Proseminar Symmetrie- und Spiegelungsgruppen 15. Vortrag: Wurzelgitter

Jens Rottmann

12.2.2001

Inhaltsverzeichnis

1 Wurzelsysteme und Wurzelgitter 2 1.1 Definitionen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Klassifikation der Wurzelsysteme . . . . . . . . . . . . . . . . . . . . 3

2 Wurzelgitter und Codes 8 2.1 Die Weylgruppe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Gitter und Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

15. Gitter und Codes

1 Wurzelsysteme und Wurzelgitter

Im letzten Vortrag wurde als Beispiel eines durch einen Code erzeugten Gitters das Gitter E_{8} behandelt. Es ist gezeigt worden, dass dieses Gitter gerade (sogar doppelt gerade) ist und es von den Elementen der Länge \sqrt{2} erzeugt wird. Im Folgenden soll eine Verbindung zwischen derartigen Gittern und binären Codes hergestellt werden.

1.1 Definitionen

Definition 1.1. Sei Γ ⊂ ℝ^{n} ein gerades Gitter (d.h. ⟨x, x⟩ ∈ 2ℤ, ∀x ∈ Γ). Sei Φ := {x ∈ Γ : ⟨x, x⟩ = 2}. Falls Φ das Gitter Γ erzeugt, so heißt Γ ein Wurzelgitter und Φ ist das zugehörige Wurzelsystem. Die Elemente von Φ heißen Wurzeln.

Bemerkung 1.2.

1. Ein gerades Gitter Γ ⊂ ℝ^{n} ist ganzzahlig, denn für x, y ∈ Γ beliebig gilt:

2ℤ ∋ ⟨x + y, x + y⟩ = \underbrace{⟨x, x⟩}_{∈2ℤ}+2 ⟨x, y⟩ + \underbrace{⟨y, y⟩}_{∈2ℤ} ⇒ ⟨x, y⟩ ∈ ℤ.

2. Für alle x, y ∈ Φ gilt

• ⟨x, y⟩ ∈ {−2, −1, 0, 1, 2}, denn die Ungleichung von Cauchy-Schwarz liefert | ⟨x, y⟩ | ≤ ∥x∥ ∥y∥ = 2 und mit (1) folgt die Behauptung.

• ⟨x, y⟩ = ±2 ⇒ x = ±y, denn in der Formel von Cauchy-Schwarz gilt die Gleichheit genau dann, wenn x und y parallel sind.

• ⟨x, y⟩ = 1 ⇔ x − y ∈ Φ, denn

⟨x − y, x − y⟩ = ⟨x, x⟩ − 2 ⟨x, y⟩ + ⟨y, y⟩ = 4 − 2 ⟨x, y⟩ .

3. In diesem Spezialfall handelt es sich bei Φ tatsächlich um ein abstraktes Wurzelsystem. Im Folgenden ist mit σ_{α} die Spiegelung an der Hyperebene α^{⊥} gemeint.

• Φ erzeugt ℝ^{n} ist trivial.

• σ_{α}Φ = Φ, ∀α ∈ Φ, denn σ_{α} ist bijektiv, damit genügt es zu zeigen, dass σ_{α}β ∈ Φ, ∀β ∈ Φ gilt:

σ_{α}β = β − 2\frac{⟨α, β⟩}{⟨α, α⟩}α = β − ⟨α, β⟩ α.

Natürlich ist mit α ∈ Φ auch −α ∈ Φ. 1.Fall: α∥β ⇒ σ_{α}β = β − 2β = −β ∈ Φ. 2.Fall: α⊥β ⇒ σ_{α}β = β − 0α = β ∈ Φ. 3.Fall: ⟨α, β⟩ = 1 ⇒ σ_{α}β = β − α ∈ Φ. 4.Fall: ⟨α, β⟩ = −1 ⇒ ⟨−α, β⟩ = 1 ⇒ σ_{α}β = β − (−α) ∈ Φ.

15. Gitter und Codes  1.2 Klassifikation der Wurzelsysteme

• α ∈ Φ, sα ∈ Φ mit s ∈ ℝ. Es folgt
⟨α, α⟩ = 2 = ⟨sα, sα⟩ = s^{2} ⟨α, α⟩ ⇒ s = ±1.
• α, β ∈ Φ. Dann ist 2\frac{⟨α,β⟩}{⟨α,α⟩} = ⟨α, β⟩ ∈ ℤ, weil Γ ganzzahlig ist.

Definition 1.3. Eine Teilmenge Δ ⊆ Φ heißt Wurzelbasis, falls

1. Δ ist Basis von Γ,

2. jedes Element β ∈ Φ läßt sich schreiben als β = ∑_{α∈Δ} k_{α}α mit k_{α} ∈ ℕ_{0}, ∀α oder −k_{α} ∈ ℕ_{0}, ∀α.

Satz 1.4. Jedes Wurzelsystem besitzt eine Wurzelbasis.

Beweis. Für einen Beweis siehe [Hum72, Kapitel 10.1].

Bemerkung 1.5.

1. Eine derartige Wurzelbasis erfüllt die Eigenschaften

(a) ⟨α, α⟩ = 2, ∀α ∈ Δ,

(b) ⟨α, β⟩ ∈ {0, −1}, ∀α, β ∈ Δ, α ≠ β denn nach Bemerkung 1.2 gilt ⟨α, β⟩ ∈ {−1, 0, 1}, ∀α, β ∈ Φ, α ≠ β und nach Definition 1.3 muß gelten α−β ∉ Φ. Gemäß Bemerkung 1.2 ist also ⟨α, β⟩ = 1 für α, β ∈ Δ nicht möglich.

2. ⟨α, β⟩ ∈ {−1, 0}, ∀α, β ∈ Δ, α ≠ β besagt gerade, dass zwei Basiswurzeln entweder orthogonal zu einander sind, oder einen Winkel von 120^{∘} einschließen, denn

cos ∢\{α, β\} = \frac{⟨α, β⟩}{∥α∥ ∥β∥} = \begin{cases}0 & , falls ⟨α, β⟩ = 0 ⇒ ∢\{α, β\} = 90^{∘}, \\ −\frac{1}{2} & , falls ⟨α, β⟩ = −1 ⇒ ∢\{α, β\} = 120^{∘}. \end{cases}

1.2 Coxeter-Dynkin-Diagramme und die Klassifikation der Wurzelsysteme

Erinnerung (Coxeter-Dynkin-Diagramme). Man kann einem Wurzelgitter (genauer einer Wurzelbasis Δ = (e_{1}, …, e_{n}) des Wurzelgitters) ein sogenanntes Coxeter-Dynkin-Diagramm zuordnen, das das Wurzelsystem und damit das Wurzelgitter eindeutig beschreibt: Die Basiswurzeln werden durch Punkte repräsentiert. Die Punkte zweier Basiswurzeln e_{i} und e_{j} (i ≠ j) werden mit einer Linie verbunden, falls ⟨e_{i}, e_{j}⟩ = −1 ist (also, falls der Winkel zwischen ihnen ein 120^{∘} Winkel ist), sonst (nach Bemerkung 1.5 (b) also im Fall ⟨e_{i}, e_{j}⟩ = 0 bzw. im Falle eines rechten Winkels zwischen e_{i} und e_{j}) werden sie nicht verbunden.

Beispiel 1.6. Als Beispiel gebe ich noch einmal wie im letzten Vortrag E_{8} an:

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An diesem Beispiel erkennt man, dass e_{8} orthogonal zu e_{1}, e_{2}, e_{3}, e_{4}, e_{6} und e_{7} liegt und mit e_{5} einen Winkel von 120^{∘} einschließt.

Definition 1.7. Ein Wurzelsystem Φ heißt irreduzibel, falls es keine Teilmengen ∅ ≠ U, V ⊂ Φ mit U ˙∪V = Φ gibt, so dass jede Wurzel in U orthogonal zu jeder Wurzel in V ist.

Beispiel 1.8. In Coxeter-Dynkin-Diagrammen ausgedrückt bedeutet dies, dass der Graph verbunden ist, z.B. ist

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verbunden und damit irreduzibel, dagegen ist

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reduzibel, denn U⊥V.

Definition 1.9. Ein Wurzelgitter heißt irreduzibel, falls das erzeugende Wurzelsystem irreduzibel ist, sonst heißt es reduzibel.

Es wird jetzt der Hauptsatz über die Klassifikation der irreduziblen Wurzelgitter angegeben.

Satz 1.10. Jedes Wurzelgitter ist die orthogonale direkte Summe der folgenden irreduziblen Wurzelgitter:

A_{n}:

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D_{n}:

.^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.} •  •  •  •  •  • •  • •  • e_{1} e_{2}  e_{3}  e_{4}  e_{5}  e_{6}  e_{n−1}  e_{n}

E_{6}:

._{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} e_{1}  e_{2}  e_{3}  e_{4}  e_{5} e_{6} • • •  • •  • •  • •  •

E_{7}:

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E_{8}:

._{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} e_{1}  e_{2}  e_{3}  e_{4}  e_{5}  e_{6}  e_{7} e_{8} • • •  • •  • •  • •  • •  • •  •

Beweis. Im Rahmen dieses Vortrags wird der Satz nicht bewiesen werden, da er keine großen neuen Erkenntnisse bringt und sehr umfangreich ist. Außerdem handelt es sich bei den Standardbeweisen zum größten Teil um relativ einfache Konstruktionsbeweise, die sehr gut in der Literatur [Hum72, Kapitel 11.4] und [Eb94, Seiten 15-23] nachgelesen werden können.

Damit man sich etwas unter den seltsam klingenden Namen vorstellen kann, werden jetzt noch einige Beispiele angegeben:

Beispiel 1.11. In den Fällen A_{1} und A_{2} ist auch gut zu erkennen, wie man ein Wurzelgitter vom Typ A_{n} als zum Vektor (1, …, 1) ∈ ℤ^{n+1} orthogonales Untergitter des Gitters ℤ^{n+1} konstruiert.

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(a) A_{1}, konstruiert als Menge der Punkte mit Summe 0.

•^{(0, 0)} •^{e_{1}} •^{e_{2}}  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(c) Erhalten des Gitters A_{2} als

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von ℤ^{3}. Desweiter ist eine mögliche

Wurzelbasis (e_{1}, e_{2}) eingezeichnet.

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(d) Eine Möglichkeit der Beschreibung der Basiswurzeln von A_{3}.

Es werden jetzt Konstruktionsmöglichkeiten der Gitter angegeben, auf die im weiteren Text noch einmal (vorallem im Beweis von Satz 2.5) zurückgekommen wird. Es sei im Folgenden Text mit ε_{i} immer der i-te kanonische Basisvektor bezeichnet.

\underline{A_{n}:} Die Gitter zu A_{n} werden so, wie es oben in Beispiel 1.11 am Fall von A_{1} und A_{2} gezeigt wurde, konstruiert:

Γ = {x = (x_{1}, …, x_{n+1}) ∈ ℤ^{n+1} : ∑_{i=1}^{n+1} x_{i} = 0}.

Die Wurzeln sind gerade die Elemente ε_{i} − ε_{j}, 1 ≤ i, j ≤ n + 1, i ≠ j. Und die Wurzeln der Form e_{i} := ε_{i} − ε_{i+1}, 1 ≤ i ≤ n sind eine Basis von Γ mit

\begin{matrix}⟨e_{i}, e_{j}⟩ = & ⟨ε_{i} − ε_{i+1}, ε_{j} − ε_{j+1}⟩ \\ = & ⟨ε_{i}, ε_{j}⟩ − ⟨ε_{i+1}, ε_{j}⟩ − ⟨ε_{i}, ε_{j+1}⟩ + ⟨ε_{i+1}, ε_{j+1}⟩ \\ = & \begin{cases}2 & , i = j, \\ −1 & , |i − j| = 1, \\ 0 & , sonst. \end{cases} \end{matrix}

Also sind sie insbesondere eine Wurzelbasis.

\underline{D_{n}:} Das Gitter Γ zu D_{n} (n ≥ 2) ist die Menge aller ”geraden Punkte“ in dem Gitter ℤ^{n}:

Γ = {x = (x_{1}, …, x_{n}) ∈ ℤ^{n} : ∑_{i} x_{i} ≡ 0 (mod 2)}.

Γ wird von den Elementen der Form ±(ε_{i} + ε_{j}) und ±(ε_{i} − ε_{j}), 1 ≤ i, j ≤ n, i ≠ j erzeugt. Diese Elemente sind auch die Wurzeln von Γ. Die Wurzeln e_{1} := ε_{1} + ε_{2}, e_{i+1} = ε_{i+1} − ε_{i}, 1 ≤ i ≤ n − 1 bilden eine Wurzelbasis mit einer einfachen Rechnung sieht man, dass gilt

⟨e_{i}, e_{j}⟩ = \begin{cases}2 & , i = j, \\ −1 & , |i − j| = 1, i, j ≠ 1, \\ −1 & , i = 1, j = 3 oder j = 1, i = 3, \\ 0 & , sonst. \end{cases}

\underline{E_{8}:} Wurde im letzten Vortrag ”Von Gittern zu Codes“ als Gitter

Γ = Γ_{C} = \frac{1}{\sqrt{2}}ρ^{−1}(C), C = H̃

schon konstruiert. Die Wurzelbasis sei wie in Beispiel 1.6 bezeichnet.

\underline{E_{7}:} Setze v := 2e_{1} + 3e_{2} + 4e_{3} + 5e_{4} + 6e_{5} + 4e_{6} + 2e_{7} + 3e_{8}. Es gilt (wie man mit einer leichten und nicht sehr interessanten Rechnung zeigen kann) ⟨v, v⟩ = 2. Mit einer ebenso leichten und etwas kürzeren Rechnung erhält man auch, dass ⟨e_{i}, v⟩ = 0, ∀2 ≤ i ≤ 8 und ⟨e_{1}, v⟩ = 1 gilt. Als Beispiel berechne ich ⟨e_{8}, v⟩:

⟨e_{8}, v⟩ = ⟨e_{8}, 2e_{1} + 3e_{2} + 4e_{3} + 5e_{4} + 6e_{5} + 4e_{6} + 2e_{7} + 3e_{8}⟩
=2 ⟨e_{8}, e_{1}⟩ + 3 ⟨e_{8}, e_{2}⟩ + 4 ⟨e_{8}, e_{3}⟩ + 5 ⟨e_{8}, e_{4}⟩ + 6 ⟨e_{8}, e_{5}⟩
+ 4 ⟨e_{8}, e_{6}⟩ + 2 ⟨e_{8}, e_{7}⟩ + 3 ⟨e_{8}, e_{8}⟩
=6 ⟨e_{8}, e_{5}⟩ + 3 ⟨e_{8}, e_{8}⟩ = −6 + 6 = 0.

Setze Γ = {x ∈ E_{8} : ⟨x, v⟩ = 0}. Dies ist ein Gitter im Unterraum v^{⊥} ≅ ℝ^{7}, denn mit x_{1}, x_{2} ∈ Γ, α, β ∈ ℤ ist auch αx_{1} + βx_{2} ∈ Γ, weil

⟨αx_{1} + βx_{2}, v⟩ = α ⟨x_{1}, v⟩ + β ⟨x_{2}, v⟩ = 0 + 0 = 0.

Es sind e_{2}, …, e_{8} ∈ Γ, also ist auch das von ihnen erzeugte Untergitter von E_{8} in Γ. Es gibt auch keine weiteren Elemente in Γ, da z · e_{1} ∉ Γ, ∀z ∈ ℤ.

Die Elemente e_{2}, …, e_{8} bilden eine Basis des ℝ^{7} und sind wie im Gitter E_{8} im Raum orientiert. Sie bilden demnach eine Wurzelbasis des Gitters Γ im ℝ^{7}, das dann von der Form E_{7} sein muss. Der Vektor v liegt im Vektorraum Γ^{⊥} und erzeugt ein Gitter vom Typ

ℤv ≅ A_{1}.

Für die weiteren Betrachtungen wird jetzt der folgende Satz benötigt.

Satz 1.12. Sei Γ ⊆ Λ Untergitter, Λ unimodular (d.h. Λ = Λ^{∗}). Gilt dann Λ/Γ ≅ ℤ^{k}, so folgt disc Γ = disc Γ^{⊥}.

Beweis. Weil Λ/Γ ≅ ℤ^{k} ist existiert eine Basis (e_{1}, …, e_{k}) von Γ die zu einer Basis (e_{1}, …, e_{n}) von Λ ergänzt werden kann. Weil Λ nach Voraussetzung unimodular ist, ist die duale Basis (e_{1}^{∗}, …, e_{n}^{∗}) auch eine Basis von Λ. Es ist (e_{k+1}^{∗}, …, e_{n}^{∗}) eine Basis von Γ^{⊥}, denn ⟨e_{i}, e_{j}^{∗}⟩ = δ_{ij} und dies ist insbesondere dann gleich Null, wenn i ∈ {1, …, k} und j ∈ {k + 1, …, n} ist. Damit läßt sich Λ schreiben als Λ = Γ^{∗} ⊕ Γ^{⊥} = (Γ^{∗} ⊕ Γ^{⊥})^{∗} = Γ ⊕ (Γ^{⊥})^{∗} . Es ergibt sich folglich

(Γ^{⊥})^{∗} /Γ^{⊥} = Γ ⊕ (Γ^{⊥})^{∗} /Γ ⊕ Γ^{⊥} = Γ^{∗} ⊕ Γ^{⊥}/Γ ⊕ Γ^{⊥} = Γ^{∗}/Γ.

Und dies liefert schließlich

disc(Γ^{⊥}) = |(Γ^{⊥})^{∗} /Γ^{⊥}| = |Γ^{∗}/Γ| = disc(Γ).

Aus dem letzten Vortrag ist bekannt, dass E_{8} unimodular ist und wir haben gerade das Gitter Γ ⊂ E_{8} als zu ℤv ⊂ E_{8} orthogonales Gitter konstruiert, also gilt nach Satz 1.12 disc(E_{7}) = disc(ℤv) = det(⟨v, v⟩) = 2.

\underline{E_{6}:} E_{6} wird auf ganz analoge Weise konstruiert: Sei v_{1} := v und v_{2} := −e_{1}. Dann ist ⟨v_{1}, v_{2}⟩ = −1. v_{1} und v_{2} spannen demnach ein Gitter vom Typ A_{2} auf. Setze Γ = {x ∈ E_{8} : ⟨x, v_{1}⟩ = ⟨x, v_{2}⟩ = 0}. Mit einer Begründung wie oben sehen wir Γ wird von e_{3}, …, e_{8} erzeugt, weil die Winkel und Längen die selben geblieben sein müssen, bilden diese Vektoren gerade ein Wurzelgitter vom Typ E_{6}. Es gilt wieder (ℤv_{1} + ℤv_{2})^{⊥} = Γ ≅ E_{6} und nach dem Satz 1.12 muss damit folgen

disc(E_{6}) = disc(A_{2}) = det [\begin{matrix}⟨v_{1}, v_{1}⟩ & ⟨v_{1}, v_{2}⟩ \\ ⟨v_{2}, v_{1}⟩ & ⟨v_{2}, v_{2}⟩ \end{matrix}] = det [\begin{matrix}2 & −1 \\ −1 & 2 \end{matrix}] = 3.

2 Wurzelgitter und Codes

Das Wurzelgitter E_{8} haben wir aus dem erweiterten Hammingcode als

E_{8} = ΓH̃ = \frac{1}{\sqrt{2}}ρ^{−1}(H̃)

15. Gitter und Codes  2.1 Die Weylgruppe

erhalten. Es soll jetzt die Frage untersucht werden, welche weiteren binären Codes C ⊂ 𝔽_{2}^{n} existieren, für die Γ_{C} ein Wurzelgitter ist.

2.1 Die Weylgruppe

Notation. Mit σ_{α} : \begin{matrix}ℝ^{n} & → & ℝ^{n} \\ x & ↦ & x − 2\frac{⟨x,α⟩}{⟨α,α⟩}α \end{matrix}, sei die Spiegelung an der Hyperebene

α^{⊥} bezeichnet.

Definition 2.1. Sei Γ ein Wurzelgitter im ℝ^{n}. W (Γ) ist die von den Spiegelungen σ_{α}, α ∈ Φ erzeugte Untergruppe von GL(n, ℝ). Diese Gruppe heißt die Weylgruppe von Γ.

Lemma 2.2. Sei Γ ⊂ ℝ^{n} ein irreduzibles Wurzelgitter. Dann sind die einzigen W (Γ)-invarianten Unterräume U ⊆ ℝ^{n} gerade U = 0 und U = ℝ^{n}.

Beweis. Sei 0 ≠ U ⊆ ℝ^{n} W (Γ)-invarianter Unterraum. Dann ist das orthogonale Komplement U^{⊥} ⊆ ℝ^{n} ebenfalls W (Γ) invariant. Es ist ℝ^{n} = U ⊕ U^{⊥}. Sei α ∈ Φ, α ∉ U und u ∈ U beliebig gegeben. Es gilt σ_{α}(u) = u−2\frac{⟨u,α⟩}{⟨α,α⟩}α = u, weil U W (Γ)-invariant ist. Es folgt also ⟨u, α⟩ = 0, und wir erhalten damit α ∈ U^{⊥}. Den Fall α ∉ U^{⊥} betrachtet man analog und es ergibt sich damit, dass jede Wurzel α ∈ Φ entweder in U oder in U^{⊥} liegt. Weil Γ von Φ erzeugt wird, ergibt sich daher eine orthogonale Zerlegung des Gitters Γ = (U ∩Γ)⊕(U^{⊥}∩Γ). Nach Voraussetzung ist Γ jedoch irreduzibel und man erhält U^{⊥} ∩ Γ = {0} ⇒ U = ℝ^{n}.

Bemerkung 2.3. Insbesondere erzeugt der W (Γ)-Orbit einer Wurzel α den Raum ℝ^{n}.

Beweis. 0 ≠ span(W (Γ)(α)) ist ein W (Γ)-invarianter Unterraum von ℝ^{n}.

Lemma 2.4. Sei Γ ⊂ ℝ^{n} ein irreduzibles Wurzelgitter. Seien α, β ∈ Φ. Dann existiert ein w ∈ W (Γ) mit w(α) = β.

Beweis. Seien α, β ∈ Φ beliebig. Im Fall, dass α = β ist, ist die Aussage klar, sei also α ≠ β. Aus Lemma 2.2 folgt, dass wir o.B.d.A. ⟨α, β⟩ ≠ 0 annehmen können, denn nach Bemerkung 2.3 zu Lemma 2.2 können wir β schreiben als:

β = ∑_{w∈W (Γ)} k_{w}w(α), k_{w} ∈ ℝ.

Es muss demnach ein w ∈ W (Γ) existieren mit ⟨w(α), β⟩ ≠ 0. Bemerkung 1.2 liefert dann ⟨w(α), β⟩ = ±1. Indem wir dann β durch −β ersetzen können wir also annehmen ⟨α, β⟩ = +1.

15. Gitter und Codes  2.2 Gitter und Codes

Dann ist:

σ_{α}σ_{β}σ_{α}(β) = σ_{α}σ_{β}(β − 2\frac{⟨β, α⟩}{⟨α, α⟩}α)

= σ_{α}σ_{β}(β − α)

= σ_{α}(−β − α + 2\frac{⟨α, β⟩}{⟨β, β⟩}β)

= σ_{α}(−β − α + β)

= α.

2.2 Gitter und Codes

Satz 2.5. Sei Γ ⊂ ℝ^{n} ein irreduzibles Wurzelgitter. Dafür sind äquivalent:

1. Γ = Γ_{C} für einen binären Code C ⊂ 𝔽_{2}^{n}.

2. 2Γ^{∗} ⊂ Γ.

3. Γ ist vom Typ A_{1}, D_{n}, (n ≥ 4, n ≡ 0 ( mod 2)), E_{7} oder E_{8}.

Zum Beweis des Satzes benötigen wir folgendes Lemma:

Lemma 2.6. Sei C ⊂ 𝔽_{2}^{n} ein binärer Code. Dann gilt:

Γ_{C}^{∗} = Γ_{C^{⊥}}.

D.h.

Γ_{C}^{∗} = \{x ∈ ℝ^{n} : ⟨x, y⟩ ∈ ℤ, ∀y ∈ Γ_{C}\} = \frac{1}{\sqrt{2}}ρ^{−1}(\{x ∈ 𝔽_{2}^{n} : ⟨x, y⟩ = 0, ∀x ∈ C\}).

Beweis. \underline{1.) Zeige: Γ_{C^{⊥}} ⊆ Γ_{C}^{∗}.} Sei y ∈ Γ_{C^{⊥}}, x ∈ Γ_{C}. Dann ist

y = \frac{1}{\sqrt{2}}(c′ + 2z′), c′ ∈ C^{⊥}, z′ ∈ ℤ^{n} und x = \frac{1}{\sqrt{2}}(c + 2z), c ∈ C, z ∈ ℤ^{n}.

Achtung: Hierbei werden C und C^{⊥} als Teilmengen von ℤ^{n} aufgefaßt!! Man rechnet leicht nach:

\begin{matrix}⟨x, y⟩ = & \frac{1}{2} ⟨c + 2z, c′ + 2z′⟩ \\ = & \frac{1}{2}(\underbrace{⟨c, c′⟩}_{=0} + \underbrace{⟨2z, c′⟩}_{∈2ℤ}+ \underbrace{⟨c, 2z′⟩}_{∈2ℤ}+ \underbrace{4 ⟨z, z′⟩}_{∈4ℤ}) \\ = & ⟨z, c′⟩ + ⟨c, z′⟩ + 2 ⟨z, z′⟩ ∈ ℤ. \end{matrix}

Damit ist also Γ_{C^{⊥}} ⊆ Γ_{C}^{∗} = {y ∈ ℝ^{n} : ⟨y, x⟩ ∈ ℤ, ∀x ∈ Γ_{C}}.

2.) Zeige vol(ℝ^{n}/Γ_{C}^{∗}) = vol(ℝ^{n}/Γ_{C}^{⊥}). Sei k = dim C. Wir wissen aus dem Vortrag über Codes, dass n = dim C + dim C^{⊥} gilt, also dim C^{⊥} = n − k. Im Letzten Vortrag war gesagt worden

vol(ℝ^{n}/Γ_{C}) = vol(ℝ^{n}/\frac{1}{\sqrt{2}}ρ^{−1}(C)) = (\frac{1}{\sqrt{2}})^{n} vol(ℝ^{n}/ρ^{−1}(C)) = 2^{\frac{n}{2} −k}.

Und aus dem ersten Vortrag aus dieser Reihe ist bekannt:

vol(ℝ^{n}/Γ_{C}^{∗}) = \frac{1}{vol(ℝ^{n}/Γ_{C})} = 2^{k− \frac{n}{2}} .

Analog ergibt sich für den dualen Code (weil dieser die Dimension n − k hat):

vol(ℝ^{n}/Γ_{C^{⊥}}) = \frac{1}{2^{\frac{n}{2}}} · 2^{n−n+k} = 2^{k− \frac{n}{2}} = vol(ℝ^{n}/Γ_{C}^{∗}).

Jetzt folgt aus 1.) und 2.) die Behauptung des Lemmas, denn ein Untergitter eines Gitters, dass das selbe Volumen wie das Gitter hat, muss mit diesem übereinstimmen.

Beweis des Satzes 2.5. \underline{(1) ⇒ (2)} Sei Γ_{C} = Γ ein irreduzibles Wurzelgitter für einen binären Code C ⊂ 𝔽_{2}^{n}. Nach dem Lemma 2.6 ist dann Γ^{∗} = Γ_{C^{⊥}}. Sei x ∈ Γ_{C^{⊥}}, also x = \frac{1}{\sqrt{2}}(c + 2z), c ∈ C^{⊥}, z ∈ ℤ^{n}.

Dann folgt 2x = \frac{1}{\sqrt{2}}(\underbrace{2c}_{∈2ℤ^{n}}+ \underbrace{4z}_{∈2ℤ^{n}}) = \frac{1}{\sqrt{2}}(0 + 2z′), für z′ ∈ 2ℤ^{n}.

Es gilt 0 ∈ C ⇒ 2x ∈ Γ_{C} = \frac{1}{\sqrt{2}}ρ^{−1}(C). \underline{(2) ⇒ (3)} Aus dem Vortrag über Gitter ist bekannt, dass gilt |Γ^{∗}/Γ| = disc(Γ). Und nach Voraussetzung ist 2Γ^{∗} ⊆ Γ, also ist

Γ^{∗}/Γ ≅ (ℤ/2ℤ)^{l}, für ein 0 ≤ l ≤ n.

Jetzt betrachten wir die möglichen Wurzelgitter:

1.) Γ = A_{n} Γ ist dann von der Form

Γ = {(x_{1}, …, x_{n+1}) ∈ ℤ^{n+1} : ∑ x_{i} = 0}. (vgl. Seite 6)

Eine Basis dieses Wurzelgitters (keine Wurzelbasis!!) ist ε_{1} −ε_{i}, 2 ≤ i ≤ n+1. Und die hierzu duale Basis lautet

((\frac{1}{n + 1}, −\frac{n}{n + 1}, \frac{1}{n + 1}, …, \frac{1}{n + 1}), …, (\frac{1}{n + 1}, …, −\frac{n}{n + 1})).

Achtung!! Es wäre ja denkbar, dass auch für andere k als n+1 (\frac{1}{k}, −\frac{k−1}{k}, \frac{1}{k}, …, \frac{1}{k}) ein Element einer dualen Basis ist, jedoch liegen diese Elemente nicht in dem sel-

ben, von Γ erzeugten, Vektorraum, denn span Γ = (∑_{i=1}^{n+1} ε_{i})^{⊥} und

⟨(\frac{1}{k}, −\frac{k−1}{k}, \frac{1}{k}, …, \frac{1}{k}), ∑_{i=1}^{n+1} ε_{i}⟩ = \frac{n−k+1}{k} ≠ 0, ∀0 ≠ k ≠ n + 1. Es kann also kein Element einer dualen Basis sein.

Man kann die Elemente der dualen Basis aus einer Linearkombination von e_{0} := (−\frac{n}{n+1}, \frac{1}{n+1}, …, \frac{1}{n+1}) und den Elementen von Γ erhalten. Es gilt (n + 1) · e_{0} ∈ Γ, damit ist

Γ^{∗}/Γ ≅ ℤ/(n + 1)ℤ.

Also erfüllt A_{n} nur für den Fall n = 1 die Bedingung 2Γ^{∗} ⊆ Γ.

2.) Γ = D_{n}, n ≥ 3

Wie oben schon angemerkt wurde (siehe Seite 7) läßt sich Γ darstellen als

Γ = {x = (x_{1}, …, x_{n}) ∈ ℤ^{n} : ∑ x_{i} ≡ 0 ( mod 2)}.

Eine Basis ist:

(ε_{1} + ε_{2}, −ε_{1} + ε_{2}, −ε_{2} + ε_{3}, …, −ε_{n−1} + ε_{n}),

und die dazu duale Basis lautet:

((\frac{1}{2}, …, \frac{1}{2}), (−\frac{1}{2}, \frac{1}{2}, …, \frac{1}{2}), …, (−\frac{1}{2}, …, −\frac{1}{2}, \frac{1}{2})).

Wir können damit das duale Gitter Γ^{∗} auch schreiben als ℤ^{n} + ℤ(\frac{1}{2}, …, \frac{1}{2}). Jetzt muß eine Fallunterscheidung gemacht werden: 1.Fall: n ist gerade: Dann ist 2Γ^{∗} = 2ℤ^{n} + ℤ \underbrace{(1, …, 1)}_{Summe ist gerade} ⊆ Γ. Also erfüllt D_{n} für

n ≥ 4, n ≡ 0 ( mod 2) die Bedingung 2Γ^{∗} ⊆ Γ. 2.Fall: n ist ungerade: Dann ist 0+\underbrace{(1, …, 1)}_{n-mal} ∈ 2Γ^{∗}. Es ist jedoch ∑_{i=1}^{n} 1 ≡ 1 ( mod 2), also kein Element

von Γ. Damit kann D_{n} für n ungerade auch nicht die Bedingung 2Γ^{∗} ⊆ Γ erfüllen.

3.) Γ = E_{8}

Aus dem letzten Vortrag wissen wir Γ ist unimodular und damit Γ^{∗} = Γ ⇒ 2Γ^{∗} ⊆ Γ.

4.) Γ = E_{7}

Wir haben bereits gesehen (siehe Seite 7) disc(E_{7}) = disc(A_{1}) = 2 = |Γ^{∗}/Γ|. Und damit folgt 2Γ^{∗} ⊆ Γ.

5.) Γ = E_{6}

Wir haben gesehen disc(E_{6}) = disc(A_{2}) = 3 (siehe Seite 8), also kann Γ^{∗}/Γ gar nicht isomorph zu ℤ/2ℤ sein, und somit kann E_{6} auch nicht 2Γ^{∗} ⊆ Γ erfüllen. Insgesamt ist also gezeigt 2Γ^{∗} ⊆ Γ für ein Wurzelgitter Γ

⇒ Γ ∈ {A_{1}, D_{n} (n ≥ 4, n gerade), E_{7}, E_{8}}.

\underline{(3) ⇒ (1)} Im Folgenden wird für jeden Fall A_{1}, D_{n} (n ≥ 4, n gerade), E_{7} und E_{8} ein binärer Code angegeben mit Γ = Γ_{C}. \underline{A_{1}:} C = \{0\} ⊂ 𝔽_{2}. Das zugehörige Gitter ist \frac{1}{\sqrt{2}}2ℤ = \sqrt{2}ℤ.

15. Gitter und Codes  LITERATUR

\underline{D_{n} (n ≥ 4, n gerade):} C̃ = \{(u_{1}, …, u_{\frac{n}{2}} ) ∈ 𝔽_{2}^{n} : ∑ u_{i} = 0\} ⊂ 𝔽_{2}^{n}. Daraus konstruieren wir jetzt den Code

C = \{v ∈ 𝔽_{2}^{n} : v = (u_{1}, u_{1}, …, u_{\frac{n}{2}} , u_{\frac{n}{2}} ), (u_{1}, …, u_{\frac{n}{2}} ) ∈ C̃\}.

\underline{E_{8}:} C = H̃ ⊂ 𝔽_{2}^{8}, wurde schon im letzten Vortrag konstruiert. \underline{E_{7}:} C = H^{⊥} ⊂ 𝔽_{2}^{7}, H: Hammingcode. Dieser Code ist gerade C = {0} ∪ {c ∈ H : w(c) = 4}. Also

C = \begin{cases}(0, 0, 0, 0, 0, 0, 0), & (0, 0, 1, 1, 0, 1, 1), & (0, 1, 0, 1, 1, 0, 1), \\ (0, 1, 1, 0, 1, 1, 0), & (1, 0, 0, 0, 1, 1, 1), & (1, 0, 1, 1, 1, 0, 0), \\ (1, 1, 0, 1, 0, 1, 0), & (1, 1, 1, 0, 0, 0, 1) \end{cases}\} .

Literatur

[Eb94] Wolfgang Ebeling. Lattices and Codes. Vieweg, 1994.

[Hum72] James E. Humphreys. Introduction to Lie Algebras and Representation Theory. Springer, 1972.