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

• Volume: 014, Issue: 1, page 1-60
• ISSN: 1435-9855

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## Abstract

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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.

## How to cite

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Howard, Benjamin, et al. "The ideal of relations for the ring of invariants of $n$ points on the line." Journal of the European Mathematical Society 014.1 (2012): 1-60. <http://eudml.org/doc/277542>.

@article{Howard2012,
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.},
author = {Howard, Benjamin, Millson, John J., Snowden, Andrew, Vakil, Ravi},
journal = {Journal of the European Mathematical Society},
keywords = {invariant theory; geometric invariant theory; ideal of relations; configuration of points; GIT; projective embedding; invariant theory; ideal of relations; configuration of points; GIT; projective embedding},
language = {eng},
number = {1},
pages = {1-60},
publisher = {European Mathematical Society Publishing House},
title = {The ideal of relations for the ring of invariants of $n$ points on the line},
url = {http://eudml.org/doc/277542},
volume = {014},
year = {2012},
}

TY - JOUR
AU - Howard, Benjamin
AU - Millson, John J.
AU - Snowden, Andrew
AU - Vakil, Ravi
TI - The ideal of relations for the ring of invariants of $n$ points on the line
JO - Journal of the European Mathematical Society
PY - 2012
PB - European Mathematical Society Publishing House
VL - 014
IS - 1
SP - 1
EP - 60
AB - 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.
LA - eng
KW - invariant theory; geometric invariant theory; ideal of relations; configuration of points; GIT; projective embedding; invariant theory; ideal of relations; configuration of points; GIT; projective embedding
UR - http://eudml.org/doc/277542
ER -

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