Related papers: Saito duality between Burnside rings for invertibl…
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…
A generalization of Arnold's strange duality to invertible polynomials in three variables by the first author and A.Takahashi includes the following relation. For some invertible polynomials $f$ the Saito dual of the reduced monodromy zeta…
A. Takahashi suggested a conjectural method to find mirror symmetric pairs consisting of invertible polynomials and symmetry groups generated by some diagonal symmetries and some permutations of variables. Here we generalize the Saito…
An invertible polynomial in $n$ variables is a quasihomogeneous polynomial consisting of $n$ monomials so that the weights of the variables and the quasi-degree are well defined. In the framework of the construction of mirror symmetric…
In earlier work, the authors described a relation between the Poincar\'e series and the classical monodromy zeta function corresponding to a quasihomogeneous polynomial. Here we formulate an equivariant version of this relation in terms of…
We give a survey on results related to the Berglund-H\"ubsch duality of invertible polynomials and the homological mirror symmetry conjecture for singularities.
An invertible polynomial is a quasihomogeneous polynomial with the number of monomials coinciding with the number of variables and such that the weights of the variables and the quasi-degree are well defined. In the framework of the search…
We show that the Bernstein-Sato polynomial (that is, the b-function) of a hyperplane arrangement with a reduced equation is calculable by combining a generalization of Malgrange's formula with the theory of Aomoto complexes due to Esnault,…
We will show that the duality for regular weight systems introduced by K. Saito can be interpreted as the duality for orbifoldized Poincare polynomials.
We study Saito duality and Fourier-Ramanujan transform for power sums and multiplicities of monodromy roots.
We introduce the zeta function of the prehomogenous vector space of binary cubic forms, twisted by the real analytic Eisenstein series. We prove the meromorphic continuation of this zeta function and identify its poles and their residues.…
An attempt is made to formulate Gaiotto's S-duality relations in an explicit quantitative form. Formally the problem is that of evaluation of the Racah coefficients for the Virasoro algebra, and we approach it with the help of the matrix…
We state a conjecture about the zeta function of crepant resolutions of Berglund--H\"ubsch orbifold hypersurfaces over a finite field. In addition to numerical evidence, we show that our conjectural zeta function satisfies the Weil…
Conjectures of J. Igusa for p-adic local zeta functions and of J. Denef and F. Loeser for topological local zeta functions assert that (the real part of) the poles of these local zeta functions are roots of the Bernstein-Sato polynomials…
We consider the Berglund-H\"ubsch-Henningson-Takahashi duality of Landau-Ginzburg orbifolds with a symmetry group generated by some diagonal symmetries and some permutations of variables. We study the orbifold Euler characteristics, the…
In this paper we prove that the Bernstein-Sato polynomial of any free divisor for which the $D[s]$-module $D[s] h^s$ admits a Spencer logarithmic resolution satisfies the symmetry property $b(-s-2) = \pm b(s)$. This applies in particular to…
This article considers some q-analogues of classical results concerning the Ehrhart polynomials of Gorenstein polytopes, namely properties of their q-Ehrhart polynomial with respect to a good linear form. Another theme is a specific linear…
Berglund-H\"ubsch duality is an example of mirror symmetry between orbifold Landau-Ginzburg models. In this paper we study a D-module-theoretic variant of Borisov's proof of Berglund-H\"ubsch duality. In the $p$-adic case, the D-module…
We examine the relationship between the (double) Schubert polynomials of Billey-Haiman and Ikeda-Mihalcea-Naruse and the (double) theta and eta polynomials of Buch-Kresch-Tamvakis and Wilson from the perspective of Weyl group invariants. We…
We generalize the Bernstein-Sato polynomials of Budur, Mustata and Saito to ideals in normal semigroup rings. In the case of monomial ideals, we also relate the roots of the Bernstein-Sato polynomial to the jumping coefficients of the…