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On the symmetry of \(b\)-functions of linear free divisors. (English) Zbl 1202.14046
The authors introduce the concept of a prehomogeneous determinant as a generalization for linear free divisors. They show that the \(b-\)function roots are symmetric about \(-1\) for reductive prehomogeneous determinants and for the regular special free divisor. As a consequence, they get a positive answer to a question raised by Castro-Jiménez and Ucha-Enriquez, saying that \(-1\) is the only integer root of the corresponding \(b-\)function.
As an application, the authors obtain new cases when the logarithmic comparison theorem holds.

14M17 Homogeneous spaces and generalizations
20G05 Representation theory for linear algebraic groups
17B66 Lie algebras of vector fields and related (super) algebras
14F40 de Rham cohomology and algebraic geometry
Full Text: DOI arXiv
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