An algebraic interpretation of the intertwining operators associated with the discrete Fourier transform

TL;DR

This paper reveals that intertwining operators for the discrete Fourier transform form a cubic algebra related to well-known algebraic structures, providing an algebraic interpretation of these operators.

Contribution

It introduces a new algebraic framework for understanding intertwining operators of the discrete Fourier transform, connecting them to the Askey-Wilson and Heun algebras.

Findings

Intertwining operators form a cubic algebra $\\mathcal{C}_q$ at roots of unity.

The algebra $\\mathcal{C}_q$ relates to the Askey-Wilson algebra.

The algebraic structure provides new insights into the discrete Fourier transform operators.

Abstract

We show that intertwining operators for the discrete Fourier transform form a cubic algebra $\mathcal{C}_q$ with $q$ a root of unity. This algebra is intimately related to the two other well-known realizations of the cubic algebra: the Askey-Wilson algebra and the Askey-Wilson-Heun algebra.