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

Wiktor Ejsmont, Patrycja Hęćka-Jędraszczyk

Abstract

The double Fock space of type B was introduced in 2023 by Bożejko 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}.

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

Abstract

The double Fock space of type B was introduced in 2023 by Bożejko 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}.

Paper Structure

This paper contains 10 sections, 9 theorems, 103 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coxeter groups of type B
  4. Orthogonal polynomials
  5. The double Fock space of type B
  6. Gauge operators
  7. The Double Poisson operator of type B
  8. The orthogonal polynomials
  9. Partitions of type B
  10. Proof of the main theorem

Key Result

Proposition 3.2

We have the decomposition where and the adjoint ${R^{(n)}_{\alpha,q}}^*$ is taken with respect to $\langle\cdot,\cdot \rangle_{0,0}$. In BEH15, we obtain the following estimation of norm

Figures (10)

  • Figure 1: The Coxeter diagram for $B(n).$
  • Figure 2: The example of a block and corresponding arcs.
  • Figure 3: The example of $\pi \in \mathcal{P}^{B}(4)$ with a positive arc ($+$) - $((\bar{3},\bar{2} ), (2,3))$ and a negative arc ($-$) - $((\bar{4},1), (\bar{1},4))$.
  • Figure 4: The example of statistic of partition $\pi \in \mathcal{P}^{B}(10),$ i.e., $\text{\normalfont Rc}(\pi)=6$, $\text{\normalfont Na}(\pi)=3$, $\text{\normalfont Cs}(\pi)=2$.
  • Figure 5: The visualization of the action $D$ on $\mathcal{NC}_2^B(4)$.
  • ...and 5 more figures

Theorems & Definitions (39)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • proof : Proof of point (5)
  • Remark 3.1
  • Proposition 3.2
  • Remark 3.3
  • Theorem 3.4
  • Definition 3.5
  • Remark 3.6
  • ...and 29 more