[Paper Review] Discrete and continuous exponential transforms of simple Lie groups of rank two

Iryna Kashuba, J. Patera|ArXiv.org|Feb 5, 2007

Advanced Algebra and Geometry10 references4 citations

TL;DR

This paper introduces discrete and continuous exponential transforms (E-transforms) for class functions on compact simple Lie groups of rank two—SU(3), O(5)/Sp(4), and G(2)—by generalizing one-variable exponentials to multivariable special functions called E-functions. The method uses symmetrization over the even Weyl group, enabling efficient expansions via orthogonal bases on fundamental domains with uniform discretization grids of arbitrary density and symmetry.

ABSTRACT

We develop and describe continuous and discrete transforms of class functions on compact simple Lie group $G$ as their expansions into series of uncommon special functions, called here $\E$-functions in recognition of the fact that the functions generalize common exponential functions. The rank of $G$ is the number of variables in the $\E$-functions. A uniform discretization of the decomposition problem is described on lattices of any density and symmetry admissible for the Lie group $G$.

Motivation & Objective

• To generalize one-dimensional exponential functions to multivariable special functions (E-functions) on compact Lie groups of rank two.
• To develop continuous and discrete transforms of class functions using E-functions as orthogonal bases on fundamental domains.
• To provide a uniform discretization scheme for these transforms on lattices of any density and symmetry compatible with the Lie group structure.
• To establish the efficiency of E-transforms by comparing their convergence properties to traditional Fourier expansions using larger symmetry groups.
• To explore the relationship between E-functions and other special functions (C- and S-functions) via Weyl group and even Weyl group symmetries.

Proposed method

• Define E-functions as symmetrized products of U(1) characters over the even subgroup of the Weyl group of a compact semisimple Lie group.
• Construct continuous transforms by expanding class functions into series of E-functions over the fundamental domain F of the group.
• Implement discrete transforms by sampling the fundamental domain on a finite grid FM, corresponding to a finite Abelian subgroup of G.
• Ensure orthogonality of E-functions under integration over F and summation over FM, leveraging the action of the affine Weyl group and even affine Weyl group.
• Use the weight lattice of the Lie group to match the geometric structure of data lattices, enabling efficient discretization of continuous transforms.
• Relate E-functions to C- and S-functions through Weyl group orbits and symmetrization, highlighting shared properties like orthogonality and symmetry under affine Weyl groups.

Experimental results

Research questions

• RQ1 How can one-dimensional exponential functions be generalized to multivariable functions on compact Lie groups of rank two?
• RQ2 What is the role of the even Weyl group in constructing orthogonal bases of E-functions for class functions on rank-two Lie groups?
• RQ3 How does the discretization of E-transforms on lattices of varying density and symmetry affect the accuracy and efficiency of function expansions?
• RQ4 In what ways do E-transforms outperform traditional Fourier expansions in terms of convergence and harmonic content due to larger symmetry groups?
• RQ5 How do E-functions relate to C- and S-functions in terms of group-theoretic construction and functional properties?

Key findings

• E-functions are defined as symmetrized products of U(1) characters over the even Weyl group, generalizing e^{imx} to higher-rank Lie groups.
• The continuous E-transform expands class functions on the fundamental domain F into orthogonal series of E-functions, with coefficients computed via integration over F.
• The discrete E-transform uses finite grids FM within F, where M selects a finite Abelian subgroup of G, enabling exact representation of digital data.
• E-functions are orthogonal under both integration over F and summation over FM, with orthogonality preserved due to the action of the even affine Weyl group.
• For rank-two groups SU(3), O(5), and G(2), the E-transforms are explicitly constructed and shown to be efficient due to the larger symmetry group (|W| > 1), reducing the number of required harmonics.
• The method allows matching the lattice structure of data to the weight lattice of the Lie group, enabling uniform discretization with arbitrary density and symmetry.