FC11 — Peters–Sterk, Symmetric and Quadratic Forms (June 2024)
FC11 — Peters & Sterk, Symmetric and Quadratic Forms.
This chapter contains 14 definitions, 3 theorems, 6 propositions, 6 lemmas, 5 corollaries, 9 examples, 3 constructions, 3 remarks and 1 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition / construction / notation
We also saw that we may assume that \(x\) is primitive since the above condition is insensitive under scaling of \(x\) . The resulting vector is unique up to sign and so \(k : = 2 q ( x ) \in \mathbb { Z }\) is an invariant of the reflection, and \(x\) is called the \(k\) -root associated to the reflection and the reflection is called a reflection in a \(k\) - root. Note that \(k\) is even for quadratic lattices but for odd lattices it might be odd. Since \(x\) is primitive, the function on \(L\) given by \(y \mapsto 2 b ( x , y ) / k\) is integral valued and defines an element in the discriminant group \({ \mathsf { d g } } _ { L }\) . It is torsion of order \({ \scriptstyle { \frac { 1 } { 2 } } } | k |\) or \(| k |\) according to whether \(k\) is even or odd respectively. In particular, for a unimodular integral lattice one can only have \(k = \pm 1\) or \(k = \pm 2\) . The sublattice of \(L\) generated by all \(k\) -roots for varying \(k\) is called its root sublattice, and the group generated by reflections in the roots, the reflection group of \(L\) . If \(L\) is spanned by its roots, it is called a root lattice. Since roots play such a central role, it is crucial to know for which lattices there are many roots, a special case of Corollary A.3.7 from the Appendix gives an instance where this is the case:
Proposition 17.1.1
Proposition 17.1.1. Suppose \(L\) is an indefinite lattice of rank \(\geq 4\) . If \(L\) contains \(a\) \(k\) -root, it contains infinitely many \(k\) -roots.
Definition / construction
For \(\varepsilon\in\{+,-\}\), the \(\varepsilon\)-root set is \(\Delta_\varepsilon(L)=\{r\in L:b(r,r)=2\varepsilon\}\) and the \(\varepsilon\)-root sublattice is \(L_{\varepsilon\text{-root}}=\sum_{r\in\Delta_\varepsilon(L)}\mathbf Zr\).
Definition / construction
The \(\varepsilon\)-Weyl group \(W^\varepsilon(L)\) is the subgroup of \(O(L)\) generated by reflections in the \(\varepsilon\)-roots.
Construction / comparison
For an even lattice, an \(\varepsilon\)-root has quadratic value \(q(r)=\varepsilon\).
Definition / construction
The real-sign and determinant conditions induce the lattice subgroups \(O^\varepsilon(L)=\{g\in O(L):\operatorname{Nm}_{\rm spin}^\varepsilon(g_\mathbf R)=1\}\), \(SO(L)=\ker(\det)\), and \(SO^+(L)=SO(L)\cap O^+(L)\).
Definition / construction
\(O^\#(L)\) denotes the subgroup of lattice isometries acting trivially on the discriminant group.
Definition / construction
\(O^{\varepsilon,\#}(L)=O^\varepsilon(L)\cap O^\#(L)\). Reflections in \(\varepsilon\)-roots act trivially on the discriminant group, hence \(W^\varepsilon(L)\subset O^{\varepsilon,\#}(L)\).
Construction / comparison
If an indefinite lattice of rank at least \(4\) contains an \(\varepsilon\)-root, then its Weyl group \(W^\varepsilon(L)\) is infinite.
Definition / construction
The level-\(n\) congruence subgroup \(O(L)[n]\) is the kernel of \(O(L)\to\operatorname{Aut}(L/nL)\). For prime \(p\), the form on \(L\) reduces to a form on \(L/pL\), and for unimodular \(L\) this reduced form identifies with the discriminant form of the scaled lattice \(L(p)\).
Example / comparison
Here \(R = \mathbb { Z }\) and the only isometries are \(\pm\) id. If \(k \in \mathbb { Z } , k \neq 0\) , we have \(\mathsf { S O } ^ { + } ( \langle k \rangle ) = \mathsf { S O } ( \langle k \rangle ) = \{ 1 \}\) and \(0 ^ { + } ( \left. k \right. ) = \left\{ 1 \right\}\) if \(k < 0\) and \(0 ^ { + } ( \left. k \right. ) = \{ 1 , - 1 \}\) otherwise.
Example / comparison
Hyperbolic lattice \(U\) over \(\mathbb { Z }\) . Hyperbolic lattices over integral domains have been discussed in \(\ S\) 6.3.B, example 6. In the present case, there are only 4 isometries, \(\pm\) id and the reflections \(j _ { 1 } = \sigma _ { e - f }\) with matrix \(\mathbf { \Sigma } = { \binom { 0 } { 1 } } \mathbf { \Sigma } _ { 0 } ^ { 1 } { \Big ) }\) and \(- j _ { 1 } = \sigma _ { e + f }\) Their properties are assembled in the following table. 
One has \(0 ^ { \pm } ( U ) = \langle \mp j _ { 1 } \rangle\) , \({ \mathsf { S O } } ^ { + } ( U ) = \{ { \mathrm { i d } } \}\) and \({ \mathsf { S O } } ^ { - } ( U ) = \langle - { \mathrm { i d } } \rangle\) . We also see that \(U\) is not a root lattice. In fact \(\mathbb { Z } ( e + \varepsilon f )\) is the lattice spanned by the \(\varepsilon\) -roots and the full root sublattice is isometric to \(\left. 2 \right. \mathbb { O } \left. - 2 \right.\) .
Example / comparison
Reducing the A–D–E root lattices modulo \(2\) gives the stated quadratic \(\mathbf F_2\)-spaces; in particular \(A_{2m}\) has Arf invariant \(1\), while the reduction of \(E_8\) has Arf invariant \(0\).
Definition 17.2.1
Definition 17.2.1. A lattice \(L\) is called \(k\) -reflective, respectively reflective, if the group generated by the reflections in \(k\) -roots for fixed \(k\) , respectively for all \(k\) , has finite index in the group \(\mathrm { O } ( L )\) .
Lemma 17.2.2
Lemma 17.2.2. The isometry group of a positive definite irreducible quadratic 2-root lattice \(\Gamma\) (that is \(\Gamma = A _ { n }\) , \(D _ { n }\) or \(E _ { n _ { \bot } }\) ) is a direct product of its Weyl group and the group of symmetries of the Dynkin diagram. Explicitly • \(\mathsf { O } ( \Gamma ) = \mathsf { W } ^ { + } ( \Gamma )\) for \(\Gamma = A _ { 1 } , E _ { 7 } , E _ { 8 }\) . • \(0 ( \Gamma ) = \mathsf { W } ^ { + } ( \Gamma ) \times \{ \pm \mathrm { i d } \}\) for \(\Gamma = A _ { n } , n \geq 2\) , \(D _ { n } , n \geq 5\) , and for \(E _ { 6 }\) . • \({ \sf O } ( \Gamma ) = { \sf W } ^ { + } ( \Gamma ) \times \mathfrak { S } _ { 3 }\) for \(\Gamma = D _ { 4 }\) , where \({ \mathfrak { S } } _ { n }\) is the permutation group in ?? letters.
Corollary 17.2.3
Corollary 17.2.3. We have • \(0 ^ { + , \# } ( A _ { n } ) = \mathsf { W } ^ { + } ( A _ { n } )\) . • \(0 ^ { + , \# } ( D _ { n } ) = \mathsf { W } ^ { + } ( D _ { n } )\) for \(n \geq 5\) , \(\mathsf { O } ^ { + , \# } ( D _ { 4 } ) = \mathsf { W } ^ { + } ( D _ { 4 } ) \times \mathbb { Z } / 3 \mathbb { Z }\) • \(0 ^ { + , \# } ( E _ { n } ) = \mathsf { W } ^ { + } ( E _ { n } )\) , \(n = 6 , 7 , 8\) .
Remark 17.2.4
Remark 17.2.4. We saw some other classical reflection groups in Examples 4.2.3 where \(k\) -roots come up for \(k \neq \pm 2\) .
Definition / construction
A root basis is a basis of roots such that every root is an all-nonnegative or all-nonpositive integral combination of the basis roots.
Proposition 17.2.5
Proposition 17.2.5. The closure of a Weyl chamber is a fundamental domain for the action of the Weyl group \({ \sf W } ( L )\) on \(L _ { \mathbb { R } }\) . 1. Let \(\mathcal { C }\) be a Weyl chamber. The set of roots \(\Delta _ { \mathcal { C } } \ = \ \{ r \ \in \ \Delta ( L ) \ | \ b ( x , r ) \ >\) 0 for all \(x \in \mathcal { C }\) } has the property that \(\Delta ( L ) = \Delta _ { \mathcal { C } } \cup - \Delta _ { \mathcal { C } }\) . There are precisely ?? roots in \(\Delta \mathcal { C }\) that form a root basis and these give a Dynkin diagram whose root lattice is \(L\) . 2. Conversely, let \(\Delta ( L ) = \Delta \cup - \Delta\) be a partition of the set \(\Delta ( L )\) such that \(\Delta\) is closed under taking sums, i.e., any non-negative integral linear combination of roots in \(\Delta\) belongs to \(\Delta\) . Then \(\{ x \in ( \mathsf C _ { L _ { \mathbb R } } \mid b ( x , r ) > 0\) for all \(r \in \Delta \}\) is \(a\) Weyl chamber. 3. W (??) acts simply transitively on root bases and on Weyl chambers. 4. The roots of given length form one orbit under \({ \mathsf { W } } ( L )\) .
Observation
Observation. Reflections in \(k\) -roots with \(k < 0\) preserve the components of the light cone. By (16.3) the subgroup of isometries of \(L\) preserving the two components of the light cone is \(\mathrm { O } ^ { - } ( L )\) . In particular it contains the reflections in \(k\) -roots, \(k < 0\) , since these have signed spinor norm 1. The corresponding Weyl group \({ \mathsf { W } } ^ { - } ( L )\) is a normal subgroup of \(\mathrm { O } ^ { - } ( L )\) .
Definition / construction / notation
A Weyl chamber in the relevant real cone is a connected component of the complement of all reflection hyperplanes.
Proposition 17.2.6
Proposition 17.2.6. Let \(\Delta ( L )\) be the set of roots \(r \in L\) with \(b ( r , r ) ~ < ~ 0\) . The closure of \(a\) Weyl chamber is a fundamental domain for the action of the Weyl group \({ \mathsf { W } } ^ { - } ( L )\) on \(C _ { V _ { \mathbb { R } } }\) . 1. Let \(\mathcal { C }\) be a Weyl chamber. The set of roots \(\Delta _ { \mathcal { C } } \ = \ \{ r \ \in \ \Delta ( L ) \ | \ b ( x , r ) \ >\) 0 for all \(x \in \mathcal { C }\) } has the property that \(\Delta ( L ) = \Delta _ { \mathcal { C } } \cup - \Delta _ { \mathcal { C } }\) . 2. Conversely, let \(\Delta ( L ) = \Delta \cup - \Delta\) be a partition of the set \(\Delta ( L )\) such that \(\Delta\) is closed under taking sums, i.e., any non-negative integral linear combination of roots in \(\Delta\) belongs to \(\Delta\) . Then \(\{ x \in \mathsf C _ { V _ { \mathbb R } } \mid b ( x , r ) > 0\) for all \(r \in \Delta\) } is \(a\) Weyl chamber.
Corollary 17.2.7
Corollary 17.2.7. The group \(0 ^ { - } ( L )\) of isometries preserving the light cone is the semi-direct product of \({ \mathsf { W } } ^ { - } ( L )\) and the stabilizer in \(0 ^ { - } ( L )\) of \(a\) Weyl chamber. Consequently, the latter group is isomorphic to \(0 ^ { - } ( L ) / \mathsf { W } ^ { - } ( L )\) .
Remark 17.2.8
Remark 17.2.8. By item 2, the Weyl group \({ \mathsf { W } } ^ { - } ( L )\) is generated by those roots in \(\Delta \mathcal { C }\) which cannot be written as a non-trivial sum of roots in \(\Delta \boldsymbol { \mathscr { e } }\) . As in the positive definite case such roots are called indecomposable. For two different indecomposable roots \(r\) , \(s \in \Delta\) we have \(r \cdot s \geq 0\) .
Theorem 17.2.9
Theorem 17.2.9 ([239, 240, 172, 72]). A root lattice of signature \(( 1 , n )\) is reflective if and only if \(n \in \{ 1 , . . . , 1 9 , 2 1 \}\) . If \(n = 2 1\) such a lattice is even: it is the unique reflective Lorentzian lattice of rank 22 (see Example 17.2.10.5 below). In particular there are no odd reflective Lorentzian lattices of rank 22.
Example / comparison
The Lorentz lattice \(\mathbb { Z } ^ { 1 , n }\) , \(n \geq 2\) , is a root lattice. Indeed, using, as before, the standard basis \(\{ e _ { 0 } , \ldots , e _ { n } \}\) for \(\mathbb { R } ^ { n + 1 }\) it has a new basis \[ \begin{array} { r l } & { \alpha _ { 0 } = ( 1 , - 1 , - 1 , - 1 , 0 , \ldots , 0 ) , \mathrm { ~ f o r ~ } n \geq 3 , \quad = ( 1 , - 1 , - 1 ) \mathrm { ~ f o r ~ } n = 2 , } \\ & { \alpha _ { j } = e _ { j } - e _ { j + 1 } , \quad j = 1 , \ldots , n - 1 , } \\ & { \alpha _ { n } = e _ { n } . } \end{array} \] The vectors \(\alpha _ { j }\) , \(j = 1 , \dotsc , n - 1\) , are \(\left( - 2 \right)\) -roots, \(\alpha _ { n }\) is a \(( - 1 )\) -root and \(\alpha _ { 0 }\) is a \(\left( - 2 \right)\) -root for \(n \geq 3\) , but a \(( - 1 )\) -root for \(n = 2\) . Although the Lorentz lattice has a basis of roots, the corresponding reflections might not generate the full reflection group. E. Vinberg developed an algorithm which terminates if a finite set of roots exists such that the corresponding reflection group has finite index in \({ \boldsymbol { \mathrm { O } } } ( \mathbb { Z } ^ { 1 , n } )\) .1 Termination occurs if and only if \(n \leq 1 9\) and so precisely then \(\mathbb { Z } ^ { 1 , n }\) is reflective. For a proof which is more in the spirit of lattice theory see [24, §6, especially Cor. 6.3 and Lemma 6.5]. These results show in particular that for \(2 \leq n \leq 9\) the above root basis suffices to generate the full reflection group. The corresponding graphs resemble the graphs \(E _ { n + 1 } ( - 1 )\) (cf. Eqn. (4.4)): 
Here we denote the self-intersections of the roots below the vertices: among the depicted roots the black ones are \(\left( - 2 \right)\) -roots and the white ones are \(( - 1 )\) -roots. As mentioned before, for \(2 \leq n \leq 1 9\) the lattices \(\mathbb { Z } ^ { 1 , n }\) are reflective, but they might not be \(\left( - 2 \right)\) -reflective because of the presence of \(( - 1 )\) -roots in the basis. For \(2 \leq n \leq 9\) we shall show below in Example 17.3.8 that \(\mathbb { Z } ^ { 1 , n }\) is not \(\left( - 2 \right)\) -reflective.
Example / comparison
Those of the lattices \(T _ { 2 , 3 , n - 2 } = E _ { n } ( - 1 ) \subset \mathbb { Z } ^ { 1 , n } ( - 1 )\) which are Lorentzian, that is, for which \(n \geq 1 0\) , have been investigated in Subsection 4.1.C, where it was shown that they are even root lattices (of rank \(n\) ). For \(n = 1 0\) we get the Enriques lattice. For \(n = 1 0 , 1 1 , 1 2 , 1 3\) these are reflective by [172], but for \(n > 1 3\) these are not reflective.
Example / comparison
The hyperbolic \(T\) -shaped lattices from Section 4.1. Recall that the lattices \(T _ { p , q , r }\) are even root lattices which for \(p ^ { - 1 } + q ^ { - 1 } + r ^ { - 1 } < 1\) are all Lorentzian. Apart from the lattices with \(( p , q , r ) = ( 2 , 3 , r )\) , \(r = 7 , 8 , 9\) , from Example 2, only the lattices with \(( p , q , r ) = ( 2 , 4 , r )\) with \(r = 5 , 6 , 7\) and \(( p , q , r ) = ( 3 , 3 , r )\) with \(r = 4 , 5 , 6\) are reflective. The roots corresponding to the vertices give a root basis only for \(( p , q , r ) = ( 2 , 3 , 7 ) , ( 3 , 3 , 4 ) , ( 2 , 4 , 5 )\) . To obtain a generating set of roots for the other root lattices, some roots have to be added to a basis. See [172] for details. Here is an example where \(\alpha _ { 1 2 }\) needs to be added: \[ \begin{array} { r l } { { \overbrace { \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \alpha _ { 4 } } ^ { \alpha _ { 1 } } } \quad } & { \alpha _ { 5 } \quad \overbrace { \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \alpha _ { 7 } } ^ { \alpha _ { 5 } } \quad \overbrace { \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \alpha _ { 9 } } ^ { \alpha _ { 8 } } \quad } & { \alpha _ { 1 0 } \quad \alpha _ { 1 1 } \quad \alpha _ { 1 1 } \quad \alpha _ { 1 2 } \quad } \\ & { - 2 \quad \quad \mathrm { ~ ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \cdot \mathrm { ~ \Lambda ~ } \ \cdot \mathrm { ~ \Lambda ~ } \ \cdot \mathrm { ~ \Lambda ~ } \ \cdot 2 } \end{array} \] The graph for the root lattice of \(T _ { 2 , 3 , 8 }\)
Example / comparison
Recall (cf. Table 4.2.1) that the label \(\infty\) on the extreme right edge means that the angle between the roots \(\alpha _ { 1 1 }\) and \(\alpha _ { 1 2 }\) is \(\pi\) . In the present situation the two corresponding reflection hyperplanes are parallel in the sense that they meet on the boundary of the light cone2). In our case this is equivalent to \(b ( \alpha _ { 1 1 } , \alpha _ { 1 2 } ) = - 2\) . 4. Reflection groups related to the Leech lattice \(\Gamma _ { 2 4 }\) . Recall (Section 5.1) that the Leech lattice is a positive definite rank 24 unimodular lattice, the unique Niemeier lattice without roots. The even unimodular Lorentzian lattice \[ I I _ { 1 , 2 5 } : = U \oplus \Gamma _ { 2 4 } ( - 1 ) \] is by classification (cf. Corollary 2.4.3) isometric to \(U \ @ E _ { 8 } ^ { \oplus 3 } ( - 1 )\) . Since \(U \mathbb { \oplus } E _ { 8 } ( - 1 )\) and \(E _ { 8 } ( - 1 )\) are spanned by their roots (see \(\ S\) 4.1.C) so is \(I I _ { 1 , 2 5 }\) . The group generated by the reflections in these roots has a fundamental domain \(P\) described in [42] whose bounding hyperplanes are orthogonal to roots of the form \(e - ( 1 + q ( v ) ) f + v\) where \(v \in \Gamma _ { 2 4 } ( - 1 )\) and \(\{ e , f \}\) is the standard basis of \(U\) . Those roots are called the Leech roots. The lattice \(I I _ { 1 , 2 5 }\) is not reflective: the symmetry group of \(P\) turns out to be isomorphic to \(\Gamma _ { 2 4 } \triangleleft ( \Gamma _ { 2 4 } )\) , an infinite group isomorphic to \(0 ^ { - } ( L ) / \mathsf { W } ^ { - } ( L )\) .
Definition / construction / notation
is by classification (cf. Corollary 2.4.3) isometric to \(U \ @ E _ { 8 } ^ { \oplus 3 } ( - 1 )\) . Since \(U \mathbb { \oplus } E _ { 8 } ( - 1 )\) and \(E _ { 8 } ( - 1 )\) are spanned by their roots (see \(\ S\) 4.1.C) so is \(I I _ { 1 , 2 5 }\) . The group generated by the reflections in these roots has a fundamental domain \(P\) described in [42] whose bounding hyperplanes are orthogonal to roots of the form \(e - ( 1 + q ( v ) ) f + v\) where \(v \in \Gamma _ { 2 4 } ( - 1 )\) and \(\{ e , f \}\) is the standard basis of \(U\) . Those roots are called the Leech roots. The lattice \(I I _ { 1 , 2 5 }\) is not reflective: the symmetry group of \(P\) turns out to be isomorphic to \(\Gamma _ { 2 4 } \triangleleft ( \Gamma _ { 2 4 } )\) , an infinite group isomorphic to \(0 ^ { - } ( L ) / \mathsf { W } ^ { - } ( L )\) .
Example / comparison
The Borcherds lattice [24]. The sublattice \(L _ { 1 , 2 1 } \subset E _ { 2 1 } ( - 1 )\) consisting of vectors \(\sum x _ { i } e _ { i }\) with \(\textstyle \sum x _ { i }\) even defines an even lattice. It is isometric to \(U \mathbb { \oplus } D _ { 2 0 } ( - 1 )\) and turns out to be a reflective root lattice having a generating set of 210 roots. Moreover, the quotient \(0 ^ { - } ( L _ { 1 , 2 1 } ) / \mathsf { W } ^ { - } ( L _ { 1 , 2 1 } )\) is isomorphic to a finite group of order \(2 ^ { 8 } \cdot 3 ^ { 3 } \cdot 5 \cdot 7\) . It is the reflective Lorentzian lattice of maximal possible rank. We shall encounter this lattice again as an example of a supersingular K3 lattice (see Remark 19.5.3).
Theorem 17.2.11
Theorem 17.2.11 ([121, Satz 4]). Let \(( L , q )\) be an integral quadratic lattice with real Witt index \(\geq 2\) , that is \(\operatorname* { m i n } ( r , s ) \geq 2\) , where \(( r , s )\) is the signature of ??. Assume that the following conditions on \(L\) hold simultaneously: • \(L\) contains at least one root \(x\) with \(q ( x ) = 1\) ; • \(L\) contains a sublattice \(L _ { 1 }\) of rank \(\geq 5\) with \(\mathrm { d i s c } ( L _ { 1 } )\) not divisible by 3; • ?? contains a sublattice \(L _ { \mathrm { 2 } }\) of \(r a n k \ge 6\) with \(\mathrm { d i s c } ( L _ { 2 } )\) odd. Then \(\mathsf { W } ^ { + } ( L ) \cap \mathsf { S O } ^ { + } ( L ) = \mathsf { S O } ^ { + } ( L ) \cap \mathsf { O } ^ { \# } ( L )\) .
Theorem: Theorem
Theorem. If the real Witt index of \(L\) is at least 2 and a normal subgroup \(N\) of \(\mathrm { S p i n } ( L )\) is not contained in the center of \(\mathrm { S p i n } ( L )\) , then \(N\) is a congruence subgroup.
Corollary 17.2.12
Corollary 17.2.12. Under the same conditions on \(( L , q )\) , one has \(\mathsf { W } ^ { \epsilon } ( L ) = \mathsf { O } ^ { \epsilon , \# } ( L )\)
Remark 17.2.13
Remark 17.2.13. The condition on the signature is necessary as we see from Theorem 17.2.9 which demonstrates that the Weyl group in a Lorentzian lattice is rarely of finite index in the full orthogonal group. See also Theorem 17.3.7 below.
Definition 17.3.1
Definition 17.3.1. Given a pair of vectors \(\{ f , y \}\) in \(V\) where \(f\) is isotropic and \(b ( f , y ) = 0\) , the Eichler–Siegel transformation with respect to \(\{ f , y \}\) is given by \[ \begin{array} { r l r } { { \psi _ { f , y } : V \longrightarrow V \longrightarrow V } } \\ & { } & { x \longmapsto x + b ( x , y ) f - b ( x , f ) y - b ( x , f ) q ( y ) f . } \end{array} \]
Construction / comparison
The Eichler–Siegel transformation \(\psi_{f,y}\) is an isometry with inverse \(\psi_{f,-y}\); it is the identity when \(f\) and \(y\) are linearly dependent.
Lemma 17.3.2
Lemma 17.3.2. Assume that \(q ( y ) \neq 0\) . Then the Eichler–Siegel transformation \(\psi _ { f , y }\) is the unique isometry of \(V\) which satisfies \[ \psi _ { f , y } z = z + b ( z , y ) f \quad \forall z \in \mathbb { R } f ^ { \perp } . \] In particular, \(\psi _ { f , y } f = f\) .
Lemma 17.3.3
Lemma 17.3.3. Suppose \(f \in L\) is a primitive isotropic vector and suppose \(y \in L\) satisfies \(b ( y , y )\) even and \(b ( f , y ) = 0\) . Then \(\psi _ { f , y }\) induces a lattice isometry of \(L\) with the following properties. 1. \(\psi _ { f , y } \in { 0 } ^ { \in } ( L )\) for \(\epsilon = -\) as well as for \(\epsilon = +\) . 2. \(\psi _ { f , y }\) induces the identity on the discriminant group. 3. One has \(\psi _ { f , y } = \psi _ { f , y + f }\) and the map \[ \begin{array} { r } { L _ { f } = f ^ { \perp } / \mathbb { Z } f \stackrel { \psi _ { f } } { \longrightarrow } 0 ( L ) } \\ { y + \mathbb { Z } f \longmapsto \psi _ { f , y } } \end{array} \] is a well-defined injective group homomorphism. In particular \(\psi _ { f } ( L _ { f } )\) is an abelian subgroup of \(\mathrm { O } ( L )\) whose rank equals \(\operatorname { r a n k } ( L _ { f } )\) . 4. \(\psi _ { f } ( L _ { f } )\) is a normal subgroup of \({ \sf O } _ { f } ( L )\) , the stabilizer of \(f\) .
Proposition 17.3.4
Proposition 17.3.4. Let \(L\) be a non-degenerate lattice of rank \(n \geq 3\) containing an isotropic vector. Then \(\mathrm { O } ( L )\) contains a free abelian group of rank \(n - 2\) . In particular, ?? has infinitely many isometries.
Corollary 17.3.5
Corollary 17.3.5. Let \(L\) be a non-degenerate indefinite lattice of rank \(\geq 5\) , then O (??) is infinite.
Proposition 17.3.6
Proposition 17.3.6. Let \(L\) be a non-degenerate lattice of rank ?? containing a primitive isotropic vector \(f\) and \(a\) \(\left( - 2 \right)\) -root \(r \in f ^ { \perp }\) . Then 1. \(\psi _ { f , r } = \sigma _ { f - r ^ { \circ } } \sigma _ { r }\) . 2. The image of the \(\left( - 2 \right)\) -root lattice of \(L _ { f }\) under \(\psi _ { f } : L _ { f } \to { \mathsf { O } } ( L )\) is contained in the Weyl group \(W ^ { - } ( f ^ { \bot } )\) considered as the subgroup of \({ \mathsf { W } } ^ { - } ( L )\) generated by reflections in \(\left( - 2 \right)\) -roots of \(f ^ { \bot }\) (and hence preserving \(f\) ). 3. If \(y \in L _ { f }\) is orthogonal to the \(\left( - 2 \right)\) -root lattice \(L _ { f , \mathrm { r o o t } }\) of \(L _ { f }\) , then \(\psi _ { f } ( y )\) restricts to the identity on the \(\left( - 2 \right)\) -root lattice of \(f ^ { \bot }\) . In particular, \(W ^ { - } ( f ^ { \bot } ) \cap\) \(\psi _ { f } ( L _ { f , \mathrm { r o o t } } ^ { \perp } ) = \{ \mathrm { i d } \}\) .
Lemma: Lemma
Lemma. Suppose that a Lorentzian lattice \(L\) contains a (primitive) isotropic vector \(f\) . Then \[ \mathsf { W } _ { f } ^ { - } L = \mathsf { W } ^ { - } ( f ^ { \perp } ) , \] where we recall that \(\mathsf { W } _ { f } ^ { - } L\) is the subgroup of the Weyl group of \(L\) fixing \(f\) , and the right-hand side, \(W ^ { - } ( f ^ { \bot } )\) , is identified with the subgroup of \({ \mathsf { W } } ^ { - } ( L )\) generated by reflections in \(\left( - 2 \right)\) -roots of \(f ^ { \bot }\) (and hence preserving \(f\) ).
Proposition 17.3.7
Proposition 17.3.7 (compare [172, Prop. 1.3.1]). For a Lorentzian type lattice \(L\) of rank \(\geq 3\) containing a primitive isotropic vector \(f\) and associated negative definite lattice \(L _ { f } = f ^ { \perp } / \mathbb { Z } f\) , let \(\delta _ { f } = \mathrm { r a n k } ( L _ { f } ) - \mathrm { r a n k } ( L _ { f , \mathrm { r o o t } } )\) . Then \(\mathsf { O } ( L ) / \mathsf { W } ^ { - } ( L )\) contains a normal abelian subgroup of rank \(\delta _ { f }\) . In particular, if \(L _ { f }\) is not spanned by its roots, the group \(\mathsf { O } ( L ) / \mathsf { W } ^ { - } ( L )\) is infinite.
Example / comparison
Example 17.3.8. For the Lorentz lattices \(E^{1,n}\), the cases \(n=2,3\) and \(5\le n\le9\) admit primitive isotropic vectors whose quotients \(f^\perp/\mathbf Zf\) are not spanned by \((-2)\)-roots; hence these lattices are not \((-2)\)-reflective.
Definition / construction
If \(L=U\oplus L\prime\) with standard basis \(e,f\) of \(U\), then \(\psi_U(L\prime)=\langle\psi_{e,y},\psi_{f,y}:y\in L\prime\rangle\subset O(L)\); this group is independent of the chosen standard basis of \(U\).
Definition / construction / notation
As indicated by the notation, this group does not depend on the choice of basis for \(U\) (see Example 17.1.2.2). The following result shows how one can use Eichler– Siegel transformations to describe certain orbits under the isometry group.
Lemma 17.3.9
Lemma 17.3.9 ([144, Lemma 2.5]). Let \(U\) be a hyperbolic plane with standard basis \(\{ e , f \}\) . Consider the lattice \(U \odot U ^ { \prime }\) where \(U ^ { \prime }\) is another hyperbolic plane. Then \(\psi _ { U } U ^ { \prime } = \psi _ { U ^ { \prime } } U \simeq \mathrm { S L } _ { 2 } ( \mathbb { Z } ) \times \mathrm { S L } _ { 2 } ( \mathbb { Z } )\) . Moreover, every vector \(x\) is in the \(\psi _ { U } ( U ^ { \prime } )\) -orbit of some vector of the form \(a e + d f\) with \(d | a\) . In particular, if \(x\) is primitive with \(q ( x ) = a\) , we can take \(d = 1\) .
Corollary 17.3.10
Corollary 17.3.10. Suppose \(L = U \oplus L ^ { \prime }\) , with \(L ^ { \prime } = U ^ { \prime } \oplus L ^ { \prime \prime }\) , \(L ^ { \prime \prime }\) even, and let \(x \in L\) . Then 1. there exists an isometry of \(L\) belonging to \(\psi _ { U } ( U ^ { \prime } )\) which sends \(x\) to \(\alpha e { + } \beta f { + } x ^ { \prime \prime }\) with \(\beta | \alpha\) and \(x ^ { \prime \prime } \in L ^ { \prime \prime }\) ; 2. if, moreover, \(b ( x , L ) = \mathbb { Z }\) , there is an isometry belonging to \(\psi _ { U } ( L ^ { \prime } )\) sending \(x\) to \(a e + f\) , \(a = q ( x )\) .
Lemma 17.3.11
Lemma 17.3.11. Let \(L\) be an even lattice. Suppose that \(L = U \mathbb { \textcircled { 1 } L ^ { \prime } }\) and let \(\{ e , f \}\) be a standard basis for \(U\) . Then 1. The group \(\mathsf { O } ( U )\) normalizes \(\psi _ { U } ( L ^ { \prime } )\) . 2. Suppose that \(r \in L ^ { \prime }\) is \(a\) (−2)-root. Then \[ \begin{array} { r l } & { \psi _ { e , r } = \sigma _ { r ^ { \circ } } \sigma _ { r + e } } \\ & { \sigma _ { r } = \psi _ { f , r ^ { \circ } } \psi _ { e , - r ^ { \circ } } \psi _ { f , r ^ { \circ } } \sigma _ { e - f } } \end{array} \] 3. Suppose that \(L ^ { \prime }\) is a lattice generated by \(\left( - 2 \right)\) -roots. Then \(\psi _ { U } ( L ^ { \prime } ) \subset \mathsf { W } ^ { - } ( L )\) .