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Some Classical Invariants, from Harmonic Quadruples to Triangle Groups

Giorgio Ottaviani, Vincenzo Galgano

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.

Some Classical Invariants, from Harmonic Quadruples to Triangle Groups

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.
Paper Structure (12 sections, 39 theorems, 157 equations, 20 figures)

This paper contains 12 sections, 39 theorems, 157 equations, 20 figures.

Table of Contents

  1. Introduction: the harmonic quadruples and the cross-ratio
  2. Binary quartics
  3. Ternary cubics
  4. Finite polyhedral groups and the ADE classification
  5. Semi-invariants of binary tetrahedral group
  6. Semi-invariants of binary octahedral group
  7. Semi-invariants of binary icosahedral group
  8. Covariants for the variety of polynomials which are powers. A short paper by Hilbert.
  9. Hyperbolic Triangle Groups and Modular Forms
  10. Appendix: Pfaffians and Exercises, by V. Galgano
  11. The Pfaffian
  12. Binary and ternary forms

Key Result

Proposition 1.1

Let $A=(a_0, a_1)$, $B=(b_0, b_1)$, $C=(c_0, c_1)$$D=(d_0, d_1)$ be four distinct points. After permuting the $4$ points, the cross-ratio $\frac{(ac)(bd)}{(ad)(bc)}=\lambda$ assumes $6$ distinct values unless

Figures (20)

  • Figure 1: chord C-E-G from an harmonic $4$-tuple.
  • Figure 2: AD is the harmonic mean between AB and AC. Recall that the sound frequencies are reciprocal to the lengths.
  • Figure 3: Apollonius construction: D is the fourth harmonic after A, B, C. The construction does not depend on the conic. The figure is from the introduction of DCG.
  • Figure 4: The harmonic property of the complete quadrilateral. Given A, B, C collinear, choose E not collinear, choose F on CE. Then G, I and D are determined successively by the picture. The point D does not depend on the choices of E, F and it is the fourth harmonic after A, B, C.
  • Figure 5: The four vertices of the square make a harmonic $4$-tuple, where $J=0$. The four vertices of the other rhombus make a equianharmonic $4$-tuple, where $I=0$. Here $a=\frac{1+i}{\sqrt{3}-1}$, $b=\frac{1-i}{\sqrt{3}+1}$.
  • ...and 15 more figures

Theorems & Definitions (87)

  • Proposition 1.1
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Theorem 2.6
  • proof
  • Theorem 2.7
  • ...and 77 more