Inverting the wedge map and Gauss composition

TL;DR

This paper develops explicit algorithms to invert the wedge map in various dimensions, connecting Gauss's composition law for quadratic forms to geometric inversion of wedge maps and metrics on Grassmannians.

Contribution

It provides a new explicit inversion algorithm for the wedge map lpha_{n,2} and links Gauss's composition law to geometric inversion problems.

Findings

Explicit inversion algorithm for lpha_{n,2}

Gauss's composition law as a special case of wedge map inversion

A natural metric on the integral Grassmannian induced by positive definite matrices

Abstract

Let $1 \le k \le n,$ and let $v_1,\ldots,v_k$ be integral vectors in $\mathbb{Z}^n$. We consider the wedge map $\alpha_{n,k} : (\mathbb{Z}^n)^k /SL_k(\mathbb{Z}) \rightarrow \wedge^k(\mathbb{Z}^n)$, $(v_1,\ldots,v_k) \rightarrow v_1 \wedge \cdots \wedge v_k $. In his Disquisitiones, Gauss proved that $\alpha_{n,2}$ is injective when restricted to a primitive system of vectors when defining his composition law for binary quadratic forms. He also gave an algorithm for inverting $\alpha_{3,2}$ in a different context on the representation of integers by ternary quadratic forms. We give here an explicit algorithm for inverting $\alpha_{n,2}$, and observe via Bhargava's composition law for $\mathbb{Z}^2 \otimes \mathbb{Z}^2 \otimes \mathbb{Z}^2 $ cube that inverting $\alpha_{4,2}$ is the main algorithmic step in Gauss's composition law for binary quadratic forms. This places Gauss's composition as a special case of the geometric problem of inverting a wedge map which may be of independent interests. We also show that a given symmetric positive definite matrix $A$ induces a natural metric on the integral Grassmannian $G_{n,k}(\mathbb{Z})$ so that the map $X \rightarrow X^TAX$ becomes norm preserving.