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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DEFINITION.<br />
Given a -orbit <strong>of</strong> regular simples, the corresp<strong>on</strong>ding tube c<strong>on</strong>sists <strong>of</strong> the<br />
indecomposable regular modules whose regular compositi<strong>on</strong> factors bel<strong>on</strong>g to<br />
this orbit.<br />
PROPERTIES.<br />
(1) Every regular indecomposable bel<strong>on</strong>gs to a unique tube.<br />
(2) Every indecomposable in a tube has the same period p under .<br />
PROOF. If X is regular uniserial with regular top T and regular length r,<br />
i i<br />
then X is regular uniserial with regular top T and regular length r. If<br />
i i<br />
T T we must have X X.<br />
i<br />
(3) If the regular simples in a tube <strong>of</strong> period p are S = S, then the<br />
i<br />
modules in the tube can be displayed as below. The symbol obtained <strong>by</strong><br />
stacking various S ’s is the corresp<strong>on</strong>ding regular uniserial. We indicate<br />
i<br />
the inclusi<strong>on</strong> <strong>of</strong> the maximal proper regular submodule Y <strong>of</strong> X <strong>by</strong> Y ¡ ¡ X, and<br />
the map <strong>of</strong> X <strong>on</strong>to the quotient Z <strong>of</strong> X <strong>by</strong> its regular socle as X ¢ Z. The<br />
translati<strong>on</strong> acts as a shift to the left, and the two vertical dotted<br />
lines must be identified.<br />
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