Inverting the wedge map and Gauss composition

Abstract

Let $1\le k\le n,$ and let $v_1,\dots ,v_k$ be integral vectors in ${\mathbb{Z}}^n$. We consider the wedge map ${\alpha }_{n,k}:({\mathbb{Z}}^n{)}^k/S{L}_k(\mathbb{Z})\to {\wedge }^k({\mathbb{Z}}^n)$, $(v_1,\dots ,v_k)\to v_1\wedge \cdots \wedge v_k$. 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 $X\to X^{T}AX$ becomes norm preserving.

10 pages