Lecture notes for Introduction to Representation Theory

on Z N restricts to the inner product B given by on L, since it takes the same values on the
basis vectors:
(ϕ i , ϕ i ) = 2
−1 i, j adjacent
(ϕ i , ϕ j ) =
0 otherwise
This means that vectors of the form
and
(0, . . . , 0, 1, 0, . . . , 0, −1, 0, . . . , 0) = ϕ i + ϕ i+1 + · · · + ϕ j−1
(0, . . . , 0, −1, 0, . . . , 0, 1, 0, . . . , 0) = −(ϕ i + ϕ i+1 + · · · + ϕ j−1 )
are the roots of L. Therefore the number of positive roots in L equals
N(N − 1) .
2
2. As a fact we also state the number of positive roots in the other Dynkin diagrams:
D N N(N − 1)
E 6 36 roots
E 7 63 roots
120 roots
E 8
Definition 5.20. Let ϕ Z n be a positive root. The reflection s is defined by the formula
s (v) = v − B(v, ϕ)ϕ.
We denote s i by s i and call these simple reflections.
Remark 5.21. As a linear operator of R n , s fixes any vector orthogonal to ϕ and
s (ϕ) = −ϕ
Therefore s is the reflection at the hyperplane orthogonal to ϕ, and in particular fixes B. The
s i generate a subgroup W ∧ O(R n ), which is called the Weyl group of . Since for every w W ,
w(ϕ i ) is a root, and since there are only finitely many roots, W has to be finite.
5.5 Gabriel’s theorem
Definition 5.22. Let Q be a quiver with any labeling 1, . . . , n of the vertices. Let V = (V 1 , . . . , V n )
be a representation of Q. We then call
the dimension vector of this representation.
d(V ) = (dim V 1 , . . . , dim V n )
We are now able to formulate Gabriel’s theorem using roots.
Theorem 5.23 (Gabriel’s theorem). Let Q be a quiver of type A n , D n , E 6 , E 7 , E 8 . Then Q has
finitely many indecomposable representations. Namely, the dimension vector of any indecomposable
representation is a positive root (with respect to B ) and for any positive root ϕ there is exactly
one indecomposable representation with dimension vector ϕ.
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