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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.

MSC:
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
Software:
Macaulay2
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References:
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