Some Classical Invariants, from Harmonic Quadruples to Triangle Groups

TL;DR

This paper explores classical invariants through harmonic quadruples and triangle groups, highlighting analogies between binary quartics, ternary cubics, and modular forms, with applications in elliptic and hyperbolic geometry.

Contribution

It provides an expanded lecture-based overview connecting invariant theory, harmonic invariants, and triangle groups in both elliptic and hyperbolic contexts.

Findings

Analogy between binary quartics and ternary cubics based on invariants

Presentation of triangle groups with associated tilings

Discussion of polynomial powers and Pfaffians

Abstract

These notes are an expanded version of the lectures held in Tromso, in May 2025 at the "Lie-Stormer Summer School : Invariant Theory from classics to modern developments", in the framework of TiME events. We emphasize the analogy between binary quartics and ternary cubics (and subsequently modular forms) based on their harmonic and equianharmonic invariants. Triangle groups are presented in both the elliptic and the hyperbolic setting with their associated tilings. The topics include the discussion of a short Hilbert paper on polynomials which are powers, that was proposed to the participants. The appendix contains some exercises, with sketches of solutions, and a section devoted to Pfaffians edited by Vincenzo Galgano.