Orbits of primitive vectors · Lattice Database

Let \(L\) be an integral lattice that is nondegenerate, with discriminant group \(A_L\).

Eight subgroups of the orthogonal group

\(SO(L)\) is the kernel of \(\det\colon O(L) \to \{\pm 1\}\). The stable orthogonal group \(\widetilde{O}(L)\) is the kernel of \(O(L) \to O(A_L)\) (GHS 2013, (6)). \(O^+(L)\) is the kernel of the real spinor norm \(\operatorname{sn}_\mathbb{R}\colon O(L \otimes \mathbb{R}) \to \mathbb{R}^*/(\mathbb{R}^*)^2 = \{\pm 1\}\), restricted to \(O(L)\). For a subgroup \(\Gamma\) of \(O(L)\), \(\Gamma^+ = \Gamma \cap O^+(L)\), \(S\Gamma = \Gamma \cap SO(L)\) and \(\widetilde{\Gamma} = \Gamma \cap \widetilde{O}(L)\); so \(\widetilde{O}^+(L) = \widetilde{O}(L) \cap O^+(L)\) (GHS 2013, (7)).

A record names the eight groups \(O\), \(SO\), \(O^+\), \(SO^+\), \(\widetilde{O}\), \(S\widetilde{O}\), \(\widetilde{O}^+\) and \(S\widetilde{O}^+\) with the keys O, SO, O+, SO+, Otilde, SOtilde, Otilde+ and SOtilde+.

Cokernel of the discriminant action

Let \(L\) be a nondegenerate even lattice, and let \(\rho_L\colon O(L)\to O(A_L,q_L)\) be its discriminant action. As a pointed set, define \(MM(L):=O(A_L,q_L)/\operatorname{im}\rho_L\), the set of left cosets pointed by the image subgroup. The sequence \[1\longrightarrow\widetilde O(L)\longrightarrow O(L)\xrightarrow{\rho_L}O(A_L,q_L)\longrightarrow MM(L)\longrightarrow 1\] is exact as a sequence of pointed sets: the preimage of the distinguished coset is \(\operatorname{im}\rho_L\). A finite computation can specify \(MM(L)\) by coset representatives. MM-triviality means that this set has one element, equivalently that \(\rho_L\) is surjective.

When \(\operatorname{im}\rho_L\) is normal, the coset set is a quotient group. It is a \(\mathbb Z\)-module only when this group is abelian. For an even indefinite \(L\) with \(\operatorname{rank}L>3\), the Miranda–Morrison exact sequence, stated in Akyol–Degtyarev, Theorem 3.8, is \[O(L)\xrightarrow{\rho_L}O(A_L,q_L)\longrightarrow E(L)\longrightarrow\mathfrak g(L)\longrightarrow 1,\] where \(E(L)\) is the Miranda–Morrison group and \(\mathfrak g(L)\) is the genus group. Under these hypotheses \(MM(L)\cong\ker(E(L)\to\mathfrak g(L))\) is an \(\mathbb F_2\)-vector space. Its isomorphism class as a torsion \(\mathbb Z\)-module is determined by \(d=\dim_{\mathbb F_2}MM(L)\): it is \((\mathbb Z/2\mathbb Z)^d\). Theorem 3.12 gives a local formula. The full group \(E(L)\) equals \(MM(L)\) only when \(\mathfrak g(L)=1\).

The reflection in \(w \in L \otimes \mathbb{R}\) with \(b(w, w) \neq 0\) is \(\sigma_w(x) = x - \frac{2 b(x, w)}{b(w, w)} w\). Every element of \(O(L \otimes \mathbb{R})\) is a product \(\sigma_{w_1} \cdots \sigma_{w_m}\) of reflections, and \[\operatorname{sn}_\mathbb{R}(\sigma_{w_1} \cdots \sigma_{w_m}) = \prod_{i=1}^m \left(-\frac{b(w_i, w_i)}{2}\right) \in \mathbb{R}^*/(\mathbb{R}^*)^2\] (Dawes 2022, (4)).

On a positive definite lattice, \(O^+(L) = SO(L)\). On a negative definite lattice, \(O^+(L) = O(L)\).

A product of \(m\) reflections has determinant \((-1)^m\). When \(b\) is positive definite, each factor \(-b(w_i, w_i)/2\) is negative, so the spinor norm is \((-1)^m\) too. When \(b\) is negative definite, each factor is positive, so the spinor norm is \(1\).

So on a definite lattice the eight groups are at most four. When \(A_L = 0\), \(\widetilde{O}(L) = O(L)\), and the eight groups are at most four again.

The series of orbits

For a subgroup \(\Gamma\) of \(O(L)\) and an integer \(n\), \(c_\Gamma(n)\) is the number of \(\Gamma\)-orbits on the primitive vectors \(v \in L\) with \(b(v, v) = n\), when it is finite. The series of orbits of primitive vectors is \[F_{L,\Gamma}(z, w) = c_\Gamma(0) + \sum_{n \geq 1} c_\Gamma(n) z^n + \sum_{n \geq 1} c_\Gamma(-n) w^n \in \mathbb{Z}[[z, w]].\]

So \(F_{L,\Gamma}(0, 0)\) is the number of \(\Gamma\)-orbits of primitive isotropic vectors, the coefficient of \(z^n\) is the number of \(\Gamma\)-orbits of primitive vectors of norm \(n\), and the coefficient of \(w^n\) is that of norm \(-n\). A record states the coefficients that are known, through \(z^4\) and \(w^4\) at first, and a coefficient that is not known is null.

Each coefficient obeys these relations, which latticedb check verifies:

\(c_\Gamma(n) = 0\) exactly when no primitive vector has norm \(n\), for every \(\Gamma\). That is so for odd \(n\) when \(L\) is even, and for \(n = 0\) and for \(n\) of the sign opposite to \(b\) when \(L\) is definite.
When \(n\) has no square factor \(k^2 > 1\), every vector of norm \(n\) is primitive, since \(b(kv, kv) = k^2 b(v, v)\). Then \(c_\Gamma(n) = 0\) exactly when the theta series has no vector of norm \(n\).
For \(H \subseteq \Gamma\), each \(\Gamma\)-orbit is a union of \(H\)-orbits, so \(c_H(n) \geq c_\Gamma(n)\).
Two keys that name one group, by the proposition above, have the same coefficients.

Computation

Let \(\varphi\colon G \to Q\) be a surjective homomorphism with kernel \(H\), and let \(G\) act on a set \(X\). Then \(x \mapsto G \cdot (x, 1)\) induces a bijection from the \(H\)-orbits on \(X\) to the \(G\)-orbits on \(X \times Q\), where \(g \cdot (x, q) = (g x, \varphi(g) q)\).

\(g \cdot (x, 1) = (y, 1)\) exactly when \(y = g x\) and \(g \in H\). Every \(G\)-orbit meets \(X \times \{1\}\): for \(g\) with \(\varphi(g) = q\), \(g^{-1} \cdot (x, q) = (g^{-1} x, 1)\).

For a definite lattice, latticedb certify computes \(c_\Gamma(n)\) for \(1 \leq |n| \leq 4\). The function qfauto of PARI/GP gives generators of \(O(L)\), and qfminim gives the vectors of norm at most 4, of which it keeps the primitive ones. For \(SO\), \(\widetilde{O}\) and \(S\widetilde{O}\), \(Q\) is the image of \(O(L)\) under \(\det\), under \(O(L) \to O(A_L)\), or under both, and the orbits are those of the proposition. \(O^+\) and its subgroups are then given by the proposition on definite lattices.

For an indefinite lattice the sets are infinite. Dawes gives algorithms for the orbits of vectors under subgroups of \(O(L)\) (Dawes 2022, Algorithms 2.1 to 2.3). When \(L\) is even and contains two orthogonal hyperbolic planes, the coefficients come from \(A_L\) and its finite quadratic form \(q_L\colon A_L \to \mathbb{Q}/2\mathbb{Z}\).

The divisor \(\operatorname{div}(v)\) of \(v \in L\) is the positive generator of the ideal \(b(v, L) \subseteq \mathbb{Z}\). Then \(v^* = v / \operatorname{div}(v)\) is in \(L^\vee\), and its image in \(A_L\) has order \(\operatorname{div}(v)\) when \(v\) is primitive (GHS 2009, Section 3.3).

Let \(L = U \oplus U_1 \oplus L_0\) be an even lattice, with \(U = \mathbb{Z}e \oplus \mathbb{Z}f\) and \(U_1\) hyperbolic planes, and let \(L_1 = U_1 \oplus L_0\). If \(u, v \in L\) are primitive, \(b(u, u) = b(v, v)\) and \(u^* \equiv v^* \bmod L\), then \(\tau(u) = v\) for some \(\tau\) in the group \(E_U(L_1)\) generated by the Eichler transvections \(t(e, a)\) and \(t(f, a)\) with \(a \in L_1\) (GHS 2009, Proposition 3.3(i), with the hypothesis of Section 1 that \(L\) is even). \(E_U(L_1)\) is a subgroup of \(S\widetilde{O}^+(L)\) (GHS 2009, (10)).

The theorem needs \(L\) even. In the odd lattice \(U^2 \oplus \langle 1 \rangle\) with \(b(x, x) = 1\), the vectors \(x\) and \(e + x\) are primitive of norm 1, and \(A_L = 0\). But \(x\) is characteristic, \(b(x, y) \equiv b(y, y) \bmod 2\) for every \(y\), and \(e + x\) is not, since \(b(e + x, f) = 1\) and \(b(f, f) = 0\). An isometry keeps this property, so no element of \(O(L)\) sends \(x\) to \(e + x\).

Let \(L = U \oplus U_1 \oplus L_0\) be even, and for an integer \(n\) let \(S_n\) be the set of \(\alpha \in A_L\) of some order \(d\) with \(q_L(\alpha) = n / d^2\) in \(\mathbb{Q}/2\mathbb{Z}\). Then \(v \mapsto v^* + L\) is a surjection from the primitive vectors of norm \(n\) onto \(S_n\), and its fibres are the \(\widetilde{\Gamma}\)-orbits for each \(\widetilde{\Gamma}\) with \(E_U(L_1) \subseteq \widetilde{\Gamma} \subseteq \widetilde{O}(L)\). So \(c_\Gamma(n) = |S_n|\) for \(\Gamma = \widetilde{O}, S\widetilde{O}, \widetilde{O}^+, S\widetilde{O}^+\). For \(\Gamma = O, SO, O^+, SO^+\), \(c_\Gamma(n)\) is the number of \(O(q_L)\)-orbits on \(S_n\).

For \(v\) primitive of norm \(n\) and \(d = \operatorname{div}(v)\), \(v^* + L\) has order \(d\) and \(q_L(v^* + L) = n / d^2\), so it is in \(S_n\). Let \(\alpha \in S_n\) have order \(d\). Since \(A_L = A_{L_0}\), \(\alpha = x + L\) for some \(x \in L_0^\vee\) with \(d x \in L_0\), and \(d^2 b(x, x) \equiv n \bmod 2 d^2\). So \(m = (n - d^2 b(x, x)) / 2d^2\) is an integer, and \(v = d(e + m f + x)\) is in \(L\) with \(b(v, v) = n\). \(b(v, f) = d\) and \(b(v, L) \subseteq d \mathbb{Z}\), so \(\operatorname{div}(v) = d\) and \(v^* + L = \alpha\). If \(v = k w\) with \(w \in L\), then \(\frac{d}{k} x \in L_0\), so \(d\) divides \(d / k\) and \(k = 1\): \(v\) is primitive. Each \(\widetilde{\Gamma}\) fixes \(v^* + L\), and by the theorem \(E_U(L_1)\) is transitive on each fibre; so the fibres are the \(\widetilde{\Gamma}\)-orbits.

For \(SO^+ \subseteq \Gamma\), the map is \(\Gamma\)-equivariant and each fibre is one \(S\widetilde{O}^+\)-orbit, so the \(\Gamma\)-orbits correspond to the orbits on \(S_n\) of the image of \(\Gamma\) in \(O(q_L)\). The reflections \(\sigma_{e+f}\) and \(\sigma_{e-f}\) are in \(O(L)\), since \(b(e \pm f, e \pm f) = \pm 2\), and act trivially on \(A_L = A_{U_1 \oplus L_0}\), so they are in \(\widetilde{O}(L)\). Their determinants are \(-1\) and \(-1\), and their real spinor norms \(-b(w, w)/2\) are \(-1\) and \(1\); so they map onto \(O(L) / SO^+(L) \subseteq \{\pm 1\}^2\), and \(O(L) = \widetilde{O}(L) \cdot SO^+(L)\). Thus \(O\), \(SO\), \(O^+\) and \(SO^+\) have one image in \(O(q_L)\). \(L\) is even and indefinite, and \(\operatorname{rank} L = 4 + \operatorname{rank} L_0 \geq l(A_L) + 4\), where \(l\) is the minimal number of generators; so \(O(L) \to O(q_L)\) is surjective (Nikulin 1980, Theorem 1.14.2).

latticedb certify computes \(c_\Gamma(n)\) for \(1 \leq |n| \leq 4\) and \(n = 0\) by this proposition, for an even lattice whose hyperbolic index, stored or bounded from below by the embeddings of \(U^k\) in its morphism files, is at least 2. The class TorsionQuadraticModule of SageMath gives \(A_L\), \(q_L\) and generators of \(O(q_L)\), and the orbits on \(S_n\) are found by breadth-first search. For an odd lattice the coefficients are stated with a reference that proves them.

Let \(L = U^2 \oplus L'\) be even and unimodular. Then \(A_L = 0\) and \(S_n = \{0\}\) for each even \(n\), so \(c_\Gamma(n) = 1\) for each of the eight groups: the vector \(e + \frac{n}{2} f\) in the first \(U\) is primitive of norm \(n\). For the K3 lattice \(U^3 \oplus E_8(-1)^2\) this is GHS 2013, Example 7.6.

GHS 2009: V. Gritsenko, K. Hulek and G. K. Sankaran, Abelianisation of orthogonal groups and the fundamental group of modular varieties, arXiv:0810.1614. GHS 2013: V. Gritsenko, K. Hulek and G. K. Sankaran, Moduli of K3 surfaces and irreducible symplectic manifolds, arXiv:1012.4155. Dawes 2022: M. Dawes, Orbits in lattices, arXiv:2205.10601.