Homology, Homotopy and Applications, vol.5(1), 2003, pp.297–324

REPRESENTATION TYPES AND 2-PRIMARY HOMOTOPY GROUPS OF CERTAIN COMPACT LIE GROUPS

DONALD M. DAVIS

(communicated by Gunnar Carlsson)

Abstract

Bousfield has shown how the 2-primary v_{1}-periodic homotopy groups of certain compact Lie groups can be obtained from their representation ring with its decomposition into types and its exterior power operations. He has formulated a Technical Condition which must be satisfied in order that he can prove that his description is valid.

We prove that a simply-connected compact simple Lie group satisfies his Technical Condition if and only if it is not E_{6} or Spin(4k + 2) with k not a 2-power. We then use his description to give an explicit determination of the 2-primary v_{1}-periodic homotopy groups of E_{7} and E_{8}. This completes a program, suggested to the author by Mimura in 1989, of computing the v_{1}-periodic homotopy groups of all compact simple Lie groups at all primes.

1. Introduction

The p-primary v_{1}-periodic homotopy groups of a topological space X, denoted v_{1}^{−1}π_{∗}(X; p), are a localization of the portion of the actual homotopy groups detected by K-theory. Each v_{1}-periodic homotopy group of X is a direct summand of some actual homotopy group of X.

In 1989, Mimura suggested to the author that the computation of v_{1}^{−1}π_{∗}(X; p) for all compact simple Lie groups X and all primes p would be an interesting project. In a series of papers over the subsequent 13-year period, the author, often in collaboration with Bendersky, had performed this computation in all cases except E_{7} and E_{8} at the prime 2.([13],[1],[7],[6], [5],[15],[12],[3],[14],[4]) In this paper, we use recent work of Bousfield to compute v_{1}^{−1}π_{∗}(E_{7}; 2) and v_{1}^{−1}π_{∗}(E_{8}; 2), thus completing the project suggested by Mimura.

The one impreciseness in these results is some extension questions involving Z_{2}’s. We write Z_{2} interchangeably with Z/2. We denote by A#B an abelian group G

The author would like to thank Pete Bousfield for his suggestions on this work and for allowing him to use his unpublished results. Received April 21, 2003, revised August 21, 2003; published on September 4, 2003. 2000 Mathematics Subject Classification: 55Q52,55T15,57T20 Key words and phrases: homotopy groups, exceptional Lie groups, representation theory \begin{matrix}c \\ ◯ \end{matrix} 2003, Donald M. Davis. Permission to copy for private use granted.

such that there is a short exact sequence of abelian groups

0 → A → G → B → 0.

Here and throughout the remainder of the paper v_{1}^{−1}π_{∗}(X) means v_{1}^{−1}π_{∗}(X; 2). Also, ν(−) denotes the exponent of 2 in an integer.

Theorem 1.1. For all integers k, there are isomorphisms

v_{1}^{−1}π_{8k+d}(E_{7}) ≈ \begin{cases}Z_{2} & d = 3 \\ Z_{2}#Z_{2} & d = 4 \\ Z_{2} ⊕ Z/2^{4} ⊕ Z/2^{g(4k+3)−2} & d = 5 \\ Z/4 ⊕ Z/2^{g(4k+3)} ⊕ Z_{2} ⊕ Z_{2} & d = 6 \\ (Z_{2} ⊕ Z_{2})#(Z_{2} ⊕ Z_{2} ⊕ Z_{2}) & d = 7 \\ (Z_{2} ⊕ Z_{2} ⊕ Z_{2})#Z_{2} & d = 8 \\ Z_{2} ⊕ Z/2^{4} ⊕ Z/2^{g(4k+5)−2} & d = 9 \\ Z/4 ⊕ Z/2^{g(4k+5)} & d = 10, \end{cases}

where

g(m) = \begin{cases}min(17, ν(m − 11 − 2^{6}) + 9) & m ≡ 3 mod 4 \\ min(18, ν(m − 13 − 2^{7}) + 9) & m ≡ 5 mod 8 \\ min(23, ν(m − 17 − 7 · 2^{11}) + 9) & m ≡ 1 mod 8. \end{cases}

Theorem 1.2. For all integers k, there are isomorphisms

v_{1}^{−1}π_{8k+d}(E_{8}) ≈ \begin{cases}Z/2^{e(4k−1)−1} ⊕ Z_{2} & d = −3 \\ Z/2^{e(4k−1)} ⊕ Z_{2} ⊕ Z_{2} & d = −2 \\ (Z_{2} ⊕ Z_{2} ⊕ Z_{2})#(Z_{2} ⊕ Z_{2}) & d = −1 \\ (Z_{2}#Z_{2}) ⊕ Z_{2} ⊕ Z_{2} & d = 0 \\ Z/2^{e(4k+1)−1} ⊕ Z_{2} ⊕ Z_{2} & d = 1 \\ Z/2^{e(4k+1)} & d = 2 \\ 0 & d = 3, 4, \end{cases}

where

e(m) = \begin{cases}min(25, ν(m − 17 − 2^{8} − 2^{11} − 2^{12}) + 12) & m ≡ 1 mod 8 \\ min(28, ν(m − 19 − 2^{11} − 2^{14} − 2^{15}) + 12) & m ≡ 3 mod 8 \\ min(39, ν(m − 29 − 2^{20} − 2^{22} − 2^{23} − 2^{25}) + 12) & m ≡ 5 mod 8 \\ min(31, ν(m − 23 − 2^{17}) + 12) & m ≡ 7 mod 8. \end{cases}

Note that the numbers m_{0} = 17, 19, 29, and 23 which occur in ν(m − m_{0} − 2^{L}) in the formula for e(m) are the largest exponents of E_{8}, and similarly for E_{7} with m_{0} = 11, 13, and 17. The exponents of a compact Lie group G are those integers m_{i} such that H^{∗}(G; Q) is an exterior algebra on classes of grading 2m_{i} + 1.([10, pp.15-16].)

In 2.2, we state a slight reformulation of a conjecture of Bousfield that would yield, for all simply-connected compact Lie groups G, the groups v_{1}^{−1}π_{∗}(G; 2) in terms of the representation ring R(G) together with its decomposition into real, complex, and quaternionic types, and its second and third exterior power operations. Bousfield has proved (see Theorem 2.5) that his conjecture is valid for those G for which R(G) satisfies a Technical Condition, which we state in 2.4. Our second main result determines which of the simply-connected compact simple Lie groups satisfy this Technical Condition.

Theorem 1.3. A simply-connected compact simple Lie group satisfies the Technical Condition 2.4 if and only if it is not E_{6} or Spin(4k + 2) with k not a 2-power.

In particular, Conjecture 2.2 is valid for E_{7} and E_{8}. It is by computing the groups and homomorphisms of 2.2 that Theorems 1.1 and 1.2 are proved.

The author has computed, for all compact simple Lie groups G, the result for v_{1}^{−1}π_{∗}(G; 2) which would be implied by Bousfield’s conjecture 2.2 and obtained remarkable agreement with the results he has obtained previously by other methods. This may be viewed both as lending credence to Conjecture 2.2 and as a check on the earlier work of the author and coworkers.

2. Bousfield’s Conjecture and Theorem

In this section, we state a slight reformulation of Bousfield’s conjecture regarding 2-primary v_{1}-periodic homotopy groups, and his Technical Condition, under which he can prove his conjecture valid.

The first step of Bousfield’s program is a real analogue of [8, 8.1,8.5]. In a November 2002 e-mail, Bousfield wrote that the following result can be proved by utilizing [9, 7.8,9.4,9.5] to adapt the argument of [8].

Theorem 2.1. (Bousfield) Let G be a simply-connected compact Lie group. There is a K/2_{∗}-local spectrum ΦG such that there is an exact sequence

→ v_{1}^{−1}π_{i+2}(G)^{#} → KO^{i}(ΦG; Z_{2}^{∧}) ⟶^{ψ^{3}−9} KO^{i}(ΦG; Z_{2}^{∧}) → v_{1}^{−1}π_{i+3}(G)^{#} →,

where (−)^{#} denotes Pontrjagin duality.

Bousfield’s conjecture expresses KO^{∗}(ΦG; Z_{2}^{∧}) in terms of the representation theory of G. For the simply-connected compact Lie group G, let R(G) be its (complex) representation ring, I ⊂ R(G) the augmentation ideal, and Q = Q(G) = I/I^{2} the group of indecomposables in I. Let R_{R}(G) (resp. R_{H}(G)) denote the real (resp. quaternionic) representation rings. We identify these with their image in R(G) under the extension homomorphisms, which are injective. Let Q_{R} ⊂ Q (resp. Q_{H} ⊂ Q) denote the image in Q of the augmentation ideal of R_{R}(G) (resp. R_{H}(G)). Let λ^{k} denote the exterior power operations on R(G).

The following conjecture uses all the above notation. We omit writing Z_{2}^{∧} as coefficient of KO^{∗}(ΦG), and a Z_{2}^{∧}⊗ which should accompany all the Q-groups. Although essential to the underlying theory, these 2-adic coefficients do not affect the subsequent calculations.

Conjecture 2.2. (Bousfield) If G is a simply-connected compact Lie group, there is an exact sequence of abelian groups

0 → KO^{0}(ΦG) → Q/(Q_{R} + Q_{H}) ⟶^{λ^{2}} Q/Q_{R} → KO^{1}(ΦG) → 0
→ Q_{H}/(Q_{R} ∩ Q_{H}) → KO^{2}(ΦG) → Q_{R} ∩ Q_{H} ⟶^{λ^{2}} Q_{H} → KO^{3}(ΦG) → 0
→ KO^{4}(ΦG) → Q/(Q_{R} ∩ Q_{H}) ⟶^{λ^{2}} Q/Q_{H} → KO^{5}(ΦG)
→ (Q_{R} + Q_{H})/(Q_{R} ∩ Q_{H}) ⟶^{λ^{2}} Q_{R}/(Q_{R} ∩ Q_{H}) → KO^{6}(ΦG)
→ Q_{R} + Q_{H} ⟶^{λ^{2}} Q_{R} → KO^{7}(ΦG) → 0.

For any integer i, the Adams operation ψ^{3} in KO^{2i}(ΦG) and KO^{2i+1}(ΦG) corresponds to 3^{−i}λ^{3} in Q under the morphisms of the exact sequence, which is expanded to all integers by Bott periodicity KO^{j}(−) ≈ KO^{j+8}(−).

Note that applying period-8 Bott periodicity to the exact sequence of 2.2 does not change the λ^{2} in Q which is being used to yield the KO^{∗}(−)-groups. We will show in Sections 4 and 5 how to compute the exact sequences of Theorem 2.1 and Conjecture 2.2 to obtain v_{1}^{−1}π_{∗}(E_{8}) and v_{1}^{−1}π_{∗}(E_{7}).

Bousfield has proved this conjecture for those G which satisfy a Technical Condition, which we now state. We begin by recalling some standard material regarding representation types, and establishing notation. All of the material in this result, as well as additional background for 2.4, may be found in [11, II.6,VI.4].

Theorem 2.3. Let G be a simply-connected compact Lie group, and let t denote conjugation on R(G). Each irreducible representation is of one of three types— real, quaternionic, or complex. Those of real or quaternionic type are self-conjugate, while those of complex type are not. There is a set B(G) of irreducible representations called basic such that R(G) is a polynomial algebra on B(G). If ρ is any representation, let

ρ̃ = ρ − dim(ρ) ∈ I(G).

Let Q, Q_{R}, and Q_{H} be as in 2.2 and its preamble. Then Q is a free abelian group with basis {ρ̃: ρ ∈ B(G)}. The set B(G) can be partitioned into subsets B_{R}(G), B_{H}(G), and B_{C}(G) of representations of real, quaternionic, and complex type, respectively. The set B_{C}(G) is composed of pairs of conjugate representations. Let B′_{C}(G) contain one element from each pair of conjugate elements of B_{C}(G). Then Q_{R} is a free abelian group with basis

{ρ̃: ρ ∈ B_{R}(G)} ∪ {2ρ̃: ρ ∈ B_{H}(G)} ∪ {ρ̃+ t(ρ̃) : ρ ∈ B′_{C}(G)},

and similarly for Q_{H} with R and H interchanged.

Now we state Bousfield’s Technical Condition.

Definition 2.4. Let G be a simply-connected compact Lie group, and let

H(G) = ker(1 − t)/ im(1 + t).

Then H(G) is an augmented Z/2-graded polynomial algebra over Z/2 on

B_{R}(G) ∪ B_{H}(G) ∪ {ρ t(ρ) : ρ ∈ B′_{C}(G)},

where gr(B_{R}(G)) = 0, gr(B_{H}(G)) = 1, and gr(ρ t(ρ)) = 0, while the augmentation ϵ satisfies ϵ(ρ) = dim(ρ) mod 2. Let H_{R}(G) denote the subgroup of H(G) of grading 0. There is an augmentation-preserving algebra homomorphism ϕ : R(G) → H(G) defined by ϕ(ρ) = ρ t(ρ). The image of ϕ is contained in H_{R}(G). This ϕ induces a morphism of indecomposables

\overline{ϕ} : I(R(G))/I^{2}(R(G)) → I(H(G))/I^{2}(H(G)),

whose image lies in the summand

I(H_{R}(G))/I^{2}(H(G)) := Ind_{R}(H(G))

of grading 0. The morphism \overline{ϕ} preserves Adams operations, and its image is automatically a ψ^{3}-submodule of Ind_{R}(H(G)). We say that G satisfies the Technical Condition if im(\overline{ϕ}) is a direct summand of Ind_{R}(H(G)) as a ψ^{3}-module; i.e., if it has a complementary ψ^{3}-submodule.

In e-mails dated January 30, 2003, and February, 8, 2003, Bousfield wrote that he has a proof of the following result, which he is in the process of writing. In fact, he will prove more; the author has just extracted from various letters from Bousfield the portion of these consequences necessary for the specific applications to v_{1}-periodic homotopy groups.

Theorem 2.5. (Bousfield) If G satisfies the Technical Condition, then Conjecture 2.2 is true for G.

In those same e-mails, Bousfield wrote that he has an idea of how he might be able to prove Conjecture 2.2 without assuming the Technical Condition, but that this is more speculative.

3. Compact simple Lie groups and the Technical Condition

In this section, we prove Theorem 1.3, which states exactly which of the simplyconnected compact simple Lie groups satisfy the Technical Condition 2.4.

We begin by tabulating for the compact simple Lie groups a set of basic representations and their division into types. For the classical groups, this information is proved in [11, VI], while for the exceptional groups it is extracted from [17].

Table 3.1. Types of basic representations

G  B_{R}(G)  B_{H}(G)  B′_{C}(G)  t(B′_{C}(G))
SU(2n + 1)  λ_{1}, …, λ_{n}  t(λ_{i}) = λ_{2n+1−i}
SU(2n), n even λ_{n}  λ_{1}, …, λ_{n−1} t(λ_{i}) = λ_{2n−i}
SU(2n), n odd  λ_{n}  λ_{1}, …, λ_{n−1} t(λ_{i}) = λ_{2n−i}
Sp(n)  λ_{i}, i even,  λ_{i}, i odd,
2 ⩽ i ⩽ n  1 ⩽ i ⩽ n
Spin(2n + 1),  λ_{1}, …, λ_{n−1},
n ≡ 0, 3 mod 4 Δ
Spin(2n + 1),  λ_{1}, …, λ_{n−1} Δ

n ≡ 1, 2 mod 4

Spin(2n), n odd λ_{1}, …, λ_{n−2} Δ_{+} t(Δ_{+}) = Δ_{−}
Spin(2n),  λ_{1}, …, λ_{n−2},
n ≡ 0 mod 4  Δ_{+}, Δ_{−}
Spin(2n),  λ_{1}, …, λ_{n−2} Δ_{+}, Δ_{−}

n ≡ 2 mod 4

G_{2}, F_{4}, E_{8}  ρ_{1}, …, ρ_{t}
E_{6} ρ_{2}, ρ_{4} ρ_{1}, ρ_{3} t(ρ_{1}) = ρ_{6},
t(ρ_{3}) = ρ_{5}
E_{7}  ρ_{1}, ρ_{3}, ρ_{4}, ρ_{6} ρ_{2}, ρ_{5}, ρ_{7}

We will say more about the specifics of the basic representations ρ_{i} of E_{6} and E_{7} when we make specific applications later.

We begin with an elementary proposition and two corollaries.

Proposition 3.2. In the notation of Definition 2.4, Ind_{R}(H(G)) is a vector space over Z/2 with basis B̃_{R}(G) ∪ B(\overline{ϕ}), where B̃_{R}(G) = \{ρ̃: ρ ∈ B_{R}(G)\}, and B(\overline{ϕ}) = \{ρ̃· t(ρ̃) : ρ ∈ B′_{C}(G)\} is a basis for im(\overline{ϕ}).

Proof. This follows immediately from the definitions.

Corollary 3.3. If either B_{R}(G) or B′_{C}(G) is empty, then G satisfies the Technical Condition.

Proof. If B_{R}(G) is empty, then im(\overline{ϕ}) = Ind_{R}(H(G)), while if B′_{C}(G) is empty, then im(\overline{ϕ}) = 0, both of which are clearly ψ^{3}-direct summands of Ind_{R}(H(G)).

Corollary 3.4. A simply-connected compact simple Lie group which does not equal SU(4m), Spin(4k + 2), or E_{6} satisfies the Technical Condition.

Proof. By Table 3.1, all other simply-connected compact simple Lie groups satisfy the hypothesis of Corollary 3.3.

We handle the three (families of) Lie groups not covered by Corollary 3.4 in separate theorems, 3.5, 3.7, and 3.32.

Theorem 3.5. For each m ⩾ 1, SU(4m) satisfies the Technical Condition.

Proof. Referring to Table 3.1, we see that

im(\overline{ϕ}) = ⟨λ̃_{i}λ̃_{4m−i} : 1 ⩽ i < 2m⟩.

We will prove

the coefficient of λ̃_{i}λ̃_{4m−i} in ψ^{3}(λ̃_{2m}) is even,  (3.6)

which, with 3.2, will imply the theorem, since ⟨λ̃_{2m}⟩ is then a ψ^{3}-submodule of Ind_{R}(H(SU(4m))) complementary to im(\overline{ϕ}).

Let T denote the maximal torus of SU(4m). Then

R(T) = Z[x_{1}, …, x_{4m}]/(x_{1} ⋯ x_{4m} − 1),

and R(SU(4m)) → R(T) is a ring homomorphism sending λ_{k} to σ_{k}, and sending ψ^{3}(λ_{2m}) to ∑ x_{i_{1}}^{3} ⋯ x_{i_{2m}}^{3}, which we call p. Here the sum is over all i_{1} < ⋯ < i_{2m}, and σ_{k} is the kth elementary symmetric polynomial in x_{1}, ⋯, x_{4m}.

This p can be written as an integer polynomial f(σ_{1}, …, σ_{4m}) in which each term has grading 6m, where gr(σ_{i}) = i. Then ψ^{3}(λ_{2m}) is the same polynomial f(λ_{1}, ⋯, λ_{4m}). To write ψ^{3}(λ̃_{2m}) in terms of λ̃_{1}, …, λ̃_{4m−1}, we replace each λ_{i} by λ̃_{i} + \binom{4m}{i}. Note λ̃_{4m} = 0.

Let i < 2m, and let τ := λ_{1}^{j_{1}} ⋯ λ_{4m}^{j_{4m}} have odd coefficient in f(λ_{1}, …, λ_{4m}). Then

τ = (λ̃_{1} + \binom{4m}{1})^{j_{1}} ⋯ (λ̃_{4m} + \binom{4m}{4m})^{j_{4m}}

contains a term βλ̃_{i}λ̃_{4m−i} with β ∈ Z if and only if j_{i} > 0 and j_{4m−i} > 0.

If we write τ = λ_{i}λ_{4m−i} ∏ λ_{k_{ℓ}}, then the coefficient β of λ̃_{i}λ̃_{4m−i} in τ is equal to j_{i}j_{4m−i} ∏ \binom{4m}{k_{ℓ}}. Note that k_{ℓ} may be repeated, and may equal i or 4m − i, but ∑ k_{ℓ} = 6m − 4m = 2m. Now we note that if ∏ \binom{4m}{k_{ℓ}} is odd, then each \binom{4m}{k_{ℓ}} is odd, hence each k_{ℓ} is at least as 2-divisible as 4m, and so 2m = ∑ k_{ℓ} is at least as 2-divisible as 4m, which is impossible. Thus the coefficient of λ̃_{i}λ̃_{4m−i} is even, proving (3.6).

The proof of the next result involves more combinatorics.

Theorem 3.7. Spin(4k + 2) satisfies the Technical Condition if and only if k is a 2-power.

Proof. It is well-known (see, e.g., [18, p.151]) that j^{∗} : R(SU(2n)) → R(Spin(2n)) satisfies

j^{∗}(λ_{i}) = j^{∗}(λ_{2n−i}).  (3.8)

Let μ_{i} = j^{∗}(λ_{i}). Then (see, e.g., [11, VI.6.2]) R(Spin(2n)) has basic representations μ_{1}, …, μ_{n−2}, Δ_{+}, Δ_{−} with

Δ_{+}Δ_{−} = μ_{n−1} + μ_{n−3} + μ_{n−5} + ⋯  (3.9)
Δ_{+}^{2} + Δ_{−}^{2} = μ_{n} + 2(μ_{n−2} + μ_{n−4} + ⋯).  (3.10)

By 3.1, B̃_{R}(Spin(2n)) = \{μ̃_{1}, …, μ̃_{n−2}\}, and B(\overline{ϕ}) = \{P \}, where P := Δ̃_{+}Δ̃_{−}. The theorem follows from the following result, in which n = 2k + 1.

Lemma 3.11. Let \overline{ϵ} = (ϵ_{1}, …, ϵ_{2k−1}) with ϵ_{i} ∈ Z/2. Let W (\overline{ϵ}) denote the subspace of the Z/2-vector space Ind_{R}(H(Spin(4k + 2))) spanned by {μ̃_{1} + ϵ_{1}P, …, μ̃_{2k−1} + ϵ_{2k−1}P \}. Every Z/2-subspace complementary to im(\overline{ϕ}) = ⟨P ⟩ is of this form.

1. If k is a 2-power, then W (\overline{0}) is a ψ^{3}-submodule of Ind_{R}(H(Spin(4k + 2))), while

2. if k is not a 2-power, W (\overline{ϵ}) can never be a ψ^{3}-submodule of Ind_{R}(H(Spin(4k+ 2))).

In order to prove Lemma 3.11, we need an explicit formula for ψ^{3} on the basis {λ̃_{1}, …, λ̃_{2n−1}} of the free abelian group

Q(SU(4k + 2)) := I(R(SU(4k + 2)))/I^{2}(R(SU(4k + 2))).

This will then be transported to Ind_{R}(H(Spin(4k + 2))) using j^{∗}, (3.8), and (3.9).

Theorem 3.12. Define integers c_{n,ℓ}^{k} by

(1 + x + ⋯ + x^{k−1})^{n} = ∑ c_{n,ℓ}^{k}x^{ℓ}.  (3.13)

In Q(SU(n)),

ψ^{k}(λ̃_{i}) = k ∑_{ℓ⩾0}(−1)^{ki+i+ℓ}c_{n,ℓ}^{k}λ̃_{ki−ℓ}.  (3.14)

Proof. Let β : I(R(G))/I^{2}(R(G)) → P K^{−1}(G) be Hodgkin’s isomorphism ([16]), where P denotes the primitives. As in [2, pp. 42-43], let B_{i} ∈ P K^{1}(SU(n)) correspond to β(λ̃_{i}) under Bott periodicity. We will prove that

ψ^{k}(B_{i}) = ∑_{ℓ⩾0}(−1)^{ki+i+ℓ}c_{n,ℓ}^{k}B_{ki−ℓ}.  (3.15)

Then (3.14) follows from the fact that ψ^{k} in K^{1}(G) corresponds to ψ^{k}/k in K^{−1}(G).

In [2, 3.2], it is shown that B_{j} = ∑(−1)^{ℓ+1}\binom{n}{j−ℓ}ξ_{ℓ}, where ξ_{ℓ} = ξ^{ℓ} − 1 satisfies ψ^{k}(ξ_{ℓ}) = ξ_{kℓ}. Thus it suffices to prove

∑_{ℓ}(−1)^{ℓ+1}\binom{n}{i−ℓ}ξ_{kℓ} = ∑_{ℓ}(−1)^{ki+i+ℓ}c_{n,ℓ}^{k} ∑_{t}(−1)^{t+1}\binom{n}{ki−ℓ−t}ξ_{t},

where the sums are taken over all values which give meaningful terms. Note that there are relations among the ξ_{i}’s when i ⩾ n, but they are the same on both sides of the equation, and hence need not be considered. The ξ_{i}’s are just formal variables, and so can be replaced by x^{−i}. Thus we wish to prove

∑_{ℓ}(−1)^{ℓ}\binom{n}{i−ℓ}x^{−kℓ} = ∑_{ℓ}(−1)^{ki+i+ℓ}c_{n,ℓ}^{k}x^{ℓ} ∑_{t}(−1)^{t}\binom{n}{ki−ℓ−t}x^{−ℓ−t}.

Multiplying both sides by (−1)^{i}x^{ki}, it is equivalent to show

∑_{ℓ}(−1)^{i−ℓ}\binom{n}{i−ℓ}(x^{k})^{i−ℓ} = ∑_{ℓ} c_{n,ℓ}^{k}x^{ℓ} ∑_{t}(−1)^{ki−ℓ−t}\binom{n}{ki−ℓ−t}x^{ki−ℓ−t}.

The left hand side equals (1 − x^{k})^{n}, while the right hand side equals (1 + x + ⋯ + x^{k−1})^{n}(1 − x)^{n}, and these are equal, establishing (3.15).

Remark 3.16. Another approach to Theorem 3.12 is illustrated by

ψ^{2}(λ̃_{1}) = −2λ̃_{2} + 2nλ̃_{1} + λ̃_{1}^{2}

in R(SU(n)), which is readily verified to be consistent with one case of 3.12.

Proof. Using R(SU(n)) → R(T) as in the proof of 3.5, λ̃_{1} corresponds to x_{1} + ⋯ + x_{n} − n, while λ̃_{2} corresponds to ∑_{i<j} x_{i}x_{j} − \binom{n}{2}. Then ψ^{2}(λ̃_{1}) corresponds to

x_{1}^{2} + ⋯ + x_{n}^{2} − n = (x_{1} + ⋯ + x_{n})^{2} − n − 2 ∑_{i<j} x_{i}x_{j},

which corresponds to (λ̃_{1} + n)^{2} − n − 2(λ̃_{2} + \binom{n}{2}) = λ̃_{1}^{2} + 2nλ̃_{1} − 2λ̃_{2}.

Proof of Lemma 3.11. We work in the Z/2-vector space V := Ind_{R}(H(Spin(4k + 2))). Since dim(Δ_{±}) is even, (3.9) becomes

P = μ̃_{2k} + μ̃_{2k−2} + · · · .  (3.17)

Since Δ_{+}^{2} + Δ_{−}^{2} = (1 + t)(Δ_{+}^{2}), (3.10) implies μ̃_{2k+1} = 0 in V. ¿From (3.14) and (3.8), we obtain

ψ^{3}(μ̃_{j}) = ∑ c_{4k+2,3j−ℓ}^{3}μ̃_{ℓ} = ∑_{ℓ=1}^{2k}(c_{4k+2,3j−ℓ}^{3} + c_{4k+2,3j−(4k+2−ℓ)}^{3})μ̃_{ℓ}.  (3.18)

We will use (3.17) to express μ̃_{2k} as P − μ̃_{2k−2} − μ̃_{2k−4} − · · ·.

By (3.13), since we work mod 2,

c_{4k+2,i}^{3} = \begin{cases}c_{2k+1,i′}^{3} & if i = 2i′ \\ 0 & if i odd. \end{cases}

Thus (3.18) implies that there is a splitting as ψ^{3}-modules,

V = V_{od} ⊕ V_{ev},

where V_{od} = ⟨μ̃_{2i−1} : 1 ⩽ i ⩽ k⟩ and V_{ev} = ⟨P, μ̃_{2i} : 1 ⩽ i ⩽ k − 1⟩. Thus im(\overline{ϕ}) = ⟨P ⟩ has a complementary ψ^{3}-submodule in V if and only if it has a complementary ψ^{3}-submodule in V_{ev}, and so we focus our attention on the latter.

Let v_{i} := μ̃_{2i} ∈ V_{ev} and c_{i} := c_{2k+1,i}^{3}. Then (3.18) becomes

ψ^{3}(v_{j}) = ∑_{t=1}^{k−1}(c_{3j−t}+c_{3j−2k−1+t})v_{t}+(c_{3j−k}+c_{3j−k−1})(P +v_{k−1}+v_{k−2}+⋯). (3.19)

One can use these sorts of formulas to prove ψ^{3}(P) = P, but this also follows from the fact that im(\overline{ϕ}) is a ψ^{3}-submodule by naturality.

Now we can easily prove 3.11(1). If k = 2^{e}, then c_{i} = 1 implies i = 0, 1, or 2, or i ⩾ 2^{e+1}. By (3.19), we must prove that if 1 ⩽ j ⩽ k − 1, then c_{3j−k} + c_{3j−k−1} = 0. Note that for such j, we have 3j − k ⩽ 2k − 3 < 2^{e+1}. Thus the only way to have c_{3j−k} + c_{3j−k−1} = 1 is if 3j − k = 0 or 3. But this is impossible, since k ≢ 0 mod 3.

To prove part (2), let

B = (\begin{matrix}A & \overline{0} \\ \overline{r} & 1 \end{matrix})

denote the matrix of ψ^{3} with respect to the ordered basis {v_{1}, …, v_{k−1}, P }. Thus A = (a_{i,j}) is a (k −1)-by-(k −1) matrix. The last column of B is due to ψ^{3}(P) = P, observed earlier. We will prove

a. If k is not a 2-power, then \overline{r} ≠ 0.

b. A + I is singular.

Lemma 3.11(2) then follows. Indeed, suppose k is not a 2-power and W (\overline{ϵ}) is a ψ^{3}-submodule. Then, for 1 ⩽ j ⩽ k − 1,

ψ^{3}(v_{j} + ϵ_{j}P) = ∑_{i=1}^{k−1} a_{i,j}v_{i} + r_{j}P + ϵ_{j}P

= ∑_{i=1}^{k−1} a_{i,j}(v_{i} + ϵ_{i}P) + (∑_{i=1}^{k−1} a_{i,j}ϵ_{i} + r_{j} + ϵ_{j})P.

The coefficient of P must be 0 for each j. Thus, with (−)^{T} denoting the transpose,

(A + I)\overline{ϵ}^{T} = \overline{r}^{T} ,

and since \overline{r} ≠ 0 by (a), A + I must be nonsingular, contradicting (b).

It remains to prove (a) and (b). Part (a) follows from (3.19) and Lemma 3.20, while (b) follows from Lemma 3.24.

Lemma 3.20. Let (1 + x + x^{2})^{2k+1} = ∑ c_{i}x^{i}. If k is not a 2-power, there exists j satisfying 1 ⩽ j ⩽ k − 1 and c_{3j−k} + c_{3j−k−1} ≡ 1 mod 2.

Proof. We write k = 2^{e}u with u > 1 odd, and divide into cases depending on the mod 6 value of u.

Case 1. If u = 3a, then j = 2^{e}a works, since c_{0} + c_{−1} = 1.

Case 2. If u = 6a+1, then j = 2^{e}(2a+1) works. (Note that if a = 0, then j = k, explaining the failure of the lemma when k is a 2-power.) To see this, note that

(1 + x + x^{2})^{2k+1} ≡ 1 + x + x^{2} + x^{2^{e+1}} mod (x^{2^{e+1}+1}),

and hence c_{2^{e+1}} + c_{2^{e+1}−1} = 1.

Case 3. If u = 3 · 2^{f}α − 1 with f ⩾ 1 and α odd, then j = 2^{e}(2^{f}α + 2^{f} − 1) works. To see this, let c(p, i) denote the coefficient of x^{i} in the polynomial or power series p = p(x). Then c_{3j−k} + c_{3j−k−1} equals

c((1 + x + x^{2})^{2k+1}, 3 · 2^{e+f} − 2^{e+1}) + c((1 + x + x^{2})^{2k+1}, 3 · 2^{e+f} − 2^{e+1} − 1). (3.21)

Mod x^{2^{e+f+2}}, we have

(1 + x + x^{2})^{2k+1} ≡ (1 + x + x^{2})(1 + x^{2^{e+f+1}})(1 + x + x^{2})^{−2^{e+1}}.

Thus (3.21) equals

c(g, 3 · 2^{e+f} − 2^{e+1}) + c(g, 3 · 2^{e+f} − 2^{e+1} − 1)
+ c(g, 2^{e+f} − 2^{e+1}) + c(g, 2^{e+f} − 2^{e+1} − 1),  (3.22)

where

g = (1+x+x^{2})(1+x^{2^{e+1}} +x^{2^{e+2}})^{−1} = (1+x+x^{2}) ∑_{i⩾0}(x^{3i2^{e+1}} +x^{2^{e+1}(3i+1)}). (3.23)

Here we have used (mod 2, as always)

(1 + z + z^{2})^{−1} = \frac{1 + z}{1 + z^{3}} = ∑_{i⩾0}(z^{3i} + z^{3i+1}).

If e = 0, then g = ∑_{j⩾0}(x^{3j} + x^{3j+1}), and so c(g, 3 · 2^{f} − 2) = c(g, 3 · 2^{f} − 3) = 1, while c(g, 2^{f} − 2) + c(g, 2^{f} − 3) = 1 since 2^{f} − 2 ≢ 1 mod 3. Thus (3.22) equals 1 in this case. If e > 0, c(g, j) = 1 if and only if j ≡ 0, 1, 2, 2^{e+1}, 2^{e+1} + 1, or 2^{e+1} + 2 mod 3·2^{e+1}. The mod 3·2^{e+1} values of the four exponents of x in (3.22) are −2^{e+1}, −2^{e+1} − 1, 0 or 2^{e+1}, and −1 or 2^{e+1} − 1. Hence the third coefficient is 1 while the others are 0.

Like the above lemma, Lemmas 3.24, 3.30, and 3.31 below all deal with mod 2 polynomials.

Lemma 3.24. Let (1+x+x^{2})^{2k+1} = ∑ c_{i}x^{i}. Let A = (a_{i,j}) be the (k−1)-by-(k−1) matrix with

a_{i,j} = c_{3j−i} + c_{3j−2k−1+i} + c_{3j−k} + c_{3j−k−1}.

Then A + I is singular.

Proof. Let γ_{j} = 1 if, for e ⩾ 0,

k = 2^{e}(4m − 1)  and 1 ⩽ (j mod 2^{e+2}) ⩽ 2^{e+1}
k = 2^{e}(4m + 1), m > 0, and 3 · 2^{e} ⩽ (k − j mod 2^{e+3}) ⩽ 5 · 2^{e} − 1,
k = 2^{e+1}  and 1 ⩽ j ⩽ 2^{e}

and γ_{j} = 0 otherwise. We will show that, for 1 ⩽ i ⩽ k − 1,

(∑_{j=1}^{k−1} γ_{j}a_{i,j}) + γ_{i} = 0,

which establishes the linear dependence of some of the columns of A + I. Observing that γ_{k} = 0, it is equivalent to show that, for 1 ⩽ i ⩽ k,

∑_{j=1}^{k−1} γ_{j}(c_{3j−i} + c_{3j−2k−1+i}) = γ_{i}.  (3.25)

We will, in fact, prove that (3.25) is true for all i ⩾ 1.

We begin with the case k = 2^{e}(4m − 1). Since k ≡ −2^{e} mod 2^{e+2}, we have

∑_{j=1}^{k−1} γ_{j}c_{3j−i} = ∑_{ℓ⩾0} ∑_{d=0}^{2^{e+1}−1} c_{3(k−2^{e}−d−ℓ2^{e+2})−i}
= c(g_{k}(x), 3(k − 2^{e}) − i),

where

g_{k}(x) = (1 + x + x^{2})^{2k+1} \frac{1 + x^{3} + ⋯ + x^{3(2^{e+1}−1)}}{1 + x^{3·2^{e+2}}}

and c(−, −) is as in the previous proof. Similarly,

∑_{j=1}^{k−1} γ_{j}c_{3j−2k−1+i} = c(g_{k}(x), k − 3 · 2^{e} − 1 + i).

Thus we must show

c(g_{k}(x), 3(2^{e+2}m − 2^{e+1}) − i) + c(g_{k}(x), 2^{e+2}m − 2^{e+2} + i − 1)
= \begin{cases}1 & if 1 ⩽ (i mod 2^{e+2}) ⩽ 2^{e+1} \\ 0 & otherwise. \end{cases}  (3.26)

We have

g_{k}(x) = (1 + x + x^{2})^{2k+1} \frac{1 + x^{3·2^{e+1}}}{(1 + x^{3})(1 + x^{3·2^{e+2}})}

= (1 + x + x^{2})^{2k+2^{e+1}} \frac{(1 + x)^{2^{e+1}−1}}{1 + x^{3·2^{e+2}}}
= (1 + x^{2^{e+3}} + x^{2^{e+4}})^{m} \frac{(1 + x)^{2^{e+1}−1}}{1 + x^{3·2^{e+2}}} .

If 2^{e+1} + 1 ⩽ (i mod 2^{e+2}) ⩽ 2^{e+2}, then both exponents of x in (3.26) are in the mod 2^{e+2} range from 2^{e+1} up to 2^{e+2} − 1. Such terms have coefficient 0 in g_{k}(x), since it is of the form f(x^{2^{e+2}}) ∑_{ℓ=0}^{2^{e+1}−1} x^{ℓ}. Thus the “otherwise” part of (3.26) is verified.

Now let i = 2^{e+2}t + ϵ with 1 ⩽ ϵ ⩽ 2^{e+1}. The left hand side of (3.26) equals

c(p_{1}(x), 3 · 2^{e+2}m − 2 · 2^{e+2} − 2^{e+2}t) + c(p_{1}(x), 2^{e+2}m − 2^{e+2} + 2^{e+2}t), (3.27)

where

p_{1}(x) = \frac{(1 + x^{2^{e+3}} + x^{2^{e+4}})^{m}}{1 + x^{3·2^{e+2}}}

because the (1+x)^{2^{e+1}−1} in g_{k}(x) corresponds to the range of values of i. Replacing x^{2^{e+2}} by x, that (3.27) has the desired value of 1 follows from Lemma 3.30. (Note i in 3.30 corresponds to 3m − 2 − t above.)

The proof when k = 2^{e}(4m + 1) and m > 0 is similar. We now have

∑_{j=1}^{k−1} γ_{j}c_{3j−i} = ∑_{ℓ⩾0} ∑_{d=0}^{2^{e+1}−1} c_{3(k−3·2^{e}−d−ℓ2^{e+3})−i}
= c(h_{k}(x), 3(k − 3 · 2^{e}) − i),

where

h_{k}(x) = (1 + x + x^{2})^{2k+1} \frac{1 + x^{3} + ⋯ + x^{3(2^{e+1}−1)}}{1 + x^{3·2^{e+3}}}
= (1 + x^{2^{e+3}} + x^{2^{e+4}})^{m}(1 + x^{2^{e+2}} + x^{2^{e+3}})\frac{(1 + x)^{2^{e+1}−1}}{1 + x^{3·2^{e+3}}} ,

and similarly

∑_{j=1}^{k−1} γ_{j}c_{3j−2k−1+i} = c(h_{k}(x), k − 9 · 2^{e} − 1 + i).

Thus we must show

c(h_{k}(x), 3(2^{e+2}m − 2^{e+1}) − i) + c(h_{k}(x), 2^{e+2}m − 2^{e+3} − 1 + i)
= \begin{cases}1 & if i = 2^{e+2}t + ϵ, 1 ⩽ ϵ ⩽ 2^{e+1}, m + t odd \\ 0 & otherwise. \end{cases}  (3.28)

If 2^{e+1} + 1 ⩽ (i mod 2^{e+2}) ⩽ 2^{e+2}, then both exponents in (3.28) are in a range (2^{e+1} to 2^{e+2} − 1 mod 2^{e+2}) where all coefficients of h_{k}(x) are 0. If, on the other hand, i = 2^{e+2}t + ϵ with 1 ⩽ ϵ ⩽ 2^{e+1}, then the left hand side of (3.28) equals

c(p_{2}(x), 3 · 2^{e+2}m − 2^{e+2}t − 2^{e+3}) + c(p_{2}(x), 2^{e+2}m − 2^{e+3} + 2^{e+2}t)  (3.29)

with

p_{2}(x) = \frac{(1 + x^{2^{e+3}} + x^{2^{e+4}})^{m}(1 + x^{2^{e+2}} + x^{2^{e+3}})}{1 + x^{3·2^{e+3}}}.

Replacing x^{2^{e+2}} by x, and letting ℓ = m + t, (3.29) equals

c(q_{m}(x), 4m − ℓ − 2) + c(q_{m}(x), ℓ − 2)

where q_{m}(x) is as in 3.31, and so (3.28) follows from 3.31, which is proved similarly to Lemma 3.30.

The proof when k = 2^{e+1} is similar and easier. It is also less important, since we don’t need the result in this case, and hence is omitted.

Lemma 3.30. Let

f_{m}(x) = \frac{(1 + x^{2} + x^{4})^{m}}{1 + x^{3}} = ∑ α_{m,i}x^{i}.

Then α_{m,i} + α_{m,4m−3−i} = 1 for all integers i.

Proof. The proof is by induction on m. It is true for m = 1 since f_{1}(x) = 1 + ∑_{i⩾2} x^{i}. Assume true for m − 1. Note that f_{m}(x) = (1 + x^{2} + x^{4})f_{m−1}(x). Thus

α_{m,i} + α_{m,4m−3−i}
= α_{m−1,i} + α_{m−1,i−2} + α_{m−1,i−4} + α_{m−1,4m−3−i} + α_{m−1,4m−5−i} + α_{m−1,4m−7−i}
= 1 + 1 + 1 = 1.

Lemma 3.31. Let

q_{m}(x) = \frac{(1 + x^{2} + x^{4})^{m}(1 + x + x^{2})}{1 + x^{6}} = ∑ β_{m,i}x^{i}

with m ⩾ 1. Then β_{m,4m−ℓ−2} + β_{m,ℓ−2} = δ_{ℓ}, where δ_{ℓ} = \begin{cases}1 & ℓ odd \\ 0 & ℓ even. \end{cases}

The following result completes the proof of Theorem 1.3.

Theorem 3.32. The exceptional Lie group E_{6} does not satisfy the Technical Hypothesis.

Proof. We use the order of the generators in the computer package LiE ([19]). Information about the types of the representations can be found in [17], although a different ordering of the generators is used there. Then R(E_{6}) has basic representations ρ_{1}, …, ρ_{6} of dimension 27, 78, 351, 2925, 351, and 27, respectively, such that ρ_{2} and ρ_{4} are real, while t(ρ_{1}) = ρ_{6} and t(ρ_{3}) = ρ_{5}. Thus V := Ind_{R}(H(E_{6})) is a Z/2-vector space with basis {ρ̃_{2}, ρ̃_{4}, ρ̃_{1}ρ̃_{6}, ρ̃_{3}ρ̃_{5}}. We will show that no subspace complementary to ⟨ρ̃_{1}ρ̃_{6}, ρ̃_{3}ρ̃_{5}⟩ is a ψ^{3}-submodule.

By Lemma 3.33(ii), a complementary subspace which is a ψ^{3}-submodule must have basis

{v_{2} = ρ̃_{2} + aρ̃_{1}ρ̃_{6} + bρ̃_{3}ρ̃_{5}, v_{4} = ρ̃_{4} + cρ̃_{1}ρ̃_{6} + dρ̃_{3}ρ̃_{5}}

satisfying ψ^{3}(v_{2}) = v_{4} and ψ^{3}(v_{4}) = v_{2}. From Lemma 3.33(i), we have, in V, ψ^{3}(ρ̃_{1}ρ̃_{6}) = ρ̃_{3}ρ̃_{5} and ψ^{3}(ρ̃_{3}ρ̃_{5}) = ρ̃_{1}ρ̃_{6}. Thus, using 3.33 again,

ψ^{3}(v_{2}) − v_{4} = (a − d + 1)ρ̃_{3}ρ̃_{5} + (b − c + 1)ρ̃_{1}ρ̃_{6}
ψ^{3}(v_{4}) − v_{2} = (c − b)ρ̃_{3}ρ̃_{5} + (d − a)ρ̃_{1}ρ̃_{6}.

It is clearly impossible to choose the scalars so that both of these are 0.

Lemma 3.33. i. Mod (2, I^{2}),

ψ^{3}(ρ̃_{1}) = ρ̃_{3} + ρ̃_{4}
ψ^{3}(ρ̃_{3}) = ρ̃_{3} + ρ̃_{4} + ρ̃_{5} + ρ̃_{6}
ψ^{3}(ρ̃_{5}) = ρ̃_{1} + ρ̃_{3} + ρ̃_{4} + ρ̃_{5}
ψ^{3}(ρ̃_{6}) = ρ̃_{4} + ρ̃_{5}.

ii. In V,

ψ^{3}(ρ̃_{2}) = ρ̃_{4} + ρ̃_{1}ρ̃_{6} + ρ̃_{3}ρ̃_{5}
ψ^{3}(ρ̃_{4}) = ρ̃_{2}.

Proof. Part (i) can be proved by the LiE methods used in the proof of (ii). It can also be obtained from [12, 3.9] by conjugating the matrix of ψ^{3} given there by the change-of-basis matrix given there. The B_{i} in [12, 3.9] correspond to ρ̃_{i}. Indeed the matrix of ψ^{3} on the basis {ρ̃_{1}, …, ρ̃_{6}} of I/I^{2} is

(\begin{matrix}378 & −2079 & −143856 & 6062445 & −75843 & 0 \\ 0 & 2430 & 46656 & −3109185 & 46656 & 0 \\ −27 & 351 & 18225 & −873261 & 11961 & 0 \\ 1 & −77 & −2405 & 146586 & −2405 & 1 \\ 0 & 351 & 11961 & −873261 & 18225 & −27 \\ 0 & −2079 & −75843 & 6062445 & −143856 & 378 \end{matrix})

The proof of part (ii) requires LiE, and an algorithm somewhat similar to that used in [12]. Irreducible representations are represented in LiE by their highest weight, which, for E_{6}, is a 6-tuple of integers. We adopt a notation that ρ_{i_{1},...,i_{r}} with 1 ⩽ i_{1} ⩽ ⋯ ⩽ i_{r} ⩽ 6 has highest weight (e_{1}, …, e_{6}) where e_{j} equals the number of k for which i_{k} = j. For example, ρ_{1,6} has highest weight (1, 0, 0, 0, 0, 1) and ρ_{4,4} has highest weight (0, 0, 0, 2, 0, 0). The following information is obtained from LiE.

ρ_{1}ρ_{6} = ρ_{1,6} + ρ_{2} + 1
ρ_{1}ρ_{2} = ρ_{1,2} + ρ_{1} + ρ_{5}
ρ_{1}ρ_{3} = ρ_{1,3} + ρ_{1,6} + ρ_{2} + ρ_{4}
ρ_{2}^{2} = ρ_{2,2} + ρ_{1,6} + ρ_{2} + ρ_{4} + 1
ρ_{5}ρ_{6} = ρ_{5,6} + ρ_{1,6} + ρ_{2} + ρ_{4}
ρ_{1,2}ρ_{6} = ρ_{1,2,6} + ρ_{1,3} + ρ_{1,6} + ρ_{2,2} + ρ_{2} + ρ_{4}
ρ_{3}ρ_{5} = ρ_{3,5} + ρ_{1,2,6} + ρ_{1,3} + ρ_{2,2} + 2ρ_{1,6} + ρ_{2} + ρ_{4} + ρ_{5,6} + 1
ρ_{2}ρ_{4} = ρ_{2,4} + ρ_{1,2,6} + ρ_{1,3} + ρ_{1,6} + ρ_{2,2} + ρ_{2} + ρ_{3,5} + ρ_{4} + ρ_{5,6}
ρ_{2}ρ_{2,2} = ρ_{2,2,2} + ρ_{1,2,6} + ρ_{2,2} + ρ_{2,4} + ρ_{2} + ρ_{4}
ψ^{3}(ρ_{2}) = 1 + ρ_{5,6} + ρ_{4} + ρ_{3,5} + ρ_{2} + ρ_{2,4} + ρ_{2,2,2} + ρ_{1,6} + ρ_{1,3}

We use these successively to express multisubscripted ρ’s as products of ρ_{i}’s. For example, the first one yields

ρ_{1,6} = ρ_{1}ρ_{6} − ρ_{2} − 1,

and the third then yields

ρ_{1,3} = ρ_{1}ρ_{3} − ρ_{2} − ρ_{4} − (ρ_{1}ρ_{6} − ρ_{2} − 1).

Ultimately we obtain, mod 2,

ψ^{3}(ρ_{2}) ≡ 1 + ρ_{1}ρ_{6} + ρ_{1}ρ_{2}ρ_{6} + ρ_{3}ρ_{5} + ρ_{2}ρ_{4} + ρ_{4} + ρ_{2}^{3}.  (3.34)

We have, mod 2, ρ̃_{2} ≡ ρ_{2}, while if i ≠ 2, then ρ̃_{i} ≡ ρ_{i} + 1. Substituting these into (3.34) yields the first equation of part (ii).

The second equation of part (ii) is proved by the same algorithm, but there are so many terms that it is only feasible to have a computer program manage all the details. We list the LiE program for completeness. The program just computes the coefficients of 1, ρ̃_{1}, ρ̃_{3}, ρ̃_{5}, ρ̃_{6}, ρ̃_{1}ρ̃_{6}, and ρ̃_{3}ρ̃_{5} in ψ^{3}(ρ_{i}). The first five coefficients are needed in the determination of the last two. They also serve as a check on the validity of our program. We obtain that, mod 2,

ψ^{3}(ρ_{4}) = 1 + ρ̃_{1} + ρ̃_{3} + ρ̃_{5} + ρ̃_{6} + 0ρ̃_{1}ρ̃_{6} + 0ρ̃_{3}ρ̃_{5}  (3.35)

plus other terms. The last two coefficients of (3.35) yield the quadratic part of part (ii). (The linear part is contained in the matrix above.)

Note that coefficients 2 to 5 of (3.35) agree with those of the matrix above. (The first coefficient equals dim(ρ_{4}) mod 2. Note also that (3.35) is not concerned with the coefficient of ρ̃_{2}.) An additional check of our program was provided by modifying it to compute the same seven coefficients in ψ^{3}(ρ_{2}). The values obtained, 0 + 0ρ̃_{1} + ρ̃_{3} + ρ̃_{5} + ρ̃_{6} + ρ̃_{1}ρ̃_{6} + ρ̃_{3}ρ̃_{5}, agreed with results obtained independently above.

The LiE program is listed below. The reader may profit by comparing with the related program in [12, §7]. Row numbers here are not part of the program; they are here for purpose of reference.

01 setdefault E6
02 on + height
03 terms=Adams(3,[0,0,0,1,0,0])
04 p2=0X null(6); x=1
05 while x==1 do x=0;
06   for i=1 to length(terms) do u=expon(terms,i);
07     if u[1]+u[2]+u[3]+u[4]+u[5]+u[6]>1 then j=1;
08       while u[j]==0 do j=j+1 od;
09       v=null(6); v[j]=1; w=u-v;
10       p1=tensor(v,w); n=length(p1); utop=expon(p1,n);
11       if terms | v==0 then x=1; p2=p2+1X v fi;
12       if terms | w==0 then x=1; p2=p2+1X w fi;
13       if utop!=u then print([u,v,utop]) fi;
14       for k=1 to n-1 do aa=expon(p1,k); if terms | aa==0 then
15         x=1; p2=p2+1X aa fi od fi od;
16     terms=terms+p2 od;
17 el=length(terms); a=null(el,7);
18 for i=1 to el do u=expon(terms,i);
19   if u==null(6) then a[i,1]=1 fi;
20   if u[1]+u[2]+u[3]+u[4]+u[5]+u[6]>0 then j=1;
21     while u[j]==0 do j=j+1 od;
22     v=null(6); v[j]=1; w=u-v;
23     p1=tensor(v,w); n=length(p1);
24     if j!=2 then k=1;
25       while w!=expon(terms,k) do k=k+1 od;
26       a[i]=a[k];
27       if j==1 then a[i,2]=a[i,2]+a[k,1]; a[i,6]=a[i,6]+a[k,5] fi;
28       if j==3 then a[i,3]=a[i,3]+a[k,1]; a[i,7]=a[i,7]+a[k,4] fi;
29       if j==5 then a[i,4]=a[i,4]+a[k,1]; a[i,7]=a[i,7]+a[k,3] fi;
30       if j==6 then a[i,5]=a[i,5]+a[k,1]; a[i,6]=a[i,6]+a[k,2] fi
fi;
31     for jj=1 to n-1 do if coef(p1,jj) % 2==1 then
32       ww=expon(p1,jj); k=1; while ww!=expon(terms,k) do k=k+1 od;
33       a[i]=a[i]+a[k] fi od fi od;
34 psit=Adams(3,[0,0,0,1,0,0]);
35 veec=null(7);
36 for i=1 to length(psit) do
37   if coef(psit,i) % 2==1 then uu=expon(psit,i);
38     k=1; while uu!=expon(terms,k) do k=k+1 od;
39     veec=veec+a[k] fi od;
40 print(veec)

For each ρ_{j_{1},...,j_{r}} that appears in ψ^{3}(ρ_{i}), we decompose it as ρ_{j_{1}} ⊗ ρ_{j_{2},...,j_{r}} minus lower terms. The first loop adds to a list of terms all those lower terms not already present in the list (which starts with the terms of ψ^{3}(ρ_{i})). This process is iterated until it stabilizes (no more terms being added to the list).

The terms in the list are ordered by height. The first term in the list will be ρ_{∅} = 1, which has coefficients a_{1} = 1 and a_{j} = 0 for j > 1 (line 19). For each ρ_{J} = ρ_{j_{1},...,j_{r}} in the list, the loop from lines 18 to 33 computes the coefficients a_{i} such that

ρ_{J} = a_{1} + a_{2}ρ̃_{1} + a_{3}ρ̃_{3} + a_{4}ρ̃_{5} + a_{5}ρ̃_{6} + a_{6}ρ̃_{1}ρ̃_{6} + a_{7}ρ̃_{3}ρ̃_{5}

plus other terms. For a term ρ_{J} in the list, let J′ = (j_{2}, …, j_{r}). Write

ρ_{j_{1}} ⊗ ρ_{J′} = ρ_{J} + ∑_{K<J} α_{K}ρ_{K}, α_{K} ∈ Z.

The vector of coefficients of ρ_{J} will include the sum of the vectors of those ρ_{K} for which α_{K} is odd (lines 31 to 33).

Noting that ρ_{j_{1}} = ρ̃_{j_{1}} + 1 if j_{1} ≠ 2, while ρ_{2} = ρ̃_{2}, the ρ_{j_{1}} ⊗ ρ_{J′} itself contributes nothing if j_{1} = 2. Otherwise, it finds ρ_{J′} earlier in the list (line 25) and contributes the vector of coefficients of ρ_{J′} (due to the 1 in ρ_{j_{1}}) (line 26) plus, if j_{1} = 1, 3, 5, or 6, appropriate terms. For example, if j_{1} = 1, the ρ̃_{1}-term is increased by the coefficient of the constant term of ρ_{J′}, and the ρ̃_{1}ρ̃_{6}-term is increased by the coefficient of ρ̃_{6} in ρ_{J′} (line 27). Finally, in lines 36 to 39, we add the coefficient sequences of those ρ_{J} which appear with odd coefficient in ψ^{3}(ρ_{i}).

4. v_{1}-periodic homotopy groups of E_{8}

In this section, we prove Theorem 1.2, the determination of the groups v_{1}^{−1}π_{∗}(E_{8}; 2). The situation for E_{8} is, at least conceptually, simpler than that for E_{7} because all of its representations are real. The following proposition is an immediate consequence of Theorem 2.5, since the hypothesis implies Q_{H} = 2Q.

Proposition 4.1. Let G be a simply-connected compact Lie group, such as G_{2}, F_{4}, E_{8}, and Spin(n) with n ≡ −1, 0, 1 mod 8, which satisfies

• the Technical Condition 2.4,

• all representations are real, and

• λ^{2} acts monomorphically on Q, where Q is as in 2.2. Let K = ker(λ^{2} : Q/2 → Q/2) and C = coker(λ^{2} : Q/2 → Q/2). Then

KO^{i}(Φ(G); Z_{2}^{∧}) ≈ \begin{cases}0 & i = 0, 1, 2 \\ 2Q/λ^{2}(2Q) & i = 3 \\ K & i = 4 \\ C#K & i = 5 \\ C & i = 6 \\ Q/λ^{2}(Q) & i = 7. \end{cases}

The # notation used here is as defined prior to Theorem 1.1.

The next result will be applied, with specific calculations, to give 1.2.

Proposition 4.2. Suppose G is as in 4.1. Let θ_{m} = λ^{3} − 3^{m} and \overline{θ} = λ^{3} − 1. Let

C_{m} = cok(θ_{m}|Q/ im(λ^{2}))
K_{m} = ker(θ_{m}|Q/ im(λ^{2}))
CK = cok(\overline{θ}| ker(λ^{2}|Q/2))
KK = ker(\overline{θ}| ker(λ^{2}|Q/2))
KC = ker(\overline{θ}| cok(λ^{2}|Q/2))
CC = cok(\overline{θ}| cok(λ^{2}|Q/2)).

Then

v_{1}^{−1}π_{8k+d}(G)^{#} ≈ \begin{cases}K_{4k−1} & d = −3 \\ C_{4k−1} ⊕ KK & d = −2 \\ (CK ⊕ KC)#KK & d = −1 \\ (CC#KC) ⊕ CK & d = 0 \\ K_{4k+1} ⊕ CC & d = 1 \\ C_{4k+1} & d = 2 \\ 0 & d = 3, 4. \end{cases}

Proof. For the most part, this result follows directly from 2.1 and 4.1. We must explain θ_{m} and the splittings.

The ψ^{3} − 9 in KO^{4j−1}(ΦG; Z_{2}^{∧}) which appears in 2.1 corresponds to ψ^{3} − 9 in K^{4j−1}(ΦG; Z_{2}^{∧}) under the realification and complexification homomorphisms, one of which is an isomorphism. Under Bott periodicity, this corresponds to 3^{1−2j}ψ^{3} −9 in K^{1}(ΦG; Z_{2}^{∧}), which is, by the proof of 2.5, isomorphic to P K^{1}(G; Z_{2}^{∧})/ im(ψ^{2}), and this is isomorphic to Q/ im(λ^{2}) with ψ^{3} in P K^{1}(G) corresponding to λ^{3} in Q. Thus, ψ^{3} − 9 in KO^{4j−1}(ΦG) corresponds to 3^{1−2j}(λ^{3} − 3^{2j+1}) in Q, as claimed in 4.2, since this ψ^{3} − 9 fits between v_{1}^{−1}π_{4j+1}(G)^{#} and v_{1}^{−1}π_{4j+2}(G)^{#}.

The exact sequences of 2.1 and 2.2 lead to extension questions. That some of these are split can probably be derived from a more careful analysis of the work in [9] that led Bousfield to 2.2. We prefer to establish these split extensions by comparison with the Bendersky-Thompson spectral sequence (BTSS) method used in [3] and [4]. This comparison is quite illuminating for both approaches.

We employ the notation of 4.2 in a BTSS chart which generalizes [3, 4.9] for G_{2} and F_{4} and [4, 1.5] for Spin(8a ± 1).

Diagram 4.3. The BTSS when all representations are real

[figure: t − s = 8k+; −3; −2; 0; 2; 4; s = 1; 2; 3; 4; C_{4k+1}; C_{4k−1}; CC; CC; CC; CC; KC; KC; KC; KC; CK; CK; CK; CK; K_{4k+1}; K_{4k−1}; KK; KK]

For G_{2}, F_{4}, and Spin(8a ± 1), the d_{3}-differentials were determined in [3] and [4] by naturality arguments. Such arguments seem to be inadequate to determine all d_{3}’s in E_{8}. But 2.2, 4.1, and 4.2 imply that the differentials must behave as displayed above in all these spaces. The most subtle part is the differentials and extensions from K_{4k+1} and C_{4k+1}. One way to see this is that the class(es) KK in position (8k, 5) do not survive by 4.2, and so must be hit by d_{3} from K_{4k+1}. But the K_{4k+1}-summand in v_{1}^{−1}π_{8k+1}(G) is isomorphic to the summand in E_{2}^{2,8k+3}(G), and so the extension into CK must be present to compensate for the class in K_{4k+1} which supported the d_{3}. A similar argument implies the differential and extension from C_{4k+1}.

This chart then implies the ⊕’s in 4.2. The one when d = −2 is due to the relation 2η = 0 in π_{∗}(−). The ⊕ when d = −1 is due to the fact that CK and KC appear in the same filtration, and the direct sum splitting there was established in [3, 3.1]. The splittings when d = 0 and d = 1 follow for similar reasons.

The reader should be aware that this BTSS comparison is only used to establish these splitting results, which should be regarded as fine tuning. Moreover, most of these splittings can probably be deduced from a deeper study of Bousfield’s approach without the BTSS. A reader who is well-versed in such spectral sequences may draw additional insight from this comparison, but one who is not may omit this portion of the paper. That is the reason that we have not explained the spectral sequence in more detail.

Next we compute the groups of 4.2 explicitly for E_{8}. For Lie groups such as E_{7}, in which the division into type is nontrivial, it is important that we use the basis of ρ̃_{i} for our Adams operations. But in E_{8} all representations are real, and so we may use any basis we like. Recall that (Q, λ^{2}, λ^{3}) of 4.2 is isomorphic to (P K^{1}(G), −ψ^{2}, ψ^{3}), which was computed in [12]. The main advantage of the K-theory perspective is that determination of ψ^{2} is enough to obtain all ψ^{k} because of the rational splitting of these Lie groups as products of spheres and the nice form of the Adams operations in spheres.

In [12, (3.2)], a basis of eigenvectors of ψ^{2} (and hence of all ψ^{k}) on P K^{1}(E_{8}; Q) is given. These are expressed there in terms of B_{i} = β(ρ̃_{i}). These eigenvectors do not span integrally, and so in [12, pp.11-19], we describe an algorithm of replacing vectors v by integral vectors (v −w)/p to reduce the exponent of p in det(M), where M is the matrix having the vectors as columns. All Adams operations on all these vectors are explicitly known, since they are explicit combinations of eigenvectors. In [12], this procedure was implemented three times for E_{8}, once (3.5) to eliminate all powers of 5 from det(M), yielding ψ^{k} on a basis of P K^{1}(E_{8})_{(5)}, once (6.1) to eliminate all powers of 3, and once (3.11) to eliminate all powers of all primes, yielding ψ^{k} on a basis of P K^{1}(E_{8}) (integral). Although we could use the latter here, the numbers are so large as to be unwieldy.

We perform the same algorithm to eliminate just the exponents of 2. We start with the eight eigenvectors of [12, (3.2)]. At each stage, we find a combination of them which is even, and replace the first vector in that combination by the combination divided by 2. Not only do we keep track of these vectors of integers, but also of the coefficients in the combinations. After 61 such steps, we have a set of eight vectors of integers such that det(M) is odd, and we have a matrix N whose columns express the new vectors in terms of the original vectors. For example, if {v_{i}} are the original vectors and {w_{i}} the final ones, then

w_{5} = \frac{1}{512}v_{5} − \frac{3}{512}v_{6} + \frac{7}{512}v_{7} − \frac{451}{512}v_{8},

the coefficients of which appear in the second half of the fifth column of N. The matrix of ψ^{k} with respect to this basis of P K^{1}(E_{8})_{(2)} equals ND_{k}N^{−1}, where D_{k} is the diagonal matrix with entries

(k, k^{7}, k^{11}, k^{13}, k^{17}, k^{19}, k^{23}, k^{29}).

We obtain the following matrices, which we write in transposed form, to fit them on one line.

(ψ^{2})^{T} = (\begin{matrix}2 & −4095 & 359169 & 516129 & 442353 & 76155 & 3045795 & −65866251 \\ 0 & 128 & −20400 & 22242 & 2241 & −436758 & −1122003 & −154822161 \\ 0 & 0 & 2048 & −17664 & −33408 & 321744 & −2942091 & −50472798 \\ 0 & 0 & 0 & 8192 & −92160 & −69120 & −2018880 & 130940625 \\ 0 & 0 & 0 & 0 & 2^{17} & −294912 & 626688 & −472681728 \\ 0 & 0 & 0 & 0 & 0 & 2^{19} & −7372800 & 57741312 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2^{23} & −66060288 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2^{29}, \end{matrix})

while (ψ^{3})^{T} equals

(\begin{matrix}3 & −70980 & 34109991 & −9760074 & −2500923 & −54666120 & 30971318970 & −8501480380389 \\ 0 & 2187 & −1858950 & 10626633 & −3418281 & −1003550877 & −8902699632 & −19780047779454 \\ 0 & 0 & 3^{11} & −4074381 & −23206257 & 754291926 & −35698131999 & −6402009498057 \\ 0 & 0 & 0 & 3^{13} & −95659380 & −71744535 & −22121231625 & 16777754459505 \\ 0 & 0 & 0 & 0 & 3^{17} & −774840978 & 9556372062 & −60452318262582 \\ 0 & 0 & 0 & 0 & 0 & 3^{19} & −87169610025 & 7495424200683 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3^{23} & −8567029273257 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3^{29} \end{matrix})

Considering these matrices mod 2, we see that K of 4.1 is ⟨w_{5}, w_{6}, w_{7}, w_{8}⟩ with \overline{θ} of 4.2 sending w_{5} ↦ 0, w_{6} ↦ w_{7} + w_{8}, w_{7} ↦ w_{8}, and w_{8} ↦ 0. Thus in 4.2, dim(KK) = dim(CK) = 2. Similarly C of 4.1 is ⟨w_{1}, w_{2}, w_{3}, w_{4}, w_{6} : w_{2} + w_{3} + w_{4} + w_{6}⟩ with w_{1} ↦ w_{3}, w_{2} ↦ w_{4} + w_{6}, w_{3} ↦ w_{4}, w_{4} ↦ w_{6}, and w_{6} ↦ 0. Hence dim(KC) = dim(CC) = 1.

We find the groups C_{4k±1} of 4.2 by using the algorithm which was applied to (E_{8}, 3) in [12, pp.35-37], to (F_{4}, 2) in [3, 4.3], and to (E_{6}, 2) in [14, 2.4]. For C_{m}, we

use Maple to pivot (\begin{matrix}(ψ^{2})^{T} \\ (ψ^{3} − 3^{m})^{T} \end{matrix} ) on odd entries and remove the corresponding row

and column. This can be done seven times, leaving a 9 × 1 matrix, corresponding to the first column of the original matrix. Thus the group presented is cyclic of order 2^{e}, with e the smallest 2-exponent of the nine relations.

If m ≡ 1 mod 8, we eventually find it convenient to let R = 3^{m} − 3^{2^{11}+273}; that is, we consider (ψ^{3} − R − 3^{2^{11}+273})^{T}. Then ν(R) = ν(m − 2^{11} − 273) + 2 ⩾ 5. We can reduce 3^{2^{11}+273} mod a suitable power of 2, such as 2^{28}, to keep the numbers manageable. The nine relations are expressed as polynomials in R. Two of them, corresponding to the 15th and 14th^{1} rows of the original matrix, have lower 2-powers than the others. Up to odd multiples, these are

2^{25} + 2^{14}R + 2^{10}R^{2} + 2^{8}R^{3} + 2^{4}R^{4} + 2^{5}R^{5}
2^{24} + 2^{10}R + 2^{6}R^{2} + 2^{4}R^{3} + R^{4} + 2R^{5}.

The minimal 2-exponent is thus min(25, ν(R − 2^{14}) + 10), which yields the first case of e(m) in 1.2.

The same two rows provide the minimal 2-exponents e in the other congruences of m. They are

• If m ≡ 3 mod 8, let R = 3^{m} −3^{2^{14}+2^{11}+19}. Then e = min(28, ν(R−2^{17})+10), due to relations

2^{28} + 2^{14}R + 2^{10}R^{2} + 2^{8}R^{3} + 2^{4}R^{4} + 2^{5}R^{5}
2^{27} + 2^{10}R + 2^{6}R^{2} + 2^{4}R^{3} + R^{4} + 2R^{5}.

• If m ≡ 5 mod 8, let R = 3^{m} − 3^{2^{25}+2^{23}+2^{22}+2^{20}+29}. Then e = min(39, ν(R) + 10) due to relations

2^{39} + 2^{14}R + 2^{10}R^{2} + 2^{8}R^{3} + 2^{4}R^{4} + 2^{5}R^{5}
2^{40} + 2^{10}R + 2^{6}R^{2} + 2^{4}R^{3} + R^{4} + 2R^{5}.

• If m ≡ 7 mod 8, let R = 3^{m} − 3^{2^{17}+23}. Then e = min(31, ν(R) + 10) due to relations

2^{31} + 2^{14}R + 2^{10}R^{2} + 2^{8}R^{3} + 2^{4}R^{4} + 2^{5}R^{5}
2^{33} + 2^{10}R + 2^{6}R^{2} + 2^{4}R^{3} + R^{4} + 2R^{5}.

This shows that the groups C_{m} are as claimed in 1.2. For the groups K_{m}, we use

Proposition 4.4 to see that K_{m} is isomorphic to the group presented by (\begin{matrix}ψ^{2} \\ ψ^{3} − 3^{m} \end{matrix});

i.e., the two component matrices are no longer in transposed form. When pivoting is performed on this matrix, after six steps of pivoting on an odd element, all remaining

1The 12th row gives the same exponents as the 14th row.

elements are even. One of them has ν(−) = 1. Since K_{m} has the same order as C_{m}, we obtain K_{m} ≈ Z/2^{e(m)−1} ⊕ Z_{2}. This completes the proof of Theorem 1.2.

The following proposition, which was used above, is a slight generalization of a portion of [4, 11.3].

Proposition 4.4. Let G be a finitely generated free abelian group, and θ and ψ commuting injective endomorphisms of G with finite cokernels. Let

K = ker(G/ im(ψ) ⟶^{θ} G/ im(ψ)).

Then K^{#} is presented by (\begin{matrix}(ψ) \\ (θ) \end{matrix}), where (ψ) and (θ) denote matrices of ψ and θ

with respect to any fixed basis.

Proof. Since Pontrjagin duality is exact, the sequence

(G/ im(ψ))^{#} ⟶^{θ^{∗}} (G/ im(ψ))^{#} → K^{#} → 0

is exact. If F is a finite abelian group, there is a natural isomorphism F^{#} → Ext(F, Z) induced from the short exact sequence 0 → Z → Q → Q/Z → 0. Also, there is an isomorphism

Ext(G/ im(ψ), Z) ≈ Hom(G, Z)/ im(ψ^{∗}).

Thus

K^{#} ≈ Hom(G, Z)/ im(ψ^{∗}, θ^{∗}).  (4.5)

On the basis of Hom(G, Z) dual to that of G, the matrices of ψ^{∗} and θ^{∗} are (ψ)^{T} and (θ)^{T}, but when we take the presentation matrix as in (4.5), we transpose back again.

5. v_{1}-periodic homotopy groups of E_{7}

Although its numbers are not quite as large, the computation of v_{1}^{−1}π_{∗}(E_{7}) is slightly more complicated than that of v_{1}^{−1}π_{∗}(E_{8}) because not all representations are real. We must work with the basis ρ̃_{1}, …, ρ̃_{7} of I/I^{2} or its corresponding basis B_{1}, …, B_{7} of P K^{1}(E_{7}). Here B_{i} = β(ρ̃_{i}).

The ordering of the basic representations in LiE has ρ_{1}, …, ρ_{7} of dimensions 133, 912, 8645, 365750, 27664, 1539, and 56, respectively, with ρ_{2}, ρ_{5}, and ρ_{7} quaternionic, and the rest real.

Proposition 5.1. With respect to the basis {B_{1}, …, B_{7}} of P K^{1}(E_{7}), the matrices of ψ^{2} and ψ^{3} are as follows.

ψ^{2} = (\begin{matrix}132 & −264 & −399664 & 186729664 & −1545576 & 0 & 0 \\ 0 & 856 & 27664 & −28553504 & 276696 & 0 & 0 \\ −1 & 2 & 7372 & −3803952 & 33306 & 0 & 0 \\ 0 & −1 & −133 & 110845 & −1273 & 1 & 0 \\ 0 & 0 & 968 & −1275832 & 21184 & −56 & 0 \\ 0 & 1 & −11857 & 18311345 & −350951 & 1541 & −1 \\ 0 & −912 & 42560 & −127779712 & 3514352 & −27776 & 56 \end{matrix})

ψ^{3} = (\begin{matrix}9043 & −1345048 & −239884976 & 182665852920 & −2039397624 & 1545576 & 0 \\ 0 & 261306 & 32232816 & −27497867040 & 319559499 & −276696 & 0 \\ −133 & 27664 & 4775915 & −3712222926 & 41922720 & −33306 & 0 \\ 1 & −856 & −128402 & 108783702 & −1294848 & 1329 & 0 \\ 0 & 8511 & 1404576 & −1295487792 & 16666478 & −24264 & 1 \\ 0 & −113736 & −19888326 & 19028389419 & −253946608 & 435708 & −56 \\ 0 & 564108 & 133440192 & −142676642736 & 2109092684 & −5069808 & 1597 \end{matrix})

Proof. The first matrix is obtained using the method of [12]. In [12, pp.8-9] is described the algorithm for finding λ^{2} on ρ̃_{1}, …, ρ̃_{7}. This involves running the LiE program in [12, §7] which gives λ^{2} on ρ_{1}, …, ρ_{7}, and then subtracting dim(ρ_{i}) from diagonal elements. This matrix is then negated to give ψ^{2} on {B_{1}, …, B_{7}} in P K^{1}(E_{7}).

We find eigenvectors of this matrix (ψ^{2}), and then, as described in Section 4, repeatedly replace vectors v by integral vectors (v − w)/2 until we have a spanning set of vectors which are explicit combinations of eigenvectors, and hence on which we know all ψ^{k}. The change of basis matrix is

M = (\begin{matrix}−95186445 & 2454179 & −27954 & 9434 & −12456 & 1800 & 1672 \\ −3705699 & 87516 & 3654 & 1221 & −1119 & −75 & −252 \\ −156975 & 3856 & −90 & −131 & 195 & −30 & −34 \\ −474 & 11 & 0 & −1 & 0 & 0 & 1 \\ −22225 & 534 & 18 & 0 & −3 & 15 & −12 \\ −297427 & 5403 & 693 & 501 & −123 & −330 & 177 \\ 517780118 & −12819489 & −53820 & −16146 & 5820 & 4920 & −1344 \end{matrix})

Thus the columns of M are combinations of B_{i}’s with respect to which all ψ^{k} have a known triangular form, similar to [12, 3.10]. One can check that det(M) = 3^{16}5^{7}7^{4}11^{2}13^{2}17, reflecting that it spans localized at 2, but not at the other primes relevant to E_{7}. We find that ψ^{3} with respect to the basis given by the columns of M is given by the following matrix P_{3}.

(\begin{matrix}3 & 0 & 0 & 0 & 0 & 0 & 0 \\ −9600 & 243 & 0 & 0 & 0 & 0 & 0 \\ 2316 & −4617 & 2187 & 0 & 0 & 0 & 0 \\ 28272 & 18711 & −34992 & 19683 & 0 & 0 & 0 \\ 66963 & 78246 & −188082 & −157464 & 177147 & 0 & 0 \\ −520551 & 279207 & 1264086 & 98415 & −1771470 & 1594323 & 0 \\ 52172712 & −55209114 & −14020857 & −92313270 & −20371905 & −15943230 & 3^{17} \end{matrix})

The matrix of ψ^{3} in the statement of this theorem is obtained as MP_{3}M^{−1}.

Now we embark on the application of 2.1 and 2.2, which holds for E_{7} by 2.5 and 3.4, to E_{7}. By 3.1 and 2.3,

Q_{R} = ⟨ρ̃_{1}, ρ̃_{3}, ρ̃_{4}, ρ̃_{6}, 2ρ̃_{2}, 2ρ̃_{5}, 2ρ̃_{7}⟩
Q_{H} = ⟨2ρ̃_{1}, 2ρ̃_{3}, 2ρ̃_{4}, 2ρ̃_{6}, ρ̃_{2}, ρ̃_{5}, ρ̃_{7}⟩.

Using 2.2 and the ψ^{2} matrix of 5.1, we obtain

Proposition 5.2. Let B = P K^{1}(E_{7}), B_{R} = ⟨B_{1}, B_{3}, B_{4}, B_{6}⟩, and B_{H} = ⟨B_{2}, B_{5}, B_{7}⟩.

Let D = ⟨B_{2} − B_{3}, B_{3} − B_{4}, B_{4} − B_{5}, B_{5} − B_{6}⟩. There are isomorphisms

KO^{i}(ΦE_{7}) ≈ \begin{cases}0 & i = 0 \\ ⟨B_{2}, B_{5}, B_{7}⟩/2 & i = 1, 2 \\ ⟨2B_{R}, B_{H}⟩/ψ^{2}(2B) & i = 3 \\ D/2 & i = 4 \\ ⟨B_{1}⟩/2#D/2 & i = 5 \\ ⟨B_{1}⟩/2 & i = 6 \\ ⟨B_{R}, 2B_{H}⟩/ψ^{2}(B) & i = 7. \end{cases}

We substitute this proposition into 2.1, using the matrix (ψ^{3}) of 5.1 and obtain the following result.

Proposition 5.3. Let θ_{m} = ψ^{3} − 3^{m}. The groups v_{1}^{−1}π_{8k+d}(E_{7})^{#} are isomorphic to

Z_{2}(B_{2} + B_{5})  d = 3
Z_{2}(B_{7})#Z_{2}(B_{2} + B_{5})  d = 4
Z_{2}(B_{7})# ker(θ_{4k+3}|⟨2B_{R}, B_{H}⟩/ψ^{2}(2B))  d = 5
⟨2B_{R}, B_{H}⟩/(ψ^{2}(2B), θ_{4k+3}⟨2B_{R}, B_{H}⟩)#(Z_{2}(B_{4} − B_{6}) ⊕ Z_{2}(B_{2} − B_{5})) d = 6
(Z_{2}(B_{2} − B_{3}) ⊕ Z_{2}(B_{3} − B_{4}))#(Z_{2}(B_{1})#(Z_{2}(B_{4} − B_{6}) ⊕ Z_{2}(B_{2} − B_{5}))) d = 7
(Z_{2}(B_{1})#(Z_{2}(B_{2} − B_{3}) ⊕ Z_{2}(B_{3} − B_{4})))#Z_{2}(B_{1})  d = 8
Z_{2}(B_{1})# ker(θ_{4k+5}|⟨B_{R}, 2B_{H}⟩/ψ^{2}(B))  d = 9
⟨B_{R}, 2B_{H}⟩/(ψ^{2}(B), θ_{4k+5}⟨B_{R}, 2B_{H}⟩)  d = 10

As we did with E_{8}, we can resolve some of the extension questions by comparing with the BTSS. The study of the extension questions is facilitated by first comparing the large summands of 5.3 with the E_{2}-term of the BTSS. As with E_{8}, the BTSS is only needed for resolving the extension questions and providing possible insight.

We first consider the group ⟨2B_{R}, B_{H}⟩/(ψ^{2}(2B), θ_{4k+3}⟨2B_{R}, B_{H}⟩) which occurs in 5.3 when d = 6. The corresponding group E_{2}^{1,8k+7}(E_{7})^{#} has the same form, with the three 2’s omitted. We compute this latter group first.

The desired group E_{2}^{1,8k+7}(E_{7})^{#} is presented by the matrix

M = (\begin{matrix}(ψ^{2})^{T} \\ (ψ^{3} − 3^{4k+3})^{T} \end{matrix} ) ,

where (ψ^{2}) and (ψ^{3}) are as in 5.1. Five times we pivot on odd elements, removing rows and columns, to leave a 9 × 2 matrix. The smallest ν(−) of remaining terms is ν = 2, and so we split off a Z/4 and pivot on that element. We let R = 3^{4k+3} − 3^{75}, and so ν(R) = ν(k−18)+4. Of the eight relations remaining, those with the smallest 2-exponents are, with odd multiples omitted,

\begin{matrix}2^{16} + 2^{7}R + 2^{6}R^{2} + 2R^{3} \\ 2^{17} + 2^{7}R + 2^{4}R^{2} + 2R^{3}. \end{matrix}  (5.4)

We obtain

E_{2}^{1,8k+7}(E_{7}) ≈ Z/4 ⊕ Z/2^{min(16,ν(k−18)+11)}.  (5.5)

Note that when we express a finite group G^{#} as a sum of cyclic summands, we may write it as G, since the two are abstractly isomorphic.

The effect of the three 2’s that occur in the first part of v_{1}^{−1}π_{8k+6}(E_{7})^{#} in 5.3 is to multiply by 2 rows 1-8, 10, 11, and 13 of the matrix M of the preceding paragraph, and then divide by 2 columns 1, 3, 4, and 6. The five odd entries on which we pivoted in the previous paragraph are in positions^{2} which were unaffected by the multiplying and dividing by 2 that took place here, and we pivot on them as before. The Z/4 is in position (8,1) and also remains unchanged by the multiplying and dividing by 2 and is still a Z/4 after the five steps of pivoting. After pivoting on the Z/4, and with R as before, the first relation of (5.4) is now twice as large, while the second is not changed.^{3} Thus we obtain that the first part of v_{1}^{−1}π_{8k+6}(E_{7})^{#} in 5.3 is Z/4 ⊕ Z/2^{min(17,ν(k−18)+11)}.

A similar analysis applies to the group in 5.3 with d = 10. We first look at the corresponding BTSS group E_{2}^{1,8k+11}(E_{7})^{#}, which has the same form without the two 2’s. The pivoting up to and including splitting off the Z/4 is the same as in the case d = 6 just considered. If k is even, we let R = 3^{4k+5} − 3^{141}, and so ν(R) = ν(k − 34) + 4. The relations with smallest 2-exponents are

\begin{matrix}2^{20} + 2^{8}R + 2^{4}R^{2} + 2R^{3} \\ 2^{19} + 2^{8}R + 2^{4}R^{2} + 2R^{3}. \end{matrix}  (5.6)

If k is odd, we let R = 3^{4k+5} − 3^{7·2^{11}+17}, and so ν(R) = ν(k − 7 · 2^{9} − 3) + 4. The relations with smallest 2-exponent are

\begin{matrix}2^{24} + 2^{8}R + 2^{4}R^{2} + 2R^{3} \\ 2^{24} + 2^{10}R + 2^{6}R^{2} + 2^{3}R^{3}. \end{matrix}  (5.7)

Thus we obtain

E_{2}^{1,8k+11}(E_{7}) ≈ Z/4 ⊕ \begin{cases}Z/2^{min(19,ν(k−34)+12)} & k even \\ Z/2^{min(24,ν(k−7·2^{9}−3)+12)} & k odd. \end{cases}  (5.8)

To obtain v_{1}^{−1}π_{8k+10}(E_{7})^{#}, we multiply rows 9, 12, and 14 by 2, and then divide columns 2, 5, and 7 by 2. The pivoting through the Z/4 goes as before, as the pivoted-upon elements were unchanged. Both the relevant relations ((5.6) or (5.7)) are divided by 2, as they are in original row 3 or 4 and column 7. Thus v_{1}^{−1}π_{8k+10}(E_{7}) is obtained from (5.8) by decreasing the exponent of the large summand by 1.

In order to determine the ker part of v_{1}^{−1}π_{8k+9}(E_{7})^{#} in 5.3 explicitly, we need the following slight modification of Proposition 4.4, which is proved by the same method.

2(1,3),(2,4),(7,6),(14,5), and (12,2) 3The first relation occurs three times (with different odd coefficients) and is due to original rows 3, 4, and 6, while the second relation comes from original row 9. The generator for these relations is that of column 7.

Proposition 5.9. Suppose G, ψ, and θ are as in 4.4. Suppose the basis of G is partitioned as B_{1} ∐ B_{2}. Let G′ be the subgroup of G spanned by B_{1} and 2B_{2}, and assume ψ(G) ⊂ G′. Then θ induces θ′ : G′/ψ(G) → G′/ψ(G). Let K′ = ker(θ′). Then (K′)^{#} is presented by

(\begin{matrix}A_{1} & A_{2} \\ \frac{1}{2}A_{3} & \frac{1}{2}A_{4} \\ C_{1} & 2C_{2} \\ \frac{1}{2}C_{3} & C_{4} \end{matrix})

where (\begin{matrix}A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix}) and (\begin{matrix}C_{1} & C_{2} \\ C_{3} & C_{4} \end{matrix}) are the matrices of ψ and θ with respect to

B_{1} ∐ B_{2}.

When this is computed using ψ = ψ^{2} and θ = ψ^{3} − 3^{4k+5} of 5.1 using the pivoting methods described above, we obtain that the ker part of v_{1}^{−1}π_{8k+9}(E_{7})^{#} is Z/2^{4} ⊕ Z/2^{e−2}, where e is the 2-exponent of the large summand of v_{1}^{−1}π_{8k+10}(E_{7}) described above. Note that the two groups must have the same order since they are the kernel and cokernel of an endomorphism of a finite abelian group.

A result similar to 5.9 applies to the large summand of v_{1}^{−1}π_{8k+5}(E_{7}), and when it is computed we obtain similarly that the ker part of v_{1}^{−1}π_{8k+5}(E_{7})^{#} is Z/2^{4} ⊕ Z/2^{e−2}, where e = min(17, ν(k − 18) + 11), the large summand in v_{1}^{−1}π_{8k+6}(E_{7}) obtained earlier. We emphasize that in performing the pivoting to obtain this group, we need find only that it has two summands and that the smaller has order 2^{4}.

As we shall use the BTSS to settle some extension questions, it is useful to know the following.

Proposition 5.10. The BTSS 2-line groups are given by

E_{2}^{2,8k+d+2}(E_{7}) ≈ Z_{2} ⊕ \begin{cases}Z/2^{4} ⊕ Z/2^{t_{1}−2} & d = 5 \\ Z/2^{5} ⊕ Z/2^{t_{2}−3} & d = 9, \end{cases}

where t_{1} is the large exponent in (5.5) and t_{2} the large exponent in (5.8).

Proof. As in [3, 3.1b], there is a short exact sequence

0 → B/(2, ψ^{2}, θ_{4k+(d+1)/2}) → E_{2}^{2,8k+d+2}(E_{7})^{#} → ker(θ_{4k+(d+1)/2}|B/ im(ψ^{2})) → 0.

The first group is Z_{2} generated by B_{1} (d = 9) or B_{7} (d = 5), while the second is obtained by the algorithm used to find K_{m} for E_{8} near the end of Section 4 or that used to find the groups v_{1}^{−1}π_{8k+d}(E_{7}) above, without the 2’s that complicated the argument there. The matrix pivots down to two columns, and the smallest 2exponent is 4 (d = 5) or 5 (d = 9). The other exponent is then forced by the equality of the orders of the kernel and cokernel of an endomorphism.

We can incorporate the above information into the BTSS chart for E_{7} below, and then use this chart, as we did with E_{8}, to draw inferences about some extensions in 5.3. The facts that we use to make these inferences are that 2η = 0 in π_{∗}(−) and that relationships between elements in the same bidegree is the same in π_{∗}(−) as it is in E_{2}(−). Theorem 1.1 follows from 5.3, the specific calculations of portions of v_{1}^{−1}π_{8k+d}(E_{7}) made above when d = 5, 6, 9, and 10, and the extension inferences obtained from the chart. The dotted differential and extension take place unless ν(k − 18) ⩾ 6, in which case the exponent of the homotopy group is 1 larger than the E_{2}-exponent.

Diagram 5.11. The BTSS of E_{7}

[figure: t − s = 8k+; −1; 0; 2; 4; 6; s = 1; 2; 3; 4; 2^{5} 2^{t_{2}−3}; 2^{2}; 2^{t_{2}}; 2^{4} 2^{t_{1}−2}; 2^{t_{1}}; 2^{2}]

References

[1] M. Bendersky and D. M. Davis, 2-primary v_{1}-periodic homotopy groups of SU(n), Amer Jour Math 114 (1991) 529-544.

[2] , The 1-line of the K-theory Bousfield-Kan spectral sequence for Spin(2n + 1), Contemp Math AMS, 279 (2001) 37-56.

[3] , A stable approach to an unstable homotopy spectral sequence, Topology 42 (2003) 1261-1287.

[4] , v_{1}-periodic homotopy groups of SO(n), to appear in Memoirs Amer Math Soc.

[5] , 3-primary v_{1}-periodic homotopy groups of F_{4} and E_{6}, Trans Amer Math Soc 344 (1994) 291-306.

[6] M. Bendersky, D. M. Davis, and M. Mahowald, v_{1}-periodic homotopy groups of Sp(n), Pac Jour Math 170 (1995) 319-378.

[7] M. Bendersky, D. M. Davis, and M. Mimura, v_{1}-periodic homotopy groups of exceptional Lie groups—torsion-free cases, Trans Amer Math Soc 333 (1992) 115-136.

[8] A. K. Bousfield, The K-theory localization and v_{1}-periodic homotopy groups of H-spaces, Topology 38 (1999) 1239-1264.

[9] , A classification of K-local spectra, Jour Pure Appl Alg 66 (1990) 121-163.

[10] M. R. Bremner, R. V. Moody, and J. Patera, Tables of dominant weight multiplicities for representations of simple Lie algebras, Marcel Dekker (1985).

[11] T. Bröcker and T. tom Dieck, Representations of compact Lie groups, Springer-Verlag (1985).

[12] D. M. Davis, From representation theory to homotopy groups, Mem Amer Math Soc 759 (2002).

[13] , v_{1}-periodic homotopy groups of SU(n) at odd primes, Proc London Math Soc 43 (1991) 529-541.

[14] , The K-completion of E_{6}, Progress in Mathematics 215 (2003) 107-124, Birkhauser.

[15] , 3-primary v_{1}-periodic homotopy groups of E_{7}, New York Jour Math 4 (1998) 185-221.

[16] L. Hodgkin, On the K-theory of Lie groups, Topology 6 (1967) 1-36.

[17] W. G. McKay and J. Patera, Tables of dimensions, indices, and branching rules for representations of simple Lie algebras, Marcel Dekker (1981).

[18] C. M. Naylor, Cohomology operations in the K-theory of the classical groups, Port Math 38 (1979) 145-153.

[19] M. vanLeeuven, LiE, a computer algebra package for Lie algebra computa-

     tions, http://wwwmathlabo.univ-poitiers.fr/∼maavl/LiE/.

This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at

ftp://ftp.rmi.acnet.ge/pub/hha/volumes/2003/n1a13/v5n1a13.(dvi,ps,pdf)

Donald M. Davis dmd1@lehigh.edu  http://www.lehigh.edu/∼dmd1

Lehigh University Bethlehem, PA 18015