ORBITS ON TWISTED BHARGAVA BOXES

Abstract

In a series of papers by Manjul Bhargava, he studies generic group orbits on the spaces of integer cubes and boxes and describes a bijection between generic orbits and rings with balanced tuples of ideals defined up to equivalence. This problem has also been considered by Akihiko Yukie in the context of prehomogeneous vector spaces, using techniques from Galois cohomology to produce a classification of generic orbits. In a paper by Wee Teck Gan and Gordan Savin, a slight variant of the group action on the space of 2x2x2 cubes are considered, and generic orbits correspond to isomorphism classes of twisted composition algebras over some cubic algebra.

This paper performs similar analysis for the space of 2x3x3 boxes, producing a bijection between generic orbits and isomorphism classes of twisted composition algebras with a fixed quadratic invariant, and makes the bijection explicit at the level of boxes relating it to Bhargava's result on integer boxes. Thus the space of twisted composition algebras over the base field unifies solutions to two different orbit problems on prehomogeneous vector spaces.