Lectures on Representations of Quivers by William Crawley-Boevey ...
Lectures on Representations of Quivers by William Crawley-Boevey ...
Lectures on Representations of Quivers by William Crawley-Boevey ...
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REMARKS.<br />
~<br />
(1) For the Kr<strong>on</strong>ecker quiver A<br />
1<br />
• •,<br />
¡<br />
¡<br />
the regular simples all have period <strong>on</strong>e. They are<br />
1<br />
¡ ¡<br />
¡ k k : .<br />
¡<br />
~<br />
(2) For the 4-subspace quiver, D with the following orientati<strong>on</strong><br />
4<br />
¡ ¡ •¥ ¡¦¥¦ ¡<br />
¡<br />
¢ ¢ ¢<br />
¢<br />
• • • •<br />
the real regular simples have period 2, and have dimensi<strong>on</strong> vectors<br />
1 1 1 1 1 1<br />
1100 0011 1001 0110 1010 0101<br />
The regular simples <strong>of</strong> dimensi<strong>on</strong> vector § are<br />
¡ ¡ k¥ ¡¦¥¦ 2<br />
¡<br />
k k k k<br />
¢ ¢ ¢ ¢<br />
¨<br />
<br />
1©<br />
¨<br />
with maps , , , where<br />
0<br />
<br />
0©<br />
1<br />
¨<br />
<br />
1©<br />
1<br />
¨<br />
<br />
1©<br />
¡<br />
<br />
¡<br />
¡ k,<br />
¡<br />
£ 0,1.<br />
(3) One can find lists <strong>of</strong> regular simples in the tables in the back <strong>of</strong><br />
V.Dlab & C.M.Ringel, Indecomposable representati<strong>on</strong>s <strong>of</strong> graphs and algebras,<br />
Mem. Amer. Math. Soc., 173 (1976). For the different graphs the tubes<br />
with period £ 1 have period as follows<br />
~¡<br />
~¡<br />
~¢<br />
~¢<br />
~¢<br />
m<br />
m<br />
6<br />
7<br />
8<br />
p,q if p>0 arrows go clockwise and q>0 go anticlockwise.<br />
m-2,2,2<br />
3,3,2<br />
4,3,2<br />
5,3,2<br />
One always has ¤ (period-1) = n-2, which can be proved with a little<br />
tubes<br />
more analysis.<br />
35
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