66,571 research outputs found
Quivers, Geometric Invariant Theory, and Moduli of Linear Dynamical Systems
We use geometric invariant theory and the language of quivers to study
compactifications of moduli spaces of linear dynamical systems. A general
approach to this problem is presented and applied to two well known cases: We
show how both Lomadze's and Helmke's compactification arises naturally as a
geometric invariant theory quotient. Both moduli spaces are proven to be smooth
projective manifolds. Furthermore, a description of Lomadze's compactification
as a Quot scheme is given, whereas Helmke's compactification is shown to be an
algebraic Grassmann bundle over a Quot scheme. This gives an algebro-geometric
description of both compactifications. As an application, we determine the
cohomology ring of Helmke's compactification and prove that the two
compactifications are not isomorphic when the number of outputs is positive.Comment: 24 pages, based on my Diplomarbeit completed in February 2005, to
appear in Linear Algebra and its Applications (LAA
Two geometric character formulas for reductive Lie groups
In this paper we prove two formulas for the characters of representations of
reductive groups. Both express the character of a representation in terms of
the same geometric data attached to it. When specialized to the case of a
compact Lie group, one of them reduces to Kirillov's character formula in the
compact case, and the other, to an application of the Atiyah-Bott fixed point
formula to the Borel-Weil realization of the representation
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