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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These two corollaries show that there are only finitely many indecomposable representations<br />
(since there are only finitely many roots) and that the dimension vec<strong>to</strong>r of each of them is a positive<br />
root. The last statement of Gabriel’s theorem follows from<br />
Corollary 5.37. For every positive root ϕ, there is an indecomposable representation V with<br />
Proof. Consider the sequence<br />
d(V ) = ϕ.<br />
s n ϕ, s n−1 s n ϕ, . . .<br />
Consider the first element of this sequence which is a negative root (this has <strong>to</strong> happen by Lemma<br />
5.33) and look at one step be<strong>for</strong>e that, calling this element α. So α is a positive root and s i α is a<br />
negative root <strong>for</strong> some i. But since the s i only change one coordinate, we get<br />
and<br />
α = ϕ i<br />
(s q . . . s n−1 s n )ϕ = ϕ i .<br />
We let C (i) be the representation having dimension vec<strong>to</strong>r ϕ i . Then we define<br />
This is an indecomposable representation and<br />
V = F n − F n<br />
−<br />
−1 . . . F q − C (i) .<br />
d(V ) = ϕ.<br />
Example 5.38. Let us demonstrate by example how reflection func<strong>to</strong>rs work. Consider the quiver<br />
D 4 with the orientation of all arrows <strong>to</strong>wards the node (which is labeled by 4). Start with the<br />
1-dimensional representation V 4 sitting at the 4-th vertex. Apply <strong>to</strong> V 4 the func<strong>to</strong>r F 3 − F 2 − F 1 − .<br />
This yields<br />
F 1 − F 2 − F 3 − V 4 = V 1 + 2 + 3 + 4<br />
.<br />
Now applying F 4 − we get<br />
F 4 − F 1 − F 2 − F 3 − V 4 = V 1 + 2 + 3 +2 4<br />
.<br />
Note that this is exactly the inclusion of 3 lines in<strong>to</strong> the plane, which is the most complicated<br />
indecomposable representation of the D 4 quiver.<br />
5.9 Problems<br />
Problem 5.39. Let Q n be the cyclic quiver of length n, i.e., n vertices connected by n oriented edges<br />
<strong>for</strong>ming a cycle. Obviously, the classification of indecomposable representations of Q 1 is given by<br />
the Jordan normal <strong>for</strong>m theorem. Obtain a similar classification of indecomposable representations<br />
of Q 2 . In other words, classify pairs of linear opera<strong>to</strong>rs A : V ⊃ W and B : W ⊃ V up <strong>to</strong><br />
isomorphism. Namely:<br />
(a) Consider the following pairs (<strong>for</strong> n ⊂ 1):<br />
1) E n, : V = W = C n , A is the Jordan block of size n with eigenvalue ∂, B = 1 (∂ C).<br />
2) E n,≤ : is obtained from E n,0 by exchanging V with W and A with B.<br />
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