The space of Dunkl monogenics associated with $\mathbb Z_2^3$ and the universal Bannai--Ito algebra

TL;DR

This paper studies Dunkl monogenics related to the group a7_2^3 and their structure as modules over the Bannai--Ito algebra, providing new insights into their dimensions and representations.

Contribution

The paper improves previous results by providing a more detailed analysis of Dunkl monogenics as modules over the Bannai--Ito algebra, including their dimensions and decomposition.

Findings

a7 M_n has dimension 2(n+1) for nonnegative multiplicity functions.

a7 M_n decomposes into two copies of an (n+1)-dimensional irreducible a7 BI-module.

Enhanced understanding of the symmetry properties of Dunkl monogenics under the Bannai--Ito algebra.

Abstract

Let $n\geq 0$ denote an integer. Let $\mathscr M_n$ denote the space of Dunkl monogenics of degree $n$ associated with the reflection group $\mathbb Z_2^3$. The universal Bannai--Ito algebra $\mathfrak{BI}$ is a unital associative algebra over $\mathbb C$ generated by $X,Y,Z$ and the relations assert that each of \begin{gather*} \{X,Y\}-Z, \qquad \{Y,Z\}-X, \qquad \{Z,X\}-Y \end{gather*} commutes with $X,Y,Z$. When the multiplicity function $k$ is real-valued the space $\mathscr M_n$ supports a $\mathfrak{BI}$-module in terms of the symmetries of the spherical Dirac--Dunkl operator. Under the assumption that $k$ is nonnegative, it was shown that $\dim \mathscr M_n=2(n+1)$ and $\mathscr M_n$ is isomorphic to a direct sum of two copies of an $(n+1)$-dimensional irreducible $\mathfrak{BI}$-module. In this paper, we improve the aforementioned result.