Chapter 7 Root systems and root lattices 7.1 Root systems of lattices and root lattices AhyperplaneHr of a n-dimensional vector space En is a (n − 1)- dimensional subspace. It is completely characterized by a normal vector r. Definition: reflection through a hyperplane. The reflection σr through the hyperplane Hr is a linear involution of En which leaves the points of Hr fixed and transforms r into −r. Reflection through a hyperplane is an automorphism of En completely 2 − 1 characterized by σr = In, Tr σr = n 2. Explicitly , (x, r) ∀x ∈ E ,σ(x)=x − 2 r. (7.1) n r N(r) Assume now that σr is a symmetry of the n-dimensional lattice L, i.e. ∀ ∈ L, σr( ) ∈ L. (7.2) Then (, r) − σ ( ) ∈ L ⇔ 2 r ∈ L, (7.3) r N(r) which shows that the 1-dimensional vector subspace {λr} contains a 1-sublattice2 of L.Fromnowonwechooser to be a generator of this 1-sublattice, so it is a visible vector3. This implies that the coefficient of the vector r in (7.3) is an integer. We call these vectors the roots of the lattice; their set is called the lattice root system, (, r) R(L)={r ∈ L, r visible : ∀ ∈ L, 2 ∈ Z}. (7.4) N(r) 1 As we should expect from the definition of a reflection, the expression of σr is inde- pendent of the normalization of r; in particular σr ≡ σ−r. 2 Concept defined in section 3.3. 3 There are no shorter collinear vectors in the lattice. 154 Introduction to lattice geometry through group action We write R instead of R(L) when L is understood. Notice that r ∈R⇔ −r ∈Rand that the dilation L → λL of the lattice changes simply R into λR. Moreover different pairs ±r of roots correspond to distinct reflections. We denote by GR the group generated by the |R|/2 reflections σr, r ∈R; z | z| it is a subgroup of the Bravais group PL of L. We know that PL is finite, so |R| is also finite. R ∪ R R z We can write = i i where the i are the different orbits of PL.The ∈R reflections σr,r i form a conjugacy class of this group; we denote by GRi the subgroup they generate. Since in a finite group G, any subgroup generated by one (or several) conjugacy classes is an invariant subgroup of G,wehave: z z GRi PL,GR PL. (7.5) When GR is R-irreducible, any of its orbit spans the space En (if it were not true, GR would leave invariant the subspace spanned by the orbit, and that contradicts its irreducibility). So each Ri spans En;thatisalsotrueof the short vectors S = S(L). Proposition 34 When GR(L) is R-irreducible, the norm of any root satisfies N(r) < 4s(L). Proof: The proposition is true for roots in S.Letr be a root not belonging to S.SinceS spans the space we can choose s ∈Ssuch that (r,s) > 0.The transformed vector sr = σr(s)=s − μr,with0 <μ=2(s, r)/N (r),isalso in S since it has the same norm as s.Sincer is visible and N(r) >N(s), Schwarz’s inequality |(s, r)| < N(s)N(r) (7.6) implies |(s, r)| <N(r).Thusμ =1, i.e. r = s − sr.ThusN(r) < 4N(s). Definition: root lattice. A root lattice is a lattice generated by its roots. As a trivial example, any one dimensional lattice L = {nr, n ∈ Z} is a root lattice; indeed σr(nr)=−nr. We recall that any one dimensional lattice can be scaled to I1. Proposition 35 The vectors of norm 1 and 2 of an integral lattice are roots of the lattice. Proof:Inanintegrallattice,v ∈ L ⇒ (,v) ∈ Z. Assume N(v)=1or 2; so v is visible. Then 2(,v)/N (v) is an integer, so v is a root. As we will see, this proposition gives important information on the sym- metry of the lattice. From the definition of the root lattice we obtain: Proposition 36 An integral lattice L generated by its vectors of norm 1 and ⊥ 2 is a root lattice which is the orthogonal sum L = L1 ⊕ L2 where L1 = Ik is generated by the norm 1 vectors and L2 is generated by its shortest vectors of norm 2. 7. Root systems and root lattices 155 Proof. From Proposition 35, L is a root lattice. If si, sj are two linearly independent norm 1 vectors of L, Schwarz’s inequality (7.6) implies (si,sj )= 0.Letk be the number of mutually orthogonal pairs ±si; these short (norm 1) vectors generate a lattice Ik.Letr be a root of norm 2; the value of ε =(r,si) is either ±1 or 0. In the former case N(r−εsi)=1so r = εsi +sj,andr visible requires that it is the sum of two orthogonal short roots, i.e. r ∈ Ik = L1. Obviously, the norm 2 roots r orthogonal to all lattice vectors of norm 1, generate L2. Note that an integral lattice which has no vectors of norm 1 and 2, may contain a root lattice;√ a trivial example is given by a non reduced integral lattice, i.e. the lattice mL, m ∈ Z with m ≥ 3 where L is an integral lattice with minimal norm s(L)=1. Let r and r be two linearly independent roots of L and φ the angle between them. From (7.4) we obtain: 4(r,r )2 4cos2(φ)= ∈ Z. (7.7) N(r)N(r ) Thus π π 2π π 3π π 5π 4cos2(φ)=0, 1, 2, 3 ⇒ φ = ; , ; , ; , . (7.8) 2 3 3 4 4 6 6 Since σrσr and its inverse σr σr are rotations by the angle 2φ in the 2-dimensional space spanned by r,r ,wehave m (σrσr ) = I, where m =2, 3, 4, 6. (7.9) The groups whose relations between generators are given by these equations are called Weyl groups. They are studied in the next subsection. To write explicitly the integer 4(r,r )2/(N(r)N(r )) as a function of m we use the Boolean function m → (m =6)whose values are 1 when m =6and 0 when m =6 .Then 4(r,r )2 = m − 2 − (m =6). (7.10) N(r)N(r ) Application to dimension 2 We have seen in section 4.3 that there are two maximal Bravais classes: p4mm (square lattices Ls)andp6mm (hexagonal lattices Lh). Their groups are irreducible (over C). So we can consider the two integral lattices. Since their shortest vectors satisfy s(Ls)=1, s(Lh)=2, and generate the lattice, Proposition 35 shows that Ls and Lh are root lattices. For each one, the root system has two orbits of roots; one of them is the set of short vectors of the lattice. We use the value i of the root norm as an index for the root orbit Ri. In the next equations we list the roots by giving their coordinates in the basis defined by the Gram matrix Q(L). For the Bravais class p4mm (square lattice), 1 0 Q(L )=I R (L )=S(L )= ± , ± , |R (L )| =4; s 2 1 s s 0 1 1 s 156 Introduction to lattice geometry through group action 1 1 R (L )= ± , ± , |R (L )| =4. (7.11) 2 s 1 −1 2 s For the Bravais class p6mm (hexagonal lattice), 2 −1 1 0 1 Q(L )= , S = R (L )= ± , ± , ± , |R (L )| =6, h −12 2 h 0 1 1 2 h 2 1 1 R (L )= ± , ± , ± , |R (L )| =6. (7.12) 6 h 1 2 −1 6 h Finally, the lattices of the other two non-generic Bravais classes, p2mm and ∼ Z2 c2mm, have the same point symmetry, 2mm 2, which is reducible. The lattices of the Bravais class p2mm are root lattices; those of c2mm are not. For the latter Bravais class, depending on the lattice, there might be 4 or 2 shortest vectors; in the latter case, these two shortest vectors are roots. The generic lattices (Bravais class p2)havenoroots. 7.1.1 Finite groups generated by reflections We will give in this subsection the list of irreducible finite groups generated by reflections, for short finite reflection groups. Those which satisfy equation (7.9) were introduced by H. Weyl in 1925 in his study of the finite-dimensional representations of the semi-simple Lie groups and they were listed by E. Cartan [30] (p. 218-224). Here we shall give the results of Coxeter, who established the complete list of finite reflection groups4 [37]. A finite reflection group G acting linearly on the orthogonal vector space En is defined by n generators and the relations: ≤ ≤ mij ≤ ∈ Z 1 i n , (σ ri σ rj ) = In,mii =1,i= j, 2 mij (7.13) and this abstract group is realizable as a finite subgroup of On with the generators σ ri represented by reflections through hyperplanes whose normal vector is denoted by ri.Iftheri span only a subspace of dimension n0 <n, the group acts trivially on its orthogonal complement. This case is equivalent to a reflection group on a space of dimension n0;fromnowon,weconsider only the action on En of the “n-dimensional” reflection groups; the number n of generators of such a group satisfies n ≥ n. We will now prove that n = n. The reflection hyperplanes of G partition En into |G| convex cones; each one is the closure of a fundamental domain for the action of G on En.