On Discretization of Tori of Compact Simple Lie Groups

TL;DR

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

Contribution

It introduces methods to compute point sets, dominant weights, and conjugate points for Lie groups, facilitating efficient discrete transforms and interpolations.

Findings

Calculated the number of points in discretized fundamental domains.

Identified the maximal sets of orthogonal $C$- and $S$-functions.

Developed an algorithm for counting conjugate points on the maximal torus.

Abstract

Three types of numerical data are provided for simple Lie groups of any type and rank. This data is indispensable for Fourier-like expansions of multidimensional digital data into finite series of $C-$ or $S-$functions on the fundamental domain $F$ of the underlying Lie group $G$. Firstly, we consider the number $|F_M|$ of points in $F$ 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 $\Lambda_M$ of dominant weights, specifying the maximal set of $C-$ and $S-$functions that are pairwise orthogonal on the point set $F_M$. Finally, we describe an efficient algorithm for finding, on the maximal torus of $G$, the number of conjugate points to every point of $F_M$. Discrete $C-$ and $S-$transforms, together with their continuous interpolations, are presented in full generality.