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Irreducible components of varieties of representations II | Birge Huisgen-Zimmermann
; Ian Shipman
; | Date: |
27 Sep 2015 | Abstract: | This article is part of a program to evolve the generic representation theory
of basic finite dimensional algebras A over an algebraically closed field K, in
other words, the goal is to determine the irreducible components of the
varieties Rep(A,d) parametrizing the finite dimensional representations with
dimension vector d, and to generically describe the representations encoded by
the components. Here we primarily target truncated path algebras, i.e.,
algebras of the form A = KQ/I for a quiver Q, where I is generated by all paths
of some fixed length in the path algebra KQ. The main result characterizes the
irreducible components of the affine (resp. projective) parametrizing variety
Rep(A,d) (resp. GRASS_d(A)) in case Q is acyclic. Our classification is in
representation-theoretic terms, permitting to list the components from the
quiver and Loewy length of A. Combined with existing theory, it moreover yields
an array of generic features of the modules parametrized by the irreducible
components, such as generic minimal projective presentations, generic skeleta
("path bases" recruited from a finite set of eligible paths), generic
dimensions of endomorphism rings, generic socles, etc.
The information on truncated path algebras with acyclic quiver supplements
the comparatively well-developed theory available in the special case where A
is hereditary, i.e., for I = 0: On one hand, we add to the classical generic
results regarding the d-dimensional KQ-modules, they address only the modules
of maximal Loewy length. On the other hand, the more general theory for I
nonzero developed here fills in generic data on the d-dimensional KQ-modules of
any fixed Loewy length. | Source: | arXiv, 1509.8051 | Services: | Forum | Review | PDF | Favorites |
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