Lecture notes for Introduction to Representation Theory
Lecture notes for Introduction to Representation Theory
Lecture notes for Introduction to Representation Theory
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Next, let φ be the edge connecting i with the next vertex <strong>to</strong>wards j and i be the vertex on the other<br />
end of φ. We then let 1 , 2 be the graphs obtained from by removing φ. Since is supposed<br />
<strong>to</strong> be a Dynkin diagram - and there<strong>for</strong>e has no cycles or loops - both 1 and 2 will be connected<br />
graphs, which are not connected <strong>to</strong> each other.<br />
• •<br />
i<br />
1<br />
• • • j<br />
• •<br />
•<br />
2<br />
Then we have i 1 , j 2 . We define<br />
With this choice we get<br />
α = k m ϕ m , ρ = k m ϕ m .<br />
m 1 m 2<br />
ϕ = α + ρ.<br />
Since k i > 0, k j < 0 we know that α ⇒= 0, ρ ⇒= 0 and there<strong>for</strong>e<br />
Furthermore,<br />
B(α, α) ⊂ 2, B(ρ, ρ) ⊂ 2.<br />
B(α, ρ) = −k i k i ⊗ ,<br />
since 1 , 2 are only connected at φ. But this has <strong>to</strong> be a nonnegative number, since k i > 0 and<br />
k i ⊗ ∗ 0. This yields<br />
B(ϕ, ϕ) = B(α + ρ, α + ρ) = B(α, α) +2 B(α, ρ) + B(ρ, ρ) ⊂ 4.<br />
<br />
∧2 ∧0 ∧2<br />
But this is a contradiction, since ϕ was assumed <strong>to</strong> be a root.<br />
Definition 5.17. We call a root ϕ = ⎨ k i ϕ i a positive root if all k i ⊂ 0. A root <strong>for</strong> which k i ∗ 0<br />
i<br />
<strong>for</strong> all i is called a negative root.<br />
Remark 5.18. Lemma 5.16 states that every root is either positive or negative.<br />
Example 5.19. 1. Let be of the type A N−1 . Then the lattice L = Z N−1 can be realized as<br />
a subgroup of the lattice Z N by letting L ∧ Z N be the subgroup of all vec<strong>to</strong>rs (x 1 , . . . , x N )<br />
such that <br />
xi = 0.<br />
The vec<strong>to</strong>rs<br />
i<br />
ϕ 1 = (1, −1, 0, . . . , 0)<br />
ϕ 2 = (0, 1, −1, 0, . . . , 0)<br />
.<br />
ϕ N−1 = (0, . . . , 0, 1, −1)<br />
naturally <strong>for</strong>m a basis of L. Furthermore, the standard inner product<br />
(x, y) = x i y i<br />
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