Complete Decompositions of Coxeter groups orbit products of $A_2, C_2, G_2$ and $H_2$ and some limits for them

TL;DR

This paper provides explicit formulas for decomposing products of orbits in Weyl groups of certain Lie groups and Coxeter groups, and discusses their limits as functions of two variables.

Contribution

It introduces a unified approach to decompose orbit products for groups $A_2, C_2, G_2$ and $H_2$, including formulas and limit analysis.

Findings

Explicit formulas for orbit product decompositions

Unified treatment across multiple groups

Analysis of limits for orbit functions

Abstract

Orbits of the Weyl reflection groups attached to the simple Lie groups $A_2, C_2, G_2$ and Coxeter group $H_2$ are considered. For each of the groups products of any two orbits are decomposed into the union of the orbits. Results are presented in a single formula for each of the groups. Orbits are considered as functions of two variables and limits for such functions are mentioned.