Lecture notes for Introduction to Representation Theory

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5.6 Reflection Functors
Definition 5.24. Let Q be any quiver. We call a vertex i Q a sink if all edges connected to i
point towards i.
i
•
We call a vertex i Q a source if all edges connected to i point away from i.
i
•
Definition 5.25. Let Q be any quiver and i Q be a sink (a source). Then we let Q i be the
quiver obtained from Q by reversing all arrows pointing into (pointing out of) i.
We are now able to define the reflection functors (also called Coxeter functors).
Definition 5.26. Let Q be a quiver, i Q be a sink. Let V be a representation of Q. Then we
define the reflection functor
+
F : RepQ ⊃ RepQ i
by the rule
i
F
i
+ (V ) k = V k if k ⇒= i
⎧
:
+
F (V ) i = ker V j ⊃ V i
⎝.
i
Also, all maps stay the same but those now pointing out of i; these are replaced by compositions
of the inclusion of ker into V j with the projections V j ⊃ V k .
Definition 5.27. Let Q be a quiver, i Q be a source. Let V be a representation of Q. Let ξ be
the canonical map
ξ : V i ⊃ V j .
Then we define the reflection functor
by the rule
j⊥i
i⊥j
F i
−
: RepQ ⊃ RepQ i
F − i
(V ) k = V k if k = i
⎧
F −
i
(V ) i = Coker (ξ) = ⎝/Imξ.
V j
i⊥j
Again, all maps stay the same but those now pointing into i; these are replaced by the compositions
of the inclusions V k ⊃ i⊥j V j with the natural map V j ⊃ V j /Imξ.
Proposition 5.28. Let Q be a quiver, V an indecomposable representation of Q.
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