Consider the preprojective algebra of type $A_n$. It is well known that this algebra is of finite representation type when $n<5$, of tame representation type when $n=5$, and of wild representation type when $n>5$. In particular, this means that there exists a family of indecomposable representations of the preprojective algebra of type $A_5$ which depends on a continuous parameter. I was wondering if anyone knew of an explicit example of such a family. Any examples of two-parameter families in type $A_n$ for $n>5$ would be also helpful.
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$\begingroup$ I decided to add a tag to bump this since I specifically would like to see a proof that the A_6 preprojective algebra is of wild representation type. $\endgroup$– Peter McNamaraCommented May 29, 2014 at 23:58
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$\begingroup$ And a two-parameter A6 example has been given at mathoverflow.net/questions/202259 by Jeremy Rickard. (I wonder what happened to the tag I apparenly added - I'll try again) $\endgroup$– Peter McNamaraCommented Apr 11, 2015 at 12:16
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1 Answer
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I only know the answer for type A5. The smallest dimension-vector for a one-parameter family of indecomposable representations is (1, 2, 2, 2, 1). All the representations in the family have socle and head both equal to S_2 \oplus S_4. (Here S_i is the one-dimensional representation attached to vertex i, with the standard labelling of the vertices.) With these informations, you will easily construct the desired representations. The paper Semicanonical bases and preprojective algebras by Geiss, Leclerc and Schröer (Ann. Sci. Ecole Norm. Sup. 38 (2005), 193-253) contains a precise description of the representations of the preprojective algebra of type A5.
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$\begingroup$ It's probably useful to add the location for a free download of the 2005 paper: numdam.org/numdam-bin/fitem?id=ASENS_2005_4_38_2_193_0. $\endgroup$ Commented Apr 11, 2015 at 17:58