TY - JOUR

T1 - Enhanced equivariant Saito duality

AU - Ebeling, Wolfgang

AU - Gusein-Zade, Sabir M.

N1 - Funding information: This work was partially supported by DFG (Mercator Fellowship, Eb 102/8-1). The second author was also partially supported by the grant RFBR-16-01-00409.

PY - 2018/10

Y1 - 2018/10

N2 - In a previous paper, the authors defined an equivariant version of the so-called Saito duality between the monodromy zeta functions as a sort of Fourier transform between the Burnside rings of an abelian group and of its group of characters. Here, a so-called enhanced Burnside ring B (G) of a finite group G is defined. An element of it is represented by a finite G-set with a G-equivariant transformation and with characters of the isotropy subgroups associated to all points. One gives an enhanced version of the equivariant Saito duality. For a complex analytic G-manifold with a G-equivariant transformation of it one has an enhanced equivariant Euler characteristic with values in a completion of B (G). It is proved that the (reduced) enhanced equivariant Euler characteristics of the Milnor fibers of Berglund-Hübsch dual invertible polynomials are enhanced dual to each other up to sign. As a byproduct, this implies the result about the orbifold zeta functions of Berglund-Hübsch-Henningson dual pairs obtained earlier.

AB - In a previous paper, the authors defined an equivariant version of the so-called Saito duality between the monodromy zeta functions as a sort of Fourier transform between the Burnside rings of an abelian group and of its group of characters. Here, a so-called enhanced Burnside ring B (G) of a finite group G is defined. An element of it is represented by a finite G-set with a G-equivariant transformation and with characters of the isotropy subgroups associated to all points. One gives an enhanced version of the equivariant Saito duality. For a complex analytic G-manifold with a G-equivariant transformation of it one has an enhanced equivariant Euler characteristic with values in a completion of B (G). It is proved that the (reduced) enhanced equivariant Euler characteristics of the Milnor fibers of Berglund-Hübsch dual invertible polynomials are enhanced dual to each other up to sign. As a byproduct, this implies the result about the orbifold zeta functions of Berglund-Hübsch-Henningson dual pairs obtained earlier.

KW - Burnside ring

KW - Group action

KW - invertible polynomial

KW - monodromy

KW - orbifold zeta function

KW - Saito duality

UR - http://www.scopus.com/inward/record.url?scp=85030315298&partnerID=8YFLogxK

UR - https://arxiv.org/abs/1506.05604

U2 - 10.48550/arXiv.1506.05604

DO - 10.48550/arXiv.1506.05604

M3 - Article

AN - SCOPUS:85030315298

VL - 17

JO - Journal of Algebra and its Applications

JF - Journal of Algebra and its Applications

SN - 0219-4988

IS - 10

M1 - 1850181

ER -