Collaborative Research Centre 701
Spectral Structures and Topological Methods in Mathematics
Friday, April 21, 2017 - 16:00 in H10
Normal forms for quiver representations induced by tree modules
A talk in the 'Seminar Representation Theory of Algebras'
||With a fixed tree module of a quiver and a fixed tree-shaped basis of its group of self-extensions, one can associate an affine space of representations with the same dimension vector. If all representations in this cell are indecomposable and pairwise non-isomorphic, this induces a normal form for these representations. Even if this is general not true, it is possible to state conditions which are sufficient to obtain cells of pairwise non-isomorphic indecomposable representations. For instance this is the case if the tree module is a torus fixed point under a certain torus action. But also recursive constructions of indecomposable representations turn out to be very useful. Now it also might happen that the same representation lies in different cells. Nevertheless, for certain roots this machinery can be used to give a full description of the set of indecomposable representations. In any case, this gives a concept which can be used to deduce normal forms for indecomposable representations of quivers based on the notion of tree modules.
We also discuss connections to a rather vague conjecture of Kac which says that the set of isomorphism classes of indecomposable representations of quivers admits a cell decomposition into locally closed subvarieties which are affine cells.