The Poisson Type Operators on the Double Fock Space of Type B

TL;DR

This paper explores Poisson type operators within the double Fock space of type B, introducing double gauge operators and providing a simplified method for calculating moments and combinatorial structures.

Contribution

It defines double gauge operators in the double Fock space of type B and simplifies the calculation of moments and combinatorial structures compared to previous methods.

Findings

Computed multidimensional moments of Poisson operators

Established compatibility with Coxeter group type B inversions

Presented a simpler method for moment calculations

Abstract

The double Fock space of type B was introduced in 2023 by Bo\.zejko and Ejsmont (\cite{BE23}). In this article, we show the acting of Poisson type operators in that space. For this purpose, we define the double gauge operators (analogous to \cite{Ans01}, \cite{Ejsmont1}) and compute the multidimensional moments of a joint distribution of Poisson operators. We show that the presented method of calculating negative arcs and restricted crossings is compatible with counting positive and negative inversions on a Coxeter group of type B. The present method is much simpler than using colored type-B set partitions in the sense of \cite{Ejsmont1}.