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Paradan’s wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator

Arzu Boysal;  Michèle Vergne — 2009

Annales de l’institut Fourier

Let P ( s ) be a family of rational polytopes parametrized by inequations. It is known that the volume of P ( s ) is a locally polynomial function of the parameters. Similarly, the number of integral points in P ( s ) is a locally quasi-polynomial function of the parameters. Paul-Émile Paradan proved a jump formula for this function, when crossing a wall. In this article, we give an algebraic proof of this formula. Furthermore, we give a residue formula for the jump, which enables us to compute it.

Multiple Bernoulli series, an Euler-MacLaurin formula, and Wall crossings

Arzu Boysal;  Michèle Vergne — 2012

Annales de l’institut Fourier

We study multiple Bernoulli series associated to a sequence of vectors generating a lattice in a vector space. The associated multiple Bernoulli series is a periodic and locally polynomial function, and we give an explicit formula (called wall crossing formula) comparing the polynomial densities in two adjacent domains of polynomiality separated by a hyperplane. We also present a formula in the spirit of Euler-MacLaurin formula. Finally, we give a decomposition formula for the Bernoulli series describing...

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