# The ideal of relations for the ring of invariants of $n$ points on the line

### Benjamin Howard

University of Michigan, Ann Arbor, USA### John J. Millson

University of Maryland, College Park, USA### Andrew Snowden

University of Michigan, Ann Arbor, USA### Ravi Vakil

Stanford University, United States

## Abstract

The ring of projective invariants of $n$ ordered points on the projective line is one of the most basic and earliest studied examples in Geometric Invariant Theory. It is a remarkable fact and the point of this paper that, unlike its close relative the ring of invariants of n unordered points, this ring can be completely and simply described. In 1894 Kempe found generators for this ring, thereby proving the First Main Theorem for it (in the terminology introduced by Weyl). In this paper we compute the relations among Kempe’s invariants, thereby proving the Second Main Theorem (again in the terminology of Weyl), and completing the description of the ring 115 years later. This paper introduces a number of new tools to the problem, and uses the graphical algebra formalism to intermediate between representation-theoretic arguments (for symmetric and Lie groups), and the symmetry-breaking of the Speyer–Sturmfels degeneration.

## Cite this article

Benjamin Howard, John J. Millson, Andrew Snowden, Ravi Vakil, The ideal of relations for the ring of invariants of $n$ points on the line. J. Eur. Math. Soc. 14 (2012), no. 1, pp. 1–60

DOI 10.4171/JEMS/295