, 3) \ar[r]^{e} & F_{4} \\ PU(3) \ar[ur]_{_{\overline{ψ}_{(u,v)}}}}

commutes up to conjugation. By construction, W (F_{4})π_{1}(T (e∘\overline{ψ}^{(u,v)}(T (PU(3))))) = W (F_{4})B(u, v).

Lemma 5.2. Let u and v be 3-adic integers with sum u + v ∈ 3ℤ_{3} and let B(u, v): P Σ_{0}(ℤ_{3}^{3}) → ℤ_{3}^{4} be the corresponding admissible homomorphism.

1. There exists a morphism f: PU(3) → F_{4} such that W (F_{4})π_{1}(T (f)) = W (F_{4})B(u, v) if and only if u = 0 = v or u, v ∈ ℤ_{3}^{∗}.

2. There exists a monomorphism f: PU(3) → F_{4} such that W (F_{4})π_{1}(T (f)) = W (F_{4})B(u, v) if and only if u, v ∈ ℤ_{3}^{∗}.

Proof. We have already seen that W (F_{4})B(u, v) is realizable by a morphism f: PU(3) → F_{4} if u = 0 = v or u, v ∈ ℤ_{3}^{∗}; if both u and v are non-zero then f is a monomorphism by (5.1.2). Conversely, if W (F_{4})B(u, v) is realizable, so is W (F_{4})A(u, v) and then (4.12) u, v ∈ ℤ_{3}^{∗} ∪ {0}, u + v ∉ ℤ_{3}^{∗}.

Alternatively, (5.2) says that any non-trivial morphism PU(3) ⟶ F_{4} is a monomorphism.

Proposition 5.3. (Cf. [1, 2.27.(ii)]) Suppose that u and v are 3-adic units with u + v ∈ 3ℤ_{3}. Then

T (PU(3)) ⟶^{B(u,v)} T (F_{4}) ⟶^{i_{2}} F_{4}

is toric if and only if (u, v) ∉ ℤ_{3}^{∗}(2, 1) ∪ ℤ_{3}^{∗}(1, −1).

Proof. Explicit (computer aided) computations of W (F_{4})^{B(u,v)} = W (F_{4})^{A(u,v)}.

The two generic non-3-toric morphisms

B(2, 1) = ( \begin{matrix}−2 & −1 \\ 2 & 1 \\ 0 & −1 \\ −2 & −1 \end{matrix} ) and B(1, −1) = ( \begin{matrix}−1 & 0 \\ 1 & 1 \\ 0 & 0 \\ 2 & 1 \end{matrix} )

are related by the equation εB(2, 1) = 2B(1, −1) where

ε = ( \begin{matrix}0 & 0 & −1 & 1 \\ 0 & 0 & −1 & −1 \\ −1 & −1 & 0 & 0 \\ −1 & 1 & 0 & 0 \end{matrix} )

is the admissible automorphism of ℤ_{3}^{4} corresponding to the exotic automorphism of F_{4}. (In general, W (F_{4})(εA(u, v)) = W (F_{4})(A(2v, −u)), cf. [1, 2.11].)

Theorem 5.4. 1. TRep(PU(3), F_{4}) is T -determined.

2. The map

⟨(−1, −1)⟩\({(u, v) ∈ (ℤ_{3}^{∗})^{2} | u + v ∈ 3ℤ_{3}}) \ (ℤ_{3}^{∗}(2, 1) ∪ ℤ_{3}^{∗}(1, −1))) → TRep(PU(3), F_{4})
±(u, v) → e ∘ \overline{ψ}^{(u,v)}

is a bijection.

Consider the set Rep(N(PU(3)), N(F_{4})) of conjugacy classes of maps from the maximal torus normalizer N(P U(3)) of PU(3) to the maximal torus normalizer N(F_{4}) of F_{4}. As we have seen (3.18), there is a map

Rep(N(PU(3)), N(F_{4})) → W (F_{4})\ Hom_{(W (PU(3)),W (F_{4}))}(T (PU(3)), T (F_{4}))

induced by the functors π_{1} and π_{2}. It is easy to calculate directly that the cohomology group H^{2}(⟨χ(σ)⟩; π_{1}(T (F_{4}))) is trivial. Then also

H_{χ}^{2}(W (PU(3)); π_{1}(T (F_{4}))) = 0 (5.5)

for ⟨σ⟩ 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(F_{4})) corresponding to the orbit W (F_{4})(χ, B(u, v)), (u, v) ∈ (ℤ_{3}^{∗})^{2}, u + v ∈ 3ℤ_{3}.

Proof of Theorem 5.4. Let f_{1}, f_{2} ∈ T Rep(PU(3), F_{4}) be two toric representations and suppose that their restrictions to the maximal torus of PU(3) agree. Under the map

TRep(PU(3), F_{4}) ⟶ TRep(N(PU(3)), N(F_{4})) ⟶ W (F_{4})\ Hom_{(W (PU(3)),W (F_{4}))}(T (PU(3)), T (F_{4}))

f_{1} and f_{2} go to the same element of the target and it follows (5.6) that the lifts (3.5) N(f_{1}) and N(f_{2}) are conjugate, i.e. that f_{1} and f_{2} have conjugate restrictions to the maximal torus normalizer N(PU(3)). In fact, N(f_{1}) = B(u, v) ⋊ χ = N(f_{2}) for some (u, v) ∈ (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, 𝔽_{3}), S_{3}) (a full subcategory of the Quillen category that may be described as formed from the inclusion of a Sylow 3-subgroup S_{3} into the special linear group SL(2, 𝔽_{3})) with just two objects, λ: E^{1} → PU(3) and ν: E^{2} → PU(3), where E^{1} and E^{2} are elementary abelian groups of order 3 and 3^{2}, respectively [12, 6.8, 7,7], see [19] for the notation used here. Since f_{1} and f_{2} agree on the centralizers, C_{PU(3)}(λE^{1}) = N_{3}(PU(3)) and C_{PU(3)}(νE^{2}) = E^{2}, it only remains to compute the relevant Wojtkowiak obstruction groups [24]. For this we need information about the centralizer C_{F_{4}}(f_{i}E^{2}) and C_{F_{4}}(f_{i}N_{3}(PU(3))).

We must have f_{1}|E^{2} = e_{5}^{2} = f_{2}|E^{2} for only e_{5}^{2} ∈ Mono(E^{2}, F_{4}) can contain in its automorphism group the automorphism group SL(2, 𝔽_{3}) of (E^{2}, ν). Thus C_{F_{4}}(f_{i}E^{2}) is a p-compact toral group of maximal rank with E^{1} as its component group (4.2).

The centralizer C_{F_{4}}(f_{i}N_{3}(PU(3))) is (3.4) the p-compact toral group

C_{Ť(F_{4})⋊W (F_{4})}(Ť(PU(3)) ⋊ ⟨σ⟩) = Ť(F_{4})^{⟨χ(σ)⟩} = t(F_{4})^{⟨χ(σ)⟩} = E^{2}

where t(F_{4}) ⊂ Ť(F_{4}) denotes the maximal elmentary abelian subgroup of the discrete approximation Ť(F_{4}) to T (F_{4}) and

The obstructions to a homotopy between the two maps Bf_{1}, Bf_{2} : BPU(3) → BF_{4} lie in the abelian groups lim_{𝕀}^{1}\underline{π}_{1} and lim_{𝕀}^{2}\underline{π}_{2} where \underline{π}_{1} and \underline{π}_{2} are the abelian 𝕀-groups

[figure: E^{2}; ℤ/2; SL(2,𝔽_{3})/S_{3}; E^{1}; SL(2,𝔽_{3})]
[figure: 0; ℤ/2; SL(2,𝔽_{3})/S_{3}; ℤ_{3}^{4}; SL(2,𝔽_{3})]

given by the homotopy groups of the above centralizers. The group SL(2, 𝔽_{3}) has no normal subgroups of index two, so it necessarily acts trivially on E^{1}. It now follows from [18, 3.1] that both obstruction groups are trivial and we conclude that f_{1} and f_{2} are conjugate. This shows that TRep(PU(3), F_{4}) is T -determined.

Let now f: PU(3) → F_{4} be any toric monomorphism. Then there is (5.1.3, 5.3) a unique, up to sign, pair of units (u, v) ∈ (ℤ_{3}^{∗})^{2}, u + v ∈ 3ℤ_{3}, (u, v) ∉ ℤ_{3}^{∗}(2, 1) ∪ ℤ_{3}^{∗}(1, −1), such that W (F_{4})π_{1}(T (f)) = W (F_{4})B(u, v) and then f = \overline{ψ}^{(u,v)} since the p-toric monomorphisms are T - determined.

Lemma 5.7. Let (u, v) ∈ (ℤ_{3}^{∗})^{2}, u + v ∈ 3ℤ_{3}, (u, v) ∉ ℤ_{3}^{∗}(2, 1) ∪ ℤ_{3}^{∗}(1, −1). Then

C_{F_{4}}(eψ^{(u,v)} SU(3)) = Ť(F_{4})^{χ(W (SU(3)))}
C_{F_{4}}(eψ^{(u,v)}T (SU(3))) = T (F_{4})

and C_{F_{4}}(ψ^{−1}) = ψ^{−1} in both cases.

Proof. Since eψ^{(u,v)} is toric, the centralizer in F_{4} of eψ^{(u,v)}T (SU(3)) equals the maximal torus of F_{4}. Proceed as in (4.16) to show that C_{F_{4}}(ψ^{−1}) = ψ^{−1}.

The centralizer BC_{F_{4}}(e\overline{ψ}^{(u,v)} PU(3)) is the homotopy colimit of the 𝕀-space

[figure: B(0); ℤ/2; SL(2,𝔽_{3})/S_{3}; B(1); SL(2,𝔽_{3})]

where B(0) = BŤ(F_{4})^{⟨χ(σ)⟩} and B(0) = BC_{F_{4}}(e_{5}^{2}). We need to be more specific about the group actions that occur here.

The 3-normalizer N_{3}(PU(3)) = C_{N(PU(3))}(Ť(PU(3))^{⟨σ⟩}) is the centralizer in N(PU(3)) of Ť(PU(3))^{⟨σ⟩} = E^{1}. Since conjugation by (0, τ) restricts to the non-trivial automorphism of Ť(PU(3))^{⟨σ⟩} we see that the induced action on N_{3}(PU(3)) = T (PU(3)) ⋊ ⟨σ⟩ is given by conjugation with (0, τ) ∈ Ň(PU(3)) = Ť(PU(3)) ⋊ W (PU(3)).

Since B̌(u, v) ⋊ χ: Ň_{3}(PU(3)) → Ň(F_{4}) is χ-equivariant whith the Weyl groups acting by conjugation, we see (4.17) that ℤ/2-acts on Ť(F_{4})^{⟨χ(σ)⟩} = C_{Ň(F_{4})}(Ň_{3}(PU(3))) as conjugation with (0, χ(τ)). With this information it is now easy to see, using [18, 3.1], that

lim_{𝕀}^{0}π_{1} = (Ť(F_{4})^{⟨χ(σ)⟩})^{⟨χ(τ)⟩} = Ť(F_{4})^{χ(W (SU(3)))}

is the only non-trivial contribution from the 𝕀-groups \underline{π}_{1} and \underline{π}_{2} to the Bousfield-Kan spectral sequence. This means that the morphisms

C_{F_{4}}(e\overline{ψ}^{(u,v)} PU(3)) ⟶ C_{F_{4}}(N(e\overline{ψ}^{(u,v)})(N(PU(3)))) ⟵ C_{N(F_{4})}(N(e\overline{ψ}^{(u,v)})(N(PU(3))))

are isomorphisms. Consider the corresponding group homomorphism μ: Ť(F_{4})^{χ(W (SU(3)))} × Ň(SU(3)) → Ň(F_{4}) which is the inclusion on the first factor and equals Ň(eψ^{(u,v)}) on the second factor. Since ψ^{−1} ⋊ 1 is inner on Ň(F_{4}), we have μ(1 × (ψ^{−1} ⋊ 1)) = (ψ^{−1} ⋊ 1)μ(1 × (ψ^{−1} ⋊ 1)) = μ(ψ^{−1} × (1 ⋊ 1)) up to inner automorphism. This shows (4.17) that C_{F_{4}}(ψ^{−1}) = ψ^{−1} is the non-trivial automorphism of C_{F_{4}}(eψ^{(u,v)} SU(3)) = E^{1}.

6. Non-toric morphisms of PU(3) to F_{4}

The non-toric morphisms of PU(3) to F_{4} require special treatment. It is the object of this section to show that also the non-toric morphisms are T -determined, i.e. to complete the proof of the following theorem.

Theorem 6.1. 1. Mono(PU(3), F_{4}) is T -determined.

2. The map

⟨(−1, −1)⟩\{(u, v) ∈ (ℤ_{3}^{∗})^{2}|u + v ∈ 3ℤ_{3}} ⟶ Mono(PU(3), F_{4})
±(u, v) ⟶ e\overline{ψ}^{(u,v)}

is a bijection.

Since the toric morphisms were dealt with in (5.4) only the non-toric ones need 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 kind of preferred lifts in certain situations.

Let G be a p-compact toral group sitting in short exact sequence S ⟶^{i_{1}} G ⟶ π_{0}(G) where S is a p-compact torus and π_{0}(G) cyclic p-group. Let j: N → X be the maximal torus normalizer of a p-compact group, X, and let i_{2} : T → N be the inclusion of the identity component. Suppose that we are given a morphisms, B and f, such that the diagram

\xymatrix{S \ar[r]^{B} \ar[d]_{i_{1}} & T \ar[d]^{ji_{2}} \\ G \ar[r]_{f} & X}

commutes up to conjugacy and B is admissible in the sense that for any ξ ∈ π_{0}(G) there exists some w in the Weyl group for X such that Bξ = wB.

Lemma 6.2. Assuming that the component group π_{0}(G) is cyclic there is a unique representation ϕ ∈ Rep(G, N) such that the diagram

\xymatrix{ &  & T \ar[d]^{i_{2}} \\  &  & N \ar[d]^{j} \\ S \ar[r]_{i_{1}} \ar[uurr]^{B} & G \ar[ur]^{ϕ} \ar[r]_{f} & X}

commutes up to conjugacy and such that the morphism

C_{j}: C_{N}(ϕG) → C_{X}(fG),

induced by j, is a maximal torus normalizer for the centralizer C_{X}(fG) of G in X.

Proof. The π_{0}(G)-map induced by j

\xymatrix{BC_{N}(i_{2}BS)_{hπ_{0}(G)} \ar[rr] \ar[dr] &  & BC_{X}(ji_{2}BS)_{hπ_{0}(G)} \ar[dl] \\  & Bπ_{0}(G)}

between the π_{0}(G)-spaces BC_{N}(i_{2}BS) = map(BS, BN)_{i_{2}B} and BC_{X}(ji_{2}BS) = map(BS, BX)_{ji_{2}B} is a maximal torus normalizer. There is an induced map

map(BG, BN)_{i_{1}→i_{2}B} = BC_{N}(i_{2}BS)^{hπ_{0}(G)} → BC_{X}(ji_{2}BS)^{hπ_{0}(G)} = map(BG, BX)_{i_{1}→ji_{2}B} (6.3)

of homotopy fixed point spaces.

According to [16, 4.6], the section Bf ∈ BC_{X}(ji_{2}BS)^{hπ_{0}(G)} admits, since π_{0}(G) is assumed to be cyclic, a unique lift Bϕ ∈ BC_{N}(i_{2}BS)^{hπ_{0}(G)} such that the restriction of (6.3) to the corresponding components,

BC_{N}(ϕG) = map(BG, BN)_{Bϕ} → map(BG, BX)_{Bf} = BC_{X}(fG)

is a maximal torus normalizer for the p-compact group C_{X}(fG).

After these general and preparatory remarks, we now return to the discussion of non-toric morphisms from PU(3) to F_{4}.

Let f: PU(3) → F_{4} be a morphism of 3-compact groups such that f|T (PU(3)) = W (F_{4})B(2, 1) ∈ [BT (PU(3)), BF_{4}]. By (6.2), there is a unique ϕ(2, 1) ∈ Rep(N_{3}(PU(3)), N(F_{4})), extending B(2, 1), such that C_{N(F_{4})}(ϕ(2, 1)N_{3}(PU(3))) is a maximal torus normalizer for C_{F_{4}}(fN_{3}(PU(3))). We shall now determine this map ϕ(2, 1).

Let Ň_{3} = Ť_{1} ⋊ ⟨σ⟩ and Ň_{2} = Ť_{2} ⋊ W_{2} be the discrete approximations to the the 3-normalizer N_{3}(PU(3)) and the maximal torus normalizer N(F_{4}), respectively. Also, let B̌(2, 1): Ť_{1} → Ť_{2} be a discrete approximation to B(2, 1). The stabilizer subgroup W (F_{4})^{B̌(2,1)} at B̌(2, 1) for the action of W (F_{4}) on Hom(Ť_{1}, Ť_{2}) is isomorphic to the permutation group Σ_{3} and generated by the two Weyl group elements

w_{1} = ( \begin{matrix}0 & 0 & 0 & 1 \\ −1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & −1 & 0 & 0 \end{matrix} ) and w_{2} = ( \begin{matrix}0 & −1 & 0 & 0 \\ −1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )

of order 3 and 2, respectively.

Lemma 6.4. The discrete approximation ϕ̌(2, 1): Ť_{1} ⋊ ⟨σ⟩ → Ť_{2} ⋊ W (F_{4}) to ϕ(2, 1) is conjugate to B̌(2, 1) ⋊ χ.

Proof. For general reasons, the discrete approximation ϕ̌(2, 1) to ϕ(2, 1) has the form ϕ̌(2, 1)(t, 1) = (B̌(2, 1)(t), 1) and ϕ̌(2, 1)(0, σ) = (a, λ(σ)) where λ: ⟨σ⟩ → W (F_{4}) is a group homomorphism, B̌(2, 1) is λ-equivariant and a ∈ Z^{1}(⟨λ(σ)⟩; Ť(F_{4})) is a 1-cocycle.

Since the homomorphism B̌(2, 1) is χ-equivariant we know that λ(σ) is an element of order 3 in the coset χ(σ)W_{2}^{B̌(2,1)}. This leaves the three possibilities χ(σ), χ(σ)w_{1}, and χ(σ)w_{1}^{2} for λ(σ). Since w_{2} conjugates χ(σ) into χ(σ)w_{1}^{2} we can ignore the third possibility. We now rule out the second possibility.

Assume for the moment that λ(σ) = χ(σ)w_{1}. Explicit computation shows that H^{0}(⟨χ(σ)w_{1}⟩; Ť(F_{4})) is a 3-discrete torus of rank 2 and that H^{0}(⟨χ(σ)w_{1}⟩; Ť(F_{4})) is cyclic of order 3 generated by the cohomology class of the 1-cocycle

a = ( \begin{matrix}0 \\ 0 \\ 1 \\ 0 \end{matrix} ) ∈ t(F_{4}) ⊂ Ť(F_{4})

which is fixed by W (F_{4})^{B̌(2,1)}. It follows that the centralizer

C_{Ť(F_{4})⋊W (F_{4})}(ϕ̌(2, 1)Ň_{3}) = C_{Ť(F_{4})⋊W (F_{4})}(B̌(2, 1)Ť(PU(3))) ∩ C_{Ť(F_{4})⋊W (F_{4})}(a, χ(σ)w_{1})
= (Ť(F_{4}) ⋊ W (F_{4})^{B̌(2,1)}) ∩ C_{Ť(F_{4})⋊W (F_{4})}(a, χ(σ)w_{1})
= C_{Ť(F_{4})⋊W (F_{4})^{B̌(2,1)}}(a, χ(σ)w_{1})
= Ť(F_{4})^{⟨χ(σ)w_{1}⟩} ⋊ W (F_{4})^{B̌(2,1)}

is the (discrete) maximal torus normalizer for SU(3) and hence (6.2) that C_{F_{4}}(fN_{3}(PU(3))) is isomorphic to the N-determined 3-compact group SU(3) [19]. Thus ϕ(2, 1): N_{3}(PU(3)) → F_{4} extends to a morphism N_{3}(PU(3)) × SU(3) ⟶ F_{4} which is a non-toric monomorphism on the second factor and we get a factorization

N_{3}(PU(3)) ⟶ C_{F_{4}}(SU(3)) = SU(3) ⟶ F_{4}

of ϕ(2, 1) through another non-toric monomorphism of SU(3) to F_{4}. The restriction of this map to the maximal tori

T (PU(3)) ⟶ T (SU(3)) ⟶ T (F_{4})

provides a factorization, up to left action by W (F_{4}), of B(2, 1) as the composition of an isomorphism followed by A(u, 0) or A(0, u), u ∈ ℤ_{3}^{∗}, and hence we have that the set

W (F_{4}) · A(2, 1) · GL(Σ_{0}(ℚ_{3}^{3})) ⊂ Hom_{ℚ_{3}}(Σ_{0}(ℚ_{3}^{3}), ℚ_{3}^{4})

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 λ(σ) = χ(σ)w_{1} can not occur and we are left with λ(σ) = χ(σ) as the only possibility. As H^{1}(⟨χ(σ)⟩; Ť(F_{4})) = 0 (5.5), ϕ̌(2, 1) = B̌(2, 1) ⋊ χ is, up to conjugation, the only extension of the pair (B̌(2, 1), χ) to a homomorphism Ť_{1} ⋊ ⟨σ⟩ → Ť(F_{4}) ⋊ W (F_{4}).

A similar statement holds for the non-toric morphism B(1, −1) which differs from B(2, 1) by an automorphism of F_{4}.

Proof of Theorem 6.1. It suffices to show that f_{1} ≃ f_{2} whenever f_{1}, f_{2} : PU(3) → F_{4} are monomorphisms such that f_{1}|T (PU(3)) = W (F_{4})B(2, 1) = f_{2}|T (PU(3)). We already know (6.4) that the two morphisms become conjugate when restricted to N_{3}(PU(3)). Therefore, the situation is now exactly as in the proof of Theorem 5.4: In order to compute the relevant Wojtkowiak obstruction groups [24] we need information about the centralizer C_{F_{4}}(f_{i}E^{2}) and C_{F_{4}}(f_{i}N_{3}(PU(3))).

Again, we must have f_{1}|E^{2} = e_{5}^{2} = f_{2}|E^{2} and C_{F_{4}}(f_{i}E^{2}) is a p-compact toral group of maximal rank with ℤ/3 as its component group (4.2).

Also, we know (6.2, 6.4) that the centralizer in Ť(F_{4})⋊W (F_{4}) of ϕ̌(2, 1) is the (discrete) maximal torus normalizer for C_{F_{4}}(f_{i}N_{3}(PU(3))). Since

C_{Ť(F_{4})⋊W (F_{4})}(ϕ̌(2, 1)Ň_{3}) = C_{Ť(F_{4})⋊W (F_{4})}(B̌(2, 1)Ť_{1}) ∩ C_{Ť(F_{4})⋊W (F_{4})}(χ(σ))
= (Ť(F_{4}) ⋊ W (F_{4})^{B̌(2,1)}) ∩ (Ť(F_{4})^{χ(σ)} ⋊ C_{W (F_{4})}(χ(σ)))
= Ť(F_{4})^{χ(σ)} ⋊ C_{W (F_{4})^{B̌(2,1)}}(χ(σ))
= t(F_{4})^{χ(σ)} ⋊ ⟨w_{1}⟩

is a finite group (of order 27 and with center of order 3) it follows that also C_{F_{4}}(f_{i}N_{3}(PU(3))) is this finite, but non-abelian, 3-group.

The obstructions to a homotopy between the two maps Bf_{1}, Bf_{2} : BPU(3) → BF_{4} lie in the set lim_{𝕀}^{1}\underline{π}_{1} and in the abelian group lim_{𝕀}^{2}\underline{π}_{2} where \underline{π}_{1} and \underline{π}_{2} are the 𝕀-groups

[figure: π; ℤ/2; SL(2,𝔽_{3})/S_{3}; E^{1}; SL(2,𝔽_{3})]
[figure: 0; ℤ/2; SL(2,𝔽_{3})/S_{3}; ℤ_{3}^{4}; SL(2,𝔽_{3})]

given by the homotopy groups of the above centralizers, e.g. π = t(F_{4})^{χ(σ)} ⋊ ⟨w_{1}⟩. The group lim_{𝕀}^{2}\underline{π}_{2} is trivial for general reasons [18, 3.1]. That also lim_{𝕀}^{1}\underline{π}_{1} = ∗ follows from (6.5) below since both the central 𝕀-subgroup

[figure: 0; ℤ/2; ℤ/3; SL(2,𝔽_{3})]

as well as the quotient 𝕀-group

[figure: π; ℤ/2; 0; SL(2,𝔽_{3})]

where SL(2, 𝔽_{3}) necessarily acts trivially, have vanishing lim^{1} by [19, 3.1] and (6.6).

The following observations were used to compute the non-abelian lim^{1}.

Let 𝕀 be a small category. Define an 𝕀-group to be a functor from the category 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 each 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

∗ ⟶ lim_{𝕀}^{0}A ⟶ lim_{𝕀}^{0}E ⟶ lim_{𝕀}^{0}G ⟶ lim_{𝕀}^{1}A ⟶ lim_{𝕀}^{1}E ⟶ lim_{𝕀}^{1}G ⟶ lim_{𝕀}^{2}A

of sets. Moreover, the fibres of the map lim_{𝕀}^{1}E ⟶ lim_{𝕀}^{1}G are precisely the orbits for an induced action of the abelian group lim_{𝕀}^{1}A on the set lim_{𝕀}^{1}E.

Corollary 6.6. Let 𝕀 be a finite group acting on a finite group π. If the π is a p-group and p does not divide the order of 𝕀, then H^{1}(𝕀; π) = ∗.

Proof. This follows, using the preceding lemma, by induction over the order of π since any nontrivial p-group has a non-trivial center.

Proof of Theorem 1.1. Modulo the action of the Weyl group W_{F_{4}}(SU(3, 3)) of order 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), F_{4}) = {0} ∪ Mono(SU(3), F_{4}) ∪ Mono(PU(3), F_{4})

are (4.13, 6.1) in correspondence.

Lemma 6.7. Let (u, v) ∈ ℤ_{3}^{∗}(2, 1) ∪ ℤ_{3}^{∗}(1, −1). Then

C_{F_{4}}(eψ^{(u,v)} SU(3)) = Ť(F_{4})^{χ(W (SU(3)))}
C_{F_{4}}(eψ^{(u,v)}T (SU(3))) = T (SU(3)) ×_{Z(SU(3))} SU(3)

and C_{F_{4}}(ψ^{−1}) = ψ^{−1} in both cases.

Proof. We shall apply the Bousfield-Kan spectral sequence [2] to map(BPU(3), BF_{4})_{e\overline{ψ}^{(u,v)}} where BPU(3) is viewed as the homotopy colimit of the 𝕀-space

[figure: B(0); ℤ/2; SL(2,𝔽_{3})/S_{3}; B(1); SL(2,𝔽_{3})] (6.8)

where B(0) = BC_{F_{4}}(e\overline{ψ}^{(u,v)}N_{3}(PU(3))) and B(1) = BC_{F_{4}}(e_{5}^{2}). 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 Ň_{3}(PU(3)) = Ť(PU(3)) ⋊ ⟨σ⟩ as conjugation with (0, τ) ∈ Ň(PU(3)) = Ť(PU(3)) ⋊ W (PU(3)). But this is again the restriction to

φ̌(2, 1)(Ň_{3}(PU(3))) ⊂ Ť(F_{4}) ⋊ W (F_{4})

of conjugation by (0, χ(τ)). Thus (4.18) the ℤ/2-action on

C_{F_{4}}(e\overline{ψ}^{(2,1)}N_{3}(PU(3))) = C_{Ť(F_{4})⋊W (F_{4})}(φ̌(2, 1)Ň_{3}) = Ť(F_{4})^{⟨σ⟩} ⋊ ⟨w_{1}⟩

is through conjugation with (0, χ(τ)).

Note also that the multiplication map μ: C_{Ť(F_{4})⋊W (F_{4})}(φ̌(2, 1)Ň_{3}) × Ň_{3} → Ť(F_{4}) ⋊ W (F_{4}) satisfies

μ(ψ^{−1} × 1) = ψ^{−1}μ(ψ^{−1} × 1) = μ(1 × N_{3}(ψ^{−1}))

up to inner automorphism. This means that the induced action on C_{F_{4}}(e\overline{ψ}^{(2,1)}N_{3}(PU(3))) is C_{F_{4}}(N_{3}(ψ^{−1})) = ψ^{−1} ⋊ 1.

Recall from [3] that there is an essentially unique monomorphism ι: DI_{2} → F_{4} inducing a monomorphism t(ι): t(DI_{2}) → t(F_{4}) and a group monomorphism χ: GL(2, 𝔽_{3}) = W (DI_{2}) → W (F_{4}) extending (4.7). Now, t(ι) is isomorphic to e_{5}^{2} and from the commutative diagram

\xymatrix{t(DI_{2}) \ar[d]_{t(ι)} \ar[r]^{w^{−1}} & t(DI_{2}) \ar[d]^{t(ι)} \\ Ť(F_{4}) ⋊ W (F_{4}) \ar[r]_{(0,χ(w))} & Ť(F_{4}) ⋊ W (F_{4})}

we see (4.18) that w ∈ GL(2, 𝔽_{3}) acts on C_{Ň(F_{4})}(t(DI_{2})) = Ť(F_{4}) ⋊ W (F_{4})^{t(DI_{2})} as conjugation with the element (0, χ(w)) of the semi-direct product. The restriction to SL(2, 𝔽_{3}) of this action gives the action on C_{N(F_{4})}(t(DI_{2})) = C_{F_{4}}(t(DI_{2})) in (6.8).

The conclusion of this is that

lim_{𝕀}^{0}\underline{π}_{1} = (Ť(F_{4})^{⟨χ(σ)⟩} ⋊ ⟨w_{1}⟩)^{⟨χ(τ)⟩} = Ť(F_{4})^{χ(W (SU(3)))}

is the only non-trivial contribution from the groups lim_{𝕀}^{−i}\underline{π}_{j}, i+j ≥ 0, of the Bousfield-Kan spectral sequence. Consequently, C_{F_{4}}(eψ^{(2,1)} SU(3)) is isomorphic to this group of order 3. The action of C_{F_{4}}(ψ^{−1}), which is the restriction of the action of C_{F_{4}}(N_{3}(ψ^{−1})), is given by ψ^{−1}.

The centralizer

C_{Ť(F_{4})⋊W (F_{4})}(eψ^{(2,1)}Ť(SU(3))) = Ť(F_{4}) ⋊ W (F_{4})^{A(2,1)}

is the (discrete) maximal torus normalizer for C_{F_{4}}(eψ^{(2,1)}Ť(SU(3))) and the centralizer

C_{Ť(F_{4})⋊W (F_{4})}(eψ^{(0,1)}Ť(SU(3))) = Ť(F_{4}) ⋊ W (F_{4})^{A(0,1)}

is the (discrete) maximal torus normalizer for C_{F_{4}}(eψ^{(0,1)}Ť(SU(3))) = SU(3) ×_{Z(SU(3))} T (SU(3)) [17, 3.4.3]. Since the two stabilizer subgroups W (F_{4})^{A(2,1)} and W (F_{4})^{A(0,1)} are conjugate in W (F_{4}), the two maximal torus normalizers are isomorphic and hence the two centralizers are isomorphic, too, by N-determinism [15] [19].

The group homomorphism μ: (Ť(F_{4}) ⋊ W (F_{4})^{A(2,1)} ⟶ Ť(F_{4}) ⋊ W (F_{4}) which is the inclusion on the first factor and equals A(2, 1) on the second factor satisfies

μ((1 ⋊ 1) × ψ^{−1}) = (ψ^{−1} ⋊ 1)μ((1 ⋊ 1) × ψ^{−1}) = μ((ψ^{−1} ⋊ 1) × 1

up to inner automorphisms. This shows (4.17) that C_{F_{4}}(ψ^{−1}) = ψ^{−1}.

7. Morphisms from G_{2} to F_{4} at the prime p = 3

Using the Jackowski-McClure decomposition of BG_{2} and the Bousfield-Kan spectral sequence we classify morphisms G_{2} ⟶ F_{4} viewed as 3-compact groups and compute their centralizers.

The Weyl group of G_{2}, W (G_{2}) < GL(Σ_{0}(ℤ_{3}^{3})) is the product of the Weyl group W (SU(3)) = ⟨σ, τ⟩ of SU(3) and the central group ⟨−1⟩ of order 2. The group morphism χ from (4.7) extends to a group homomorphism χ: W (G_{2}) → W (F_{4}) simply by putting χ(−1) = −1. Let 𝕀 = 𝕀(W (G_{2}), W (SU(3))) denote the category

[figure: 0; ⟨−1⟩; W (G_{2})/W (SU(3)); 1; W (G_{2})]

of the central inclusion of W (SU(3)) into W (G_{2}). Then BG_{2} is [18, §4] H^{∗}𝔽_{3}-equivalent to the homotopy colimit of an 𝕀^{op}-space

[figure: B(0); ⟨ψ^{−1}⟩; B(1); W (SU(3))^{op}\W (G_{2})^{op}; W (G_{2})^{op}]

(7.1)

where B(0) = BSU(3) and B(1) = BT (SU(3)).

Theorem 7.2. The restriction map

Rep(G_{2}, F_{4}) ⟶ Rep(SU(3), F_{4})

is bijective. The centralizer C_{F_{4}}(eψ^{(u,v)} G_{2}), u, v ∈ ℤ_{3}^{∗} ∪ {0}, is isomorphic to SU(2) if uv = 0 and trivial otherwise.

Proof. We must show that any morphism SU(3) ⟶ F_{4} extends uniquely to G_{2}. Since this is true for the trivial morphism by [17, 6.7], we only need here to consider non-trivial morphisms.

Let (u, v) ∈ (ℤ_{3}^{∗} ∪ {0})^{2}, (u, v) ≠ (0, 0). Since eψ^{(u,v)} : SU(3) → F_{4} is invariant under ψ^{−1}, this map eψ^{(u,v)} and its restriction to the maximal torus form a homotopy coherent set of maps out of the 𝕀^{op}-space (7.1). Thus it suffices to show that lim_{𝕀}^{−i}\underline{π}_{j}(u, v) = 0 for i + j ≥ −1 where \underline{π}_{j}(u, v) is the 𝕀-group

[figure: π_{j}(0); ℤ/2; W (G_{2})/W (SU(3)); π_{j}(1); W (G_{2})]

where the group π_{j}(0) = π_{j}(u, v)(0) = π_{j}(BC_{F_{4}}(eψ^{(u,v)} SU(3))) and the group π_{j}(1) = π_{j}(u, v)(1) = π_{j}(BC_{F_{4}}(eψ^{(u,v)}T (SU(3)))). Since the abelian 𝕀-groups \underline{π}_{j}(u, v) are in fact ℤ_{3}[𝕀]- modules and W (SU(3)) is normal in W (G_{2}), it follows from [18, 3.1] that lim_{𝕀}^{0}\underline{π}_{j}(u, v) = π_{j}(u, v)(0)^{ℤ/2} = π_{j}(BC_{F_{4}}(eψ^{(u,v)} SU(3)))^{ℤ/2} is the subgroup that is invariant under the action of ψ^{−1} and that the higher limits are automatically trivial. By (4.16, 5.7, 6.7), π_{j}(u, v)(0)^{ℤ/2} is trivial except when either u = 0 or v = 0 when it equals the invariants π_{j}(BSU(3))^{⟨Bψ^{−1}⟩}.

We now examine the case (u, v) = (0, 1) more closely. According to Dynkin [8, 9] the Lie group F_{4} contains a copy of (a central quotient of) SU(2) × G_{2}. The restriction to G_{2} of this inclusion SU(2) × G_{2} ⟶ F_{4} equals, up to an automorphism of F_{4}, the map eψ^{(0,1)} for otherwise the restriction to the other factor, the inclusion of SU(2) into F_{4}, would factor through the trivial 3-compact group. The homotopy class of the restriction

BSU(2) × BSU(3) ⟶ BSU(2) × BG_{2} ⟶ BF_{4}

to SU(2) × SU(3) is determined by its adjoint in π_{0}(map(BSU(2), map(BSU(3), BF_{4})_{B(eψ^{(0,1)})}) = π_{0}(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

\xymatrix{SU(2) × SU(3) \ar[d] \ar[r]^{Sι(2,3)×1} & SU(3) × SU(3) \ar[d] \\ SU(2) × G_{2} \ar[r] & _{F_{4}}}

commutes up to conjugacy. After taking adjoint maps we end up with

\xymatrix{BSU(2) \ar[d]_{BSι(2,3)} \ar[r] & map(BG_{2}, BF_{4})_{B(eψ^{(0,1)})} \ar[d] \\ BSU(3) \ar[r]^{≃} & map(BSU(3), BF_{4})_{B(eψ^{(0,1)})}}

which commutes up to homotopy and where the lower horizontal arrow represents (4.16) a homotopy equivalence homotopy equivariant under the action ⟨Bψ^{−1}⟩. By the above computations with the Bousfield-Kan spectral sequence,

π_{∗}(map(BG_{2}, BF_{4}), B(eψ^{(0,1)})) = π_{∗}(map(BSU(3), BF_{4}), B(eψ^{(0,1)}))^{⟨Bψ^{−1}⟩},

and linked with (7.4) this shows that the upper horizontal map is a homotopy equivalence as well.

The morphism eψ^{(u,v)} : G_{2} → F_{4} where u, v ∈ ℤ_{3}^{∗} with sum u + v ∈ 3ℤ_{3}, is an example a non-trivial non-monomorphism defined on a center-free 3-compact group.

The following two results were needed for the proof of Theorem 7.2.

Lemma 7.3. Let Sι(2, 3): SU(2) → SU(3) be the canonical inclusion. The map

Rep(SU(2), SU(2)) ⟶ Rep(SU(2), SU(3))
ψ^{u} ⟶ Sι(2, 3)ψ^{u}

is a bijection that identifies Out(SU(2)) = ℤ_{3}^{∗}/⟨−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 ℤ_{3}/⟨−1⟩.

Since ψ^{−1}Sι(2, 3) = Sι(2, 3)ψ^{−1} = Sι(2, 3), the image of π_{∗}(BSU(2)) in π_{∗}(BSU(3)) is invariant under the action of the group ⟨Bψ^{−1}⟩.

Lemma 7.4. There is an isommorphism, induced by Sι(2, 3),

π_{∗}(BSU(2)) ⟶ π_{∗}(BSU(3))^{⟨Bψ^{−1}⟩}

between the homotopy of BSU(2) and the ⟨Bψ^{−1}⟩-invariant subgroup of the homotopy of BSU(3).

Proof. There is a short exact sequence of homotopy groups

0 ⟶ π_{∗}(SU(2)) ⟶ π_{∗}(SU(3)) ⟶ π_{∗}(S^{5}) ⟶ 0

of 𝔽_{3}-complete spaces induced by the fibration of SU(3) onto S^{5} with fibre SU(2). This fibration splits since π_{4}(S^{3}) ⊗ ℤ_{3} = 0. The homomorphism ψ^{−1}, complex conjugation of matrices, restricts to the identity on the fibre and induces the degree −1-map on the base. Since this map induces multiplication by −1 on the homotopy groups of S^{5}, the claim follows.

8. Morphisms from SU(3) to G_{2} at the prime p = 3

The classification of morphisms SU 3) ⟶ G_{2} of 3-compact groups proceeds very much like the classification of morphisms SU(3) ⟶ F_{4}.

Lemma 8.1. The set Mono(E^{1}, G_{2}) contains two elements, e_{1}^{1}, e_{2}^{1}, with centralizer Weyl groups of order 2, 6, and Quillen automorphism groups of order 2, 2, respectively. The centralizer C_{G_{2}}(e_{2}^{1}) is isomorphic to SU(3).

The set Mono(E^{2}, G_{2})/ Aut(E^{2}) contains a unique element, e_{2}^{2} = t(G_{2}), with Quillen automorphism group W (G_{2}) of order 12.

Let χ_{1} : W (SU(3)) → W (G_{2}) be the inclusion and χ_{2} : W (SU(3)) → W (G_{2}) the injection given by χ_{2}(σ) = σ and χ_{2}(τ) = −τ. Then the identity map A_{1} : Σ_{0}(ℤ_{3}^{3}) → Σ_{0}(ℤ_{3}^{3}) is χ_{1}-equivariant and the ℤ_{3}-linear map A_{2} : Σ_{0}(ℤ_{3}^{3}) → Σ_{0}(ℤ_{3}^{3}) with matrix

A_{2} = ( \begin{matrix}1 & −2 \\ 2 & −1 \end{matrix} )

is χ_{2}-equivariant.

Lemma 8.2. A ℤ_{3}-linear map Σ_{0}(ℤ_{3}^{3}) ⟶ Σ_{0}(ℤ_{3}^{3}) is admissible with respect to W (SU(3)) and W (G_{2}) is and only it belongs to W (G_{2})(uA_{1}) or W (G_{2})(uA_{2}) for some scalar u ∈ ℤ_{3}.

Proof. Computerized calculations show that any admissible homomorphism must, up to inner automorphisms, be either χ_{1}- or χ_{2}-equivariant. Next, one solves the two systems of linear equations Aw = χ_{i}(w)A, w ∈ W (SU(3)), i = 1, 2.

Proposition 8.3. Any non-trivial morphism f: SU(3) → G_{2} is a monomorphism.

Proof. Let f: SU(3) → G_{2} be any non-trivial morphism and T (f): T (SU(3)) → T (G_{2}) a lift of f to the maximal tori. Then W (G_{2})π_{1}(T (f)) equals W (G_{2})(uA_{1}) or W (G_{2})(uA_{2}) for some 3adic integer, u. In fact, since the order of W (SU(3)) is divisible by 3, u must be a unit [3]. In the first case, W (G_{2})π_{1}(T (f)) = W (G_{2})(uA_{1}), f is a monomorphism. And if W (G_{2})π_{1}(T (f)) = W (G_{2})(uA_{2}), the kernel of Ť(f) equals the center of SU(3) and f factors through a monomorphism \overline{f}: PU(3) → G_{2}. However, such a monomorphism can not exist since the Quillen category of PU(3) contains an object E^{2} ↣ PU(3) with Quillen automorphism group SL(2, 𝔽_{3}) of order 24 exceeding the order of the Quillen automorphism group of e_{2}^{2} ∈ Mono(E^{2}, G_{2}).

Consider now the diagram

\xymatrix{E^{1} \ar[d]_{z} \ar[dr]_{z} \ar[drr]^{e_{2}^{1}} \\ SU(3) & SU(3) \ar[r]_{e} & _{G_{2}}}

where the SU(3) to the right stands for C_{G_{2}}(e_{2}^{1}) and z stands for center. Here, eψ^{−1} = e since C_{G_{2}}(ψ^{−1}) = ψ^{−1}.

Lemma 8.4. For any monomorphism f: SU(3) → G_{2}, fz = e_{2}^{1}.

Proof. Since π_{1}(T (f)) = uA_{1}, u ∈ 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) ∈ t(G_{2}) whose stabilizer subgroup is W (SU(3)).

It follows (3.9) that

Mono(SU(3), SU(3))_{z→z} = Mono(SU(3), G_{2})_{z→e_{2}^{1}} = Mono(SU(3), G_{2})

or, alternatively, that the map

⟨−1⟩\ℤ_{3}^{∗} ⟶ Mono(SU(3), G_{2})
±u ⟶ eψ^{u}

is a bijection. Also, any monomorphism f: SU(3) → G_{2} is 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 G_{2})_{Bf}

is a homotopy equivalence. Clearly, f is toric as well (2.10).

Theorem 8.5. 1. Rep(SU(3), G_{2}) = {0} ∪ Mono(SU(3), G_{2}) is T -determined.

2. The action

Mono(SU(3), G_{2}) × Out(SU(3)) ⟶ Mono(SU(3), G_{2})

is transitive and the stabilizer at f ∈ Mono(SU(3), G_{2}) equals W_{G_{2}}(f SU(3)) = ⟨ψ^{−1}⟩.

Proof. This is clear from the explicit description of the set Rep(SU(3), G_{2}). For instance, the restriction map

Mono(SU(3), G_{2}) ⟶ Mono(T (SU(3)), G_{2})

can be identified to the map

{u ∈ ℤ_{3}^{∗} | u ≡ 1 mod 3} ⟶ W (G_{2})(uA_{1})

which clearly is injective.

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Matematisk Institut, Universitetsparken 5, DK–2100 København Ø

E-mail address: moller@math.ku.dk