Saito duality between Burnside rings for invertible polynomials (Q2903282)

A polynomial of \(n\) variables and \(n\) monomials is determined by an \((n\times n)\)-matrix of exponents. The polynomial is called invertible if the matrix is invertible. Transposing the matrix leads to the Berglund-Hübsch dual. In this note the equivariant monodromy zeta-function for such polynomials is studied. For a finite symmetry group \(G\) this function is an element of the Burnside ring \(K_0(\mathrm{f.}G\mathrm{-sets})\), that is, the Grothendieck ring of finite \(G\)-sets. Dual polynomials have dual zeta-functions, where duality is to be understood in the sense of Saito duality. It can be regarded as a Fourier transformation on Burnside rings.NEWLINENEWLINEFinally there is an application of the present results to the \` geometric roots\'\ considered in a previous paper of the authors [Mosc. Math. J. 11, No. 3, 463--472 (2011; Zbl 1257.32028)]