A hierarchy of Palm measures for determinantal point processes with gamma kernels

The gamma kernels are a family of projection kernels $K^{(z,z')}=K^{(z,z')}(x,y)$ on a doubly infinite $1$-dimensional lattice. They are expressed through Euler's gamma function and depend on two continuous parameters $z,z'$. The gamma kernels initially arose from a model of random partitions via a limit transition. On the other hand, these kernels are closely related to unitarizable representations of the Lie algebra $\mathfrak{su}(1,1)$. Every gamma kernel $K^{(z,z')}$ serves as a correlation kernel for a determinantal measure $M^{(z,z')}$, which lives on the space of infinite point configurations on the lattice. We examine chains of kernels of the form $$\ldots, K^{(z-1,z'-1)}, \; K^{(z,z')},\; K^{(z+1,z'+1)}, \ldots,$$ and establish the following hierarchical relations inside any such chain: Given $(z,z')$, the kernel $K^{(z,z')}$ is a one-dimensional perturbation of (a twisting of) the kernel $K^{(z+1,z'+1)}$, and the one-point Palm distributions for the measure $M^{(z,z')}$ are absolutely continuous with respect to $M^{(z+1,z'+1)}$. We also explicitly compute the corresponding Radon-Nikod\'ym derivatives and show that they are given by certain normalized multiplicative functionals.Comment: Version 2: minor changes, typos fixe

Rationality problem of three-dimensional monomial group actions

Let $K$ be a field of characteristic not two and $K(x,y,z)$ the rational function field over $K$ with three variables $x,y,z$. Let $G$ be a finite group of acting on $K(x,y,z)$ by monomial $K$-automorphisms. We consider the rationality problem of the fixed field $K(x,y,z)^G$ under the action of $G$, namely whether $K(x,y,z)^G$ is rational (that is, purely transcendental) over $K$ or not. We may assume that $G$ is a subgroup of $\mathrm{GL}(3,\mathbb{Z}) and the problem is determined up to conjugacy in$\mathrm{GL}(3,\mathbb{Z})$. There are 73 conjugacy classes of$G$in$\mathrm{GL}(3,\mathbb{Z}). By results of Endo-Miyata, Voskresenski\u\i, Lenstra, Saltman, Hajja, Kang and Yamasaki, 8 conjugacy classes of 2-groups in \mathrm{GL}(3,\mathbb{Z})$have negative answers to the problem under certain monomial actions over some base field$K$, and the necessary and sufficient condition for the rationality of$K(x,y,z)^G$over$K$is given. In this paper, we show that the fixed field$K(x,y,z)^G$under monomial action of$G$is rational over$K$except for possibly negative 8 cases of 2-groups and unknown one case of the alternating group of degree four. Moreover we give explicit transcendental bases of the fixed fields over$K$. For unknown case, we obtain an affirmative solution to the problem under some conditions. In particular, we show that if$K$is quadratically closed field then$K(x,y,z)^G$is rational over$K$. We also give an application of the result to 4-dimensional linear Noether's problem.Comment: 54 page