Lecture notes for Introduction to Representation Theory

Proposition 5.31. Let Q be a quiver and V a representation of Q.
1. Let i Q be a sink and let V be surjective at i. Then
d(F i + V ) = s i (d(V )).
2. Let i Q be a source and let V be injective at i. Then
d(F i − V ) = s i (d(V )).
Proof. We only prove the first statement, the second one follows similarly. Let i Q be a sink and
let
: V j ⊃ V i
be surjective. Let K = ker . Then
Therefore we get
j⊥i
dim K = dim V j − dim V i .
j⊥i
⎩
d(F + V ) − d(V ) = dim V j − 2 dim V i = −B (d(V ), ϕ i )
i
i
j⊥i
and
This implies
⎩
d(Fi + V ) − d(V ) j
= 0, j ⇒ = i.
d(F i + V ) − d(V ) = −B (d(V ), ϕ i ) ϕ i
⊆ d(F + V ) = d(V ) − B (d(V ), ϕ i ) ϕ i = s i (d(V )) .
i
5.7 Coxeter elements
Definition 5.32. Let Q be a quiver and let be the underlying graph. Fix any labeling 1, . . . , n
of the vertices of . Then the Coxeter element c of Q corresponding to this labeling is defined as
Lemma 5.33. Let
c = s 1 s 2 . . . s n .
α = k i ϕ i
with k i ⊂ 0 for all i but not all k i = 0. Then there is N N, such that
has at least one strictly negative coefficient.
i
N
α c
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