Gauss's Composition Law in Quadratic Forms

• Gauss's composition law is a classical structure on binary quadratic forms that reinterprets composition via the inversion of the wedge map.
• It uses Plücker coordinates and gcd-based methods to explicitly recover lattice bases from primitive pairs.
• Modern generalizations, including Bhargava’s cubes, unify geometric, arithmetic, and algorithmic approaches in this framework.

Gauss's composition law is a classical structure on the set of integral binary quadratic forms of fixed discriminant, most fundamentally understood as a special case of inverting the wedge map in the theory of exterior powers and integral lattices. In modern language, this perspective places Gauss’s construction in the context of the geometry of numbers, the arithmetic of Grassmannians, and the representation theory of forms and tensors, enabling explicit algorithmic procedures and connections to recent advances such as Bhargava’s cube constructions (Chua, 27 Jun 2024).

1. The Wedge Map and Exterior Powers

Given a free abelian group $\mathbb{Z}^n$, the $k$-th exterior power $\wedge^k \mathbb{Z}^n$ is the abelian group generated by $k$-tuples of elements modulo antisymmetry; its standard basis is indexed by increasing $k$-tuples $$I = \{i_1 < \cdots < i_k\} \subset \{1, \ldots, n\}$$. The wedge map

$$\alpha_{n,k}: (\mathbb{Z}^n)^k / SL_k(\mathbb{Z}) \longrightarrow \wedge^k \mathbb{Z}^n,$$

sends $k$-tuples $(v_1, \ldots, v_k)$ (viewed as columns of an $n \times k$ matrix $X$) to their wedge product $v_1 \wedge \cdots \wedge v_k$. The image in terms of coordinates is given by the Plücker coordinates: each $I$ corresponds to the determinant of the $k \times k$ minor of $X$ consisting of rows $i_1, \ldots, i_k$. The wedge map factors through orbits under post-multiplication by $SL_k(\mathbb{Z})$, as this corresponds to changing basis within the same $k$-plane and does not alter the element in $\wedge^k \mathbb{Z}^n$.

2. Primitivity, Injectivity, and the Main Theorem

A $k$-tuple in $(\mathbb{Z}^n)^k$ is called primitive if the greatest common divisor of its Plücker coordinates is $1$. This criterion is equivalent to the tuple being extendable to an $SL_n(\mathbb{Z})$ matrix. Gauss proved that when restricted to primitive systems, the wedge map $\alpha_{n,2}$ is injective: if $(a, b)$ and $(c, d)$ are primitive pairs in $\mathbb{Z}^n$ with $a \wedge b = c \wedge d$, then there exists $H \in SL_2(\mathbb{Z})$ such that $(c, d) = (a, b) \cdot H$. Modern proofs of this result, including those by inverting explicit $2 \times 2$ matrices, reinforce its foundational nature in the theory of quadratic forms and lattice geometry (Chua, 27 Jun 2024).

3. Inversion of the Wedge Map: Algorithmic Framework

The inversion of the wedge map $\alpha_{n,2}$ is central in modern presentations of Gauss composition. For a primitive bivector $X = (X_{ij})_{1 \leq i < j \leq n}$ in the image of $\alpha_{n,2}$, the Plücker relations

$$X_{ij} X_{k\ell} - X_{ik} X_{j\ell} + X_{i\ell} X_{jk} = 0$$

for all distinct $i < j < k < \ell$ precisely characterize the image and ensure compatibility for inversion. The explicit algorithm for recovering $(x, y) \in \mathbb{Z}^n \times \mathbb{Z}^n$ with $x \wedge y = X$, as presented in [(Chua, 27 Jun 2024), Theorem 3.2], proceeds via:

• Fixing $X_{12} \neq 0$ and normalizing via $SL_2(\mathbb{Z})$;
• Computing $x_1 := \gcd(X_{12}, X_{13}, \ldots, X_{1n})$ and obtaining coefficients $A_j$ for the corresponding linear combination;
• Setting $y_1 = 0$, assigning $y_j$ via $A_j$ ($j=2,\ldots,n$);
• Solving the congruence for $x_2 \bmod y_2$ to fit the Plücker relation, followed by linear formulas to recover $x_j$ for $j \geq 3$.

This approach allows inversion by one gcd computation plus linear operations.

4. Gauss's Law in the Context of Binary Quadratic Forms

In the special case $n=4,\ k=2$, Gauss's composition of binary quadratic forms can be recast entirely as inverting $\alpha_{4,2}$. The space $\wedge^2 \mathbb{Z}^4$ is $6$-dimensional, governed by a single Pfaffian (Plücker) relation:

$X_{12} X_{34} - X_{13} X_{24} + X_{14} X_{23} = 0.$

Given two primitive binary quadratic forms of the same discriminant, their composition involves constructing bivectors $X, X'$ in $\mathrm{Im} \, \alpha_{4,2}$ and inverting $\alpha_{4,2}$ on the coordinatewise product $X \star X'$. This efficiently encodes Gauss's original composition law as a lattice-theoretic and geometric operation within the Plücker embedding framework (Chua, 27 Jun 2024).

5. Bhargava’s Cubes and Generalizations of Composition

Bhargava’s reinterpretation of classical composition employs $2 \times 2 \times 2$ integral cubes, representing elements in $\mathbb{Z}^{2\otimes 2\otimes 2}$. Each cube corresponds to three binary quadratic forms $Q_i(x,y) = -\det(M_ix - N_iy)$, with discriminants matched by construction and the faces identified as vectors $x, y$. The six-dimensional vector $x\wedge y \in \wedge^2 \mathbb{Z}^4$ satisfies the required Pfaffian relation. Given two forms $Q_2, Q_3$ of the same discriminant, the corresponding bivector $X$ is constructed and the inversion algorithm provides the pair $(x, y)$, defining the front/back faces of a unique Bhargava cube. The original forms $Q_2, Q_3$ and the composed form $Q_1$ are attached to the cube, and the group-theoretic relation $[Q_1] = -([Q_2] + [Q_3])$ (or, renormalizing, $[Q_1] = [Q_2] + [Q_3]$) realizes composition through a single inversion of $\alpha_{4,2}$. This unifies classical and modern approaches in explicit, algorithmic terms (Chua, 27 Jun 2024).

6. The Metric Structure on Integral Grassmannians

For any symmetric positive definite integer $n \times n$ matrix $A$, the pairing

$\langle X, Y \rangle_A := \det(X^T A Y)$

naturally induces a metric on $k$-planes in $\mathbb{Z}^n$. When $k=2$, each $X$ gives a $k$-ary form $\Phi_X(u) := (Xu)^T A (Xu)$ with determinant $\det(X^T A X)$. By properties of the Cauchy-Binet formula, the map $X \mapsto X^T A X$ is norm-preserving: the $A$-norm of $X$, defined as $\|X\|_A^2 := \det(X^T A X)$, is mapped to forms with the same determinant. This provides a volume-form-preserving structure on the integral Grassmannian $G_{n,k}(\mathbb{Z}) = \mathrm{Im} \, \alpha_{n,k}$, aligning algebraic, geometric, and arithmetic perspectives (Chua, 27 Jun 2024).

7. Summary and Conceptual Significance

Gauss’s composition law for binary quadratic forms finds a unified and computationally transparent description via the inversion of the wedge map in the case $(n,k) = (4,2)$. The Plücker embedding encodes algebraic relations, the gcd-based inversion algorithm provides explicit recovery of basis elements, and the framework naturally extends to geometric and group-theoretic settings through Bhargava’s cubes and integral Grassmannians. This modern interpretation situates Gauss’s law as a particularly symmetric and efficient manifestation of deeper arithmetic geometry principles, highlighting its ongoing relevance within current research (Chua, 27 Jun 2024).