Irreducible Representations of Exceptional Simple Lie Algebras

Der Landmesser P.

Donnerstag, 15. März 2001 Let G = G(F) be a group of Lie type whose Dynkin diagram 𝒟 is G_{2}, F_{4} or E_{i}, i = 6, 7, 8.

Definition 1. Let U be a Sylow p-subgroup of G = G(F), charF = p, |F | = q. Set

cc(U) = {|x^{U}| | x ∈ U},

and

cd(U) = {χ(1) | χ ∈ Irr(U)}.

If any element of cc(U) (cd(U)) is a q-power, we say that U is a q-power-size (q-power-degree) group.

Theorem 2 (P.). There exists c = c(𝒟), 𝒟 a Dynkin diagram of exceptional type, such that U is a q-power-size group, provided charF ≥ c. Definition 3. We say that a prime p is good w.r.t. a Dynkin diagram if p ∤ n_{δ}(ϕ), for any positive root ϕ = ∑_{δ∈Δ} n_{δ}(ϕ)δ.

p > 3 is a good prime for type G_{2}, F_{4}, E_{6}, E_{7}, while p > 5 is good for E_{8}.

Conjecture 4. If charF is good then U(𝒟) is a q-power -size group. Theorem 5 (P.). This conjecture holds true if 𝒟 is of type G_{2}. Theorem 6 (Karkar). If charF = p ≥ 5, then the Sylow p-subgroup U of G_{2}(F) is a q-power-degree group. Conjecture 7. If charF is good then U(𝒟) is a q-power-degree group. Remark 8 (Isaacs, P.). Both conjectures hold true if G is a classical group.

Irreducible Modules

Let L be a simple Lie algebra over ℂ, H a Cartan subalgebra of L, Φ the corresponding root system (a subset of H^{∗} = Hom(H, ℂ)), and Δ the simple roots.

Fix {x_{ϕ}, y_{ϕ} | ϕ ∈ Φ^{+}} ∪ {h_{δ} | δ ∈ Δ} a ℂ-basis of L, Let Λ be the weight lattice associated to Φ and Λ^{+} the subset of dominant weights.

Any finite dimensional irreducible L-module V is uniquely encoded by some λ ∈ Λ^{+}. We denote V by V (λ).

H acts semisimply on V and V = ⨁_{μ∈H^{∗}} V_{μ}, where V_{μ} = {v ∈ V | h.v = μ(h)v, ∀h ∈ H}.

The Structure of V (λ)

Lemma 9. Let V = V (λ). Then V_{λ} = ⟨v^{+}⟩. We call v^{+} a maximal vector. Fix a total ordering ≤ on Δ (hence on Φ). Let [ϕ_{1}, …, ϕ_{m}] be the list of positive roots in ascending order. V is generated by y_{ϕ_{1}}^{i_{1}} ⋯ y_{ϕ_{m}}^{i_{m}}.v^{+}. Moreover the former vector lies in V_{μ}, where μ = λ − ∑ i_{j}ϕ_{j} is an element of λ − ℕΔ.

Let Π(λ) be {μ ∈ H^{∗} | V (λ)_{μ} ≠ 0}. We define a partial order on Λ declaring that μ ⪯ ν iff ν − μ ∈ ℕΔ.

Denote with 𝒲 the Weyl group.

Theorem 10. Π(λ) is the union of all 𝒲-orbits of all dominant weights μ ⪯ λ. Moreover dim V_{μ} = dim V_{σμ}, for any σ ∈ 𝒲. Lemma 11. Each weight is conjugate under 𝒲 to one and only one dominant weight. Moreover if μ is dominant σμ ≺ μ, σ ∈ 𝒲. Set

Π(λ)^{+} = (λ − ℕΔ) ∩ Λ^{+}.

Then Π(λ) is the 𝒲-orbit of Π(λ)^{+}.

The dimension of eigenspaces may be obtained recursively, starting from dim V_{λ} = 1. Set m(μ) = dim V (λ)_{μ} and ρ = \frac{1}{2} ∑_{ϕ∈Φ^{+}} ϕ. Theorem 12 (Freudenthal’s Formula). For Π(λ) ∋ μ ≺ λ, a = |λ + ρ|^{2} − |μ + ρ|^{2} > 0 and am(μ) = 2 ∑_{ϕ≻0} ∑_{i=1}^{∞} m(μ + iϕ)(μ + iϕ, ϕ).

The Digraph Γ(λ)

First attempt: The vertices of Γ(λ) are the eigenspaces V_{μ}, μ ∈ Π(λ) and the edges are (V_{μ}, V_{ν}) if μ − ν = δ ∈ Δ.

If nu ≠ λ then ∃μ ∈ Π, δ ∈ Δ, and v ∈ V_{μ}, such that y_{δ}.v ≠ 0.

Warm up with G_{2}

Let L =G_{2}, Δ = {α, β}, α short and β long. Set λ = 2α + β, then dim V (λ) = 7. Π(λ) = 𝒲λ∪{0} = {±(2α+β), ±(α+β), ±α, 0}. Call 𝔛 the representation afforded by V (λ).

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In particular m(μ) = 1, for any μ ∈ Π(λ), and we may define v_{ν} = y_{δ}.v_{μ} as generators of the eigenspaces, where μ − δ = ν. After reordering,

𝔛(y_{α}) = e_{21} + e_{43} + e_{54} + e_{76},

and

𝔛(y_{β}) = e_{32} + e_{76}.

We determine 𝔛(x_{δ}) recursively observing that

x_{δ}.v_{μ} = y_{γ}x_{δ}.v_{μ+γ} + ε_{δ,γ}⟨μ + γ, δ⟩v_{μ+γ},

where γ is the unique simple root such that μ + γ ∈ Π(λ) and ε_{δ,γ} is 1 if δ = γ, 0 otherwise.

We obtain

𝔛(x_{α}) = e_{12} + 2e_{34} + 2e_{45} + e_{67},

and

𝔛(x_{β}) = e_{23} + e_{56}.

Remark that all representing matrices lie in (ℤ)_{7}

F_{4}, a more difficult case

Let 𝒟 be F_{4} of rank 4. Let Δ = {α, β, γ, δ}, with α, β short and γ, δ long, each one adjacent only to the next.

The irreducible module V of minimal dimension is realized by Λ^{+} ∋ λ = 2α + 3β + 2γ + δ and dim V = 26. Here Π(λ) = 𝒲λ ∪ {0}, all eigenspaces of dimension 1 but V_{0} which has dimension 2.

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Lemma 13. If ϵ, ζ are simple roots orthogonal to each other, then ϵ + ζ ∉ Φ. In particular [y_{ϵ}, y_{ζ}] = 0.

For example, if we call v_{4} a generator for V_{α+2β+γ+δ}, then y_{β}y_{δ}.v_{4} = y_{δ}y_{β}.v4 and this vector generates V_{α+β+γ}. This takes care of all vertices with more than one incoming edge, but V_{0}.

Remark that the subgraph starting at V_{α+β} and ending at V_{−α−β} corresponds to the adjoint module W for 𝔰𝔩_{3}.

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Consider a Chevalley basis {e_{ϕ}, f_{ϕ}, h_{δ}} for 𝔰𝔩_{3}. Wlog we may assume that V_{α+β} is generated by e_{α+β}, that is, the maximal vector for the ad-representation corresponds to the highest root. Therefore 𝔛(y_{α})_{↾W} equals

e_{11,10} + e_{14,12} + e_{15,13} + 2e_{15,14} + e_{17,16}

and 𝔛(y_{β})_{↾W}

e_{12,10} + e_{13,11} + e_{16,14} + 2e_{16,13} + e_{17,15}.

We may get rid of all incoming edges but one and call it a defining edge. In this way 𝔛(y_{α}) is

e_{21} +e_{75} +e_{98} +𝔛(y_{α})_{↾W} +e_{19,18} +e_{22,20} +e_{26,25},

𝔛(y_{β}) is

e_{32} +e_{54} +e_{86} +𝔛(y_{β})_{↾W} +e_{21,19} +e_{23,22} +e_{25,24},

𝔛(y_{γ}) = e_{43}+e_{10,8}+e_{11,9}+e_{18,16}+e_{19,17}+e_{24,23},

and

𝔛(y_{δ}) = e_{64} + e_{85} + e_{97} + e_{20,18} + e_{22,19} + e_{23,21}.

Working backwards as in the G_{2} case, we may obtain the corresponding matrices for x_{ζ}, ζ ∈ Δ. Since 𝔛 exists x_{ζ}.v_{μ} does not depend on the choice of ϵ ∈ Δ such that v_{m}u = y_{ϵ}.v_{μ+ϵ}.

Problems

• is the basis obtained a Chevalley basis, i.e. with integer structure constants?

• find a different proof of the independence of the representing matrices for x_{ζ} from the choice pf predecessors;