On E-Discretization of Tori of Compact Simple Lie Groups

TL;DR

This paper provides essential numerical data and algorithms for discretizing tori of compact simple Lie groups, enabling Fourier-like expansions of multidimensional data using $E$-functions.

Contribution

It introduces methods to determine point counts, orthogonal weights, and conjugate points for $E$-discretization of Lie group tori, with a comprehensive discrete $E$-transform.

Findings

Number of points in $F^{e}_M$ determined from lattice $P^{ u}_M$

Minimal orthogonal set of $E$-function weights identified

Efficient algorithm for conjugate point enumeration developed

Abstract

Three types of numerical data are provided for compact simple Lie groups $G$ of classical types and of any rank. This data is indispensable for Fourier-like expansions of multidimensional digital data into finite series of $E-$functions on the fundamental domain $F^{e}$. Firstly, we determine the number $|F^{e}_M|$ of points in $F^{e}$ from the lattice $P^{\vee}_M$, which is the refinement of the dual weight lattice $P^{\vee}$ of $G$ by a positive integer $M$. Secondly, we find the lowest set $\Lambda^{e}_M$ of the weights, specifying the maximal set of $E-$functions that are pairwise orthogonal on the point set $F^{e}_M$. Finally, we describe an efficient algorithm for finding the number of conjugate points to every point of $F^{e}_M$. Discrete $E-$transform, together with its continuous interpolation, is presented in full generality.