Inverting the wedge map and Gauss composition

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.