Lattices

A lattice is a free \(\mathbb{Z}\)-module \(L\) of finite rank \(n\) with a symmetric bilinear form \(b \colon L \times L \to \mathbb{Q}\). An isometry \(\varphi \colon L \to M\) is an isomorphism of modules with \(b_M(\varphi x, \varphi y) = b_L(x, y)\) for all \(x, y \in L\). \(O(L)\) is the group of the isometries \(L \to L\).

Gram tensor

Each lattice of the catalogue comes with a chosen basis \(e_1, \dots, e_n\) of \(L\); the named lattices state theirs. The Gram tensor is the form \(b\), a symmetric \((0, 2)\)-tensor, and the catalogue stores its components \(b(e_i, e_j)\) in the chosen basis: row \(i\) lists \(b(e_i, e_1), \dots, b(e_i, e_n)\).

A vector of \(L\) is written by its coordinates in the chosen basis.

Orthogonal sums

\(L = M_1 \oplus M_2\) is an orthogonal sum when \(L\) is the direct sum of the submodules \(M_1\) and \(M_2\) and \(b(x, y) = 0\) for \(x \in M_1\), \(y \in M_2\). The basis of an orthogonal sum is the union of the bases of the summands, in the order written. \(L^k\) is the orthogonal sum of \(k\) copies of \(L\).

The page of a lattice states the orthogonal decomposition that the chosen basis gives. It is the finest partition of \(e_1, \dots, e_n\) into parts with \(b(e_i, e_j) = 0\) for \(e_i\), \(e_j\) in different parts, and \(L\) is the orthogonal sum of the sublattices that the parts generate.

Twists

For \(k \in \mathbb{Q}^\times\), the twist \(L(k)\) is the module \(L\) with the form \(k b\).

Determinant and signature

The determinant of \(L\) is \(\det(b(e_i, e_j))\).

The determinant is the same for every basis of \(L\).

A change of basis by \(P \in \mathrm{GL}_n(\mathbb{Z})\) multiplies it by \(\det(P)^2 = 1\).

Let \(v_1, \dots, v_n\) be a \(b\)-orthogonal basis of \(L \otimes_{\mathbb{Z}} \mathbb{Q}\). The signature of \(L\) is \((n_+, n_-)\), where \(n_+\) is the number of \(i\) with \(b(v_i, v_i) > 0\) and \(n_-\) the number of \(i\) with \(b(v_i, v_i) < 0\).

The signature does not depend on the orthogonal basis.

Definiteness

\(L\) is positive definite when \(b(x, x) > 0\) for every nonzero \(x \in L\), negative definite when \(b(x, x) < 0\) for every nonzero \(x \in L\), and indefinite when \(b(x, x) > 0\) and \(b(y, y) < 0\) for some \(x, y \in L\). It is positive semidefinite when \(b(x, x) \geq 0\) for all \(x\), \(b \neq 0\) and the determinant is zero, negative semidefinite when the same holds with \(\leq\), and zero when \(b = 0\).

Degenerate lattices

The correlation of \(L\) is the map \(L \to \operatorname{Hom}_{\mathbb{Z}}(L, \mathbb{Q})\), \(x \mapsto b(x, -)\). Its kernel is the radical of \(b\). \(L\) is degenerate when the radical is not zero.

The radical has rank \(n - n_+ - n_-\), and \(L\) is degenerate exactly when its determinant is \(0\).

Integral lattices

\(L\) is integral when \(b(x, y) \in \mathbb{Z}\) for all \(x, y \in L\), which is when every \(b(e_i, e_j)\) is an integer. An integral lattice is even when \(b(x, x)\) is even for every \(x \in L\), and odd otherwise. The parity of \(L\) is even or odd accordingly.