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We study the moduli space of rank 2 instanton sheaves on $p3$ in terms of representations of a quiver consisting of 3 vertices and 4 arrows between two pairs of vertices. Aiming at an alternative compactification for the moduli space of instanton sheaves, we show that for each rank 2 instanton sheaf, there is a stability parameter $theta$ for which the corresponding quiver representation is $theta$-stable (in the sense of King), and that the space of stability parameters has a non trivial wall-and-chamber decomposition. Looking more closely at instantons of low charge, we prove that there are stability parameters with respect to which every representation corresponding to a rank 2 instanton sheaf of charge 2 is stable, and provide a complete description of the wall-and-chamber decomposition for representation corresponding to a rank 2 instanton sheaf of charge 1.
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We extend the scope of a former paper to vector bundle problems involving more than one vector bundle. As the main application, we obtain the solution of the well-known moduli problems of vector bundles associated with general quivers.
We study the irreducible components of the moduli space of instanton sheaves on $mathbb{P}^3$, that is rank 2 torsion free sheaves $E$ with $c_1(E)=c_3(E)=0$ satisfying $h^1(E(-2))=h^2(E(-2))=0$. In particular, we classify all instanton sheaves with $c_2(E)le4$, describing all the irreducible components of their moduli space. A key ingredient for our argument is the study of the moduli space ${mathcal T}(d)$ of stable sheaves on $mathbb{P}^3$ with Hilbert polynomial $P(t)=dcdot t$, which contains, as an open subset, the moduli space of rank 0 instanton sheaves of multiplicity $d$; we describe all the irreducible components of ${mathcal T}(d)$ for $dle4$.
We extend the results of [GVX] to the setting of a stable polar representation G|V (G connected, reductive over C), satisfying some mild additional hypotheses. Given a G-equivariant rank one local system L on the general fiber of the quotient map f : V --> V/G, we compute the Fourier transform of the corresponding nearby cycle sheaf P on the zero-fiber of f. Our main intended application is to the theory of character sheaves for graded semisimple Lie algebras over C.
This is a survey article for Handbook of Linear Algebra, 2nd ed., Chapman & Hall/CRC, 2014. An informal introduction to representations of quivers and finite dimensional algebras from a linear algebraists point of view is given. The notion of quiver representations is extended to representations of mixed graphs, which permits one to study systems of linear mappings and bilinear or sesquilinear forms. The problem of classifying such systems is reduced to the problem of classifying systems of linear mappings.
It is shown that, given a representation of a quiver over a finite field, one can check in polynomial time whether it is absolutely indecomposable.
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