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Five Lectures on Projective Invariants

Giorgio Ottaviani

TL;DR

This work develops a representation-theoretic framework for projective invariants, focusing on invariant rings of forms and of configurations of points. It integrates classical tools (Veronese and split varieties, Cayley–Sylvester theory, tableau calculus) with modern methods (Schur–Weyl duality, Reynolds operator, Molien formula) to compute invariants and covariants, obtain Hilbert series, and describe fundamental generating sets and relations. The text connects algebraic geometry, invariant theory, and combinatorics through explicit constructions (symbolic methods, graphical algebras) and culminates in concrete descriptions for binary and ternary forms, as well as six points on lines and planes, including Cremona’s hexahedral equation and related theta-characteristic phenomena. The overall contribution is a cohesive, representation-theoretic roadmap for deriving generators, relations, and numerical invariants across classical invariant problems with concrete computational schemes.

Abstract

We introduce invariant rings for forms (homogeneous polynomials) and for d points on the projective space, from the point of view of representation theory. We discuss several examples, addressing some computational issues. We introduce the graphical algebra for the invariants of d points on the line. This is an expanded version of the notes for the School on Invariant Theory and Projective Geometry, Trento, September 17-22, 2012.

Five Lectures on Projective Invariants

TL;DR

This work develops a representation-theoretic framework for projective invariants, focusing on invariant rings of forms and of configurations of points. It integrates classical tools (Veronese and split varieties, Cayley–Sylvester theory, tableau calculus) with modern methods (Schur–Weyl duality, Reynolds operator, Molien formula) to compute invariants and covariants, obtain Hilbert series, and describe fundamental generating sets and relations. The text connects algebraic geometry, invariant theory, and combinatorics through explicit constructions (symbolic methods, graphical algebras) and culminates in concrete descriptions for binary and ternary forms, as well as six points on lines and planes, including Cremona’s hexahedral equation and related theta-characteristic phenomena. The overall contribution is a cohesive, representation-theoretic roadmap for deriving generators, relations, and numerical invariants across classical invariant problems with concrete computational schemes.

Abstract

We introduce invariant rings for forms (homogeneous polynomials) and for d points on the projective space, from the point of view of representation theory. We discuss several examples, addressing some computational issues. We introduce the graphical algebra for the invariants of d points on the line. This is an expanded version of the notes for the School on Invariant Theory and Projective Geometry, Trento, September 17-22, 2012.

Paper Structure

This paper contains 38 sections, 75 theorems, 236 equations, 2 tables.

Table of Contents

  1. Introduction and first examples
  2. What is invariant theory, and the more modest aim of these lectures
  3. The Veronese variety and its equations
  4. The split variety and its equations
  5. Facts from representation theory
  6. Basics about representations
  7. Young diagrams and symmetrizers
  8. Representations of finite groups and of $\Sigma_d$
  9. Representations of $GL(n+1)$ and $SL(n+1)$, Schur functors
  10. The Lie algebra $\mathfrak{sl}(n+1)$ and the weight structure of its representations
  11. Schur-Weyl duality
  12. Invariants of forms and representation theory.
  13. Invariance for the torus
  14. Counting monomials of given weight.
  15. Lie algebra action on forms
  16. ...and 23 more sections

Key Result

Theorem 1

Theorems & Definitions (154)

  • Theorem 1
  • proof
  • Proposition 1: Lasker Lemma
  • Theorem 2
  • proof
  • Theorem 3
  • Remark 1
  • Theorem 4: BrionBrfaulkes
  • Example 1
  • Remark 2
  • ...and 144 more