Definite lattices
Invariants of definite lattices, and the sign convention for negative definite forms.
Let \(L\) be a definite lattice.
Sign convention
For \(L\) negative definite, the invariants below are those of the twist \(L(-1)\).
Minimum and kissing number
The minimum of \(L\) is the least value of \(b(x, x)\) over the nonzero \(x \in L\). The kissing number is the number of \(x\) with \(b(x, x)\) equal to the minimum.
The kissing number is finite because the form is definite.
Theta series
The theta series of \(L\) is \(\sum_{x \in L} q^{b(x, x)}\). The coefficient of \(q^k\) is the number of \(x \in L\) with \(b(x, x) = k\).
The page states the theta series when \(L\) is integral.
Isometry group
\(O(L)\) is finite, and its order is even for \(L \neq 0\).
Because the form is definite, the set \(V_c = \{x \in L : |b(x, x)| \leq c\}\) is finite for each \(c\), and for \(c\) large it contains a basis of \(L\). \(O(L)\) permutes \(V_c\), and an isometry is determined by its values on that basis, so \(O(L)\) is finite. \(-1 \in O(L)\) has order 2 for \(L \neq 0\).
The order is computed with qfauto of PARI/GP.
Vectors of norm 2
\(\Phi_{\{2\}}(L) = \{r \in L : b(r, r) = 2\}\) is a set of roots. By Witt’s theorem, \(\mathbb{Z}\Phi_{\{2\}}(L)\) is an orthogonal sum of lattices \(A_n\), \(D_n\), \(E_6\), \(E_7\) and \(E_8\), and the database states its type for each integral lattice. The full set of roots \(\Phi(L)\) is a root system, with components and simple roots.