Lecture notes for Introduction to Representation Theory

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