Affine Weyl Groups Gabriele Nebe Lehrstuhl D für Mathematik Summerschool GRK 1632, September 2015
Crystallographic root systems. Definition A crystallographic root system Φ is a finite set of non zero vectors in Euclidean space V s.t. (R1) Φ Rα = {α, α} for all α Φ (R2) s α(φ) = Φ for all α Φ, where s α : v v 2(v,α) α is the reflection along α (α,α) (R3) 2(α,β) (α,α) Z for all α, β Φ. Φ is called irreducible if for all α, β Φ there are m N, α = α 1,..., α m, α m+1 = β Φ s.t. m i=1 (α i, α i+1 ) 0. Remark Φ irreducible root system, then there is a basis = {α 1,..., α n} Φ V s.t. for all α Φ there are a 1,..., a n Z 0 with α = a 1 α 1 +... + a nα n or α = a 1 α 1... a nα n positive root negative root Φ + := Φ + ( ) denotes the set of all positive roots.! α = c i α i Φ + (the highest root) with maximal height n i=1 c i N.
The irreducible crystallographic root systems A n B n C n D n E 6 E 7 E 8 F 4 α1 α2 α n 1 αn α1 α2 α n 1 αn α1 α2 α n 1 αn αn α1 α2 α n 2 α6 α1 α2 α 3 α4 α5 α n 1 α7 α1 α2 α 3 α4 α5 α6 α8 α1 α2 α3 α4 α 5 α6 α7 α1 α2 α3 α4 G 2 α1 α2
Example a1 a2 = {α 1, α 2 } Φ + = {α 1, 2α 1 + α 2, α 1 + α 2, α 2 } Φ = Φ + Φ + order of s α1 s α2 is 4 s α1, s α2 = D 8 a1 H(a2) H(a1) a2
Finite Weyl groups Definition Let Φ be crystallographic root system. Then W (Φ) := s α : α Φ is called the Weyl group of Φ. Remark Assume that Φ is irreducible. W (Φ) = s α : α The Dynkin diagram encodes a presentation of W (Φ) W (Φ) acts irreducibly on V. F 4 α1 α2 α3 α4 W (F 4 ) = s 1, s 2, s 3, s 4 s 2 i, (s is j ) 2 ( i j > 1), (s 1 s 2 ) 3, (s 2 s 3 ) 4, (s 3 s 4 ) 3 Φ A n B n/c n D n E 6 E 7 E 8 Φ n(n + 1) 2n 2 2n(n 1) 72 126 240 W (Φ) (n + 1)! 2 n n! 2 n 1 n! 2 7 3 4 5 2 10 3 4 5 7 2 14 3 5 5 2 7 W (Φ) S n+1 C 2 S n (C n 1 2 ) : S n S 4 (3) : 2 C 2 S 6 (2) 2.O + 8 (2) : 2
Affine Weyl groups a1 s2 s1 H(a2) 2a1+a2 H(2a1+a2,1) s3 H(a1) a2 2a1+a2 a1 a2 affine Weyl group <s1,s2,s3> <s1,s2,s3 s1^2,s2^2,s3^2 (s1s2)^4,(s1s3)^4,(s2s3)^2 > Definition H α,k := {v V (v, α) = k} (affine hyperplane) α := 2 α the coroot of α Φ (α,α) s α,k : V V, v v ((v, α) k)α the reflection in the affine hyperplane H α,k
Affine Weyl groups Remark α = 2 (α,α) α, s α,k : v v ((v, α) k)α, H α,k := {v V (v, α) = k} (α ) = α (α, α) = 2, α = α if (α, α) = 2 H α,k = H α,0 + k 2 α s α,k fixes H α,k pointwise and sends 0 to kα, so it is the reflection in the affine hyperplane H α,k. Definition Let Φ be an irreducible crystallographic root system. Φ := {α α Φ} the dual root system L(Φ) := Φ Z the root lattice, L(Φ) # = {v V (v, α) Z for all α Φ} L(Φ ) # =: L(Φ) the weight lattice Proposition The affine Weyl group of Φ W a(φ) := s α,k α Φ, k Z = L(Φ ) : W (Φ) is the semidirect product of W (Φ) with the translation subgroup L(Φ ).
Proof: W a (Φ) = L(Φ ) : W (Φ) s α,k : v v ((v, α) k)α with α = 2 (α,α) α, (α, α ) = 2. W (Φ) = s α,0 α Φ W a(φ). s α,0 s α,1 = t(α ) because both map v V to (v (v, α)α )s α,1 = v (v, α)α ((v, α) (v, α) (α, α ) 1)α = v + α }{{} =2 So L(Φ ) = t(α ) α Φ W a(φ). L(Φ ) W (Φ) = {1}. L(Φ ) is normalized by W (Φ) because and W (Φ)(Φ ) = Φ. s α,k = s α,0 t(kα ) L(Φ ) : W (Φ). s α,0 t(v)s α,0 = t(s α,0 (v))
Affine Weyl groups a1 a2
Root lattice, weight lattice root lattice L(Φ) ( = α 1, α ) 2 1 1. 1 2 a1 a2 L(Φ ( ) = 2α ) 1, α 2 4 2. 2 2 weight lattice L(Φ) = α 1, 1 2 α 2
Alcoves Definition H := {H α,k α Φ, k Z} V := V H H H A connected component of V is called an alcove. A := {v V 0 < (v, α) < 1 for all α Φ + } the standard alcove. S := {s α,0 α } {s α,1 } the reflections in the faces of the standard alcove. Theorem Φ irreducible crystallographic root system. A is a simplex. W a(φ) = S. W a(φ) acts simply transitively on the set of alcoves. A is a fundamental domain for the action of W a(φ) on V.
The standard alcove is a fundamental domain a1 a2
Presentation of W a (Φ) in standard generators S a1 s2 s1 H(a2) 2a1+a2 H(2a1+a2,1) s3 H(a1) a2 2a1+a2 a1 a2 affine Weyl group <s1,s2,s3> <s1,s2,s3 s1^2,s2^2,s3^2 (s1s2)^4,(s1s3)^4,(s2s3)^2 >
The length function Definition Any w W a(φ) is a product of elements in S. We put l(w) := min{r s 1,..., s r S ; w = s 1 s 2 s r} the length of w and call any expression w = s 1 s l(w) a reduced word for w. Definition For w W a(φ) let L(w) := {H H H separates A and A w} and n(w) := L(w).
Words of minimal length s2 s3 s1 s1(s1s2s1)(s1s2s3s2s1)(s1s2s3s1s3s2s1)(s1s2s3s1s3s1s3s2s1)... =s2s1s3s1s3s2s1
Words of minimal length s2 s3 s1 w = s3s2s1s3s1s2s1 = s2s1s3s1s2s3s1
Separating hyperplanes s2 s3 s1 w = s3s2s1s3s1s2s1 = s2s1s3s1s2s3s1 L(w) contains 7 hyperplanes length of reduced word of w = 7
The length function Definition Any w W a(φ) is a product of elements in S. We put l(w) := min{r s 1,..., s r S ; w = s 1 s 2 s r} the length of w and call any expression w = s 1 s l(w) a reduced word for w. Definition For w W a(φ) let L(w) := {H H H separates A and A w} and n(w) := L(w). Theorem Let w = s 1 s l(w) and H i := H si = {v V vs i = v}. Then In particular n(w) = l(w). L(w) = {H 1, H 2 s 1, H 3 s 2 s 1,..., H l(w) s l(w) 1 s 2 s 1 }.
Coweights modulo coroots and diagram automorphisms W a(φ) = L(Φ ) : W (Φ) L(Φ ) coroot lattice L(Φ) # = {v V (v, α) Z for all α Φ} coweight lattice L(Φ ) L(Φ) # Ŵ a(φ) := L(Φ) # : W (Φ) acts on the set of alcoves. W a(φ) acts simply transitively on the set of alcoves. So Ŵa(Φ) = Wa(Φ) : Ω where Ω = Stab Ŵ a(φ) (A ). Ω = Ŵa(Φ)/Wa(Φ) = L(Φ) # /L(Φ ) acts faithfully on the simplex A. Ω acts as diagram automorphisms on the extended Dynkin diagram. Φ A n B n C n D 2n D 2n+1 E 6 E 7 E 8 F 4 G 2 Ω C n+1 C 2 C 2 C 2 C 2 C 4 C 3 C 2 1 1 1
Ŵ a (Φ) a1 a2
The order of the finite Weyl group 2a1 a2 Wa=L(Phi^):W(Phi) => W(Phi) = 8
The order of the finite Weyl group Φ irreducible crystallographic root system. W a(φ) = L(Φ ) : W (Φ) A is a fundamental domain for the action of W a(φ). P ( ) := { n i=1 λ iα i 0 < λ i < 1} is a fundamental domain for L(Φ ). L(Φ ) is a normal subgroup of W a(φ) with W a(φ)/l(φ ) = W (Φ). Theorem P ( ) is the union of W (Φ) alcoves. (up to a set of measure 0). Write the highest root α = n i=1 c iα i. Then So W (Φ) = n! Ω c 1 c n. vol(p ( ))/ vol(a ) = n! Ω c 1 c n.
The order of the finite Weyl group W (Φ) = n! Ω c 1 c n. Φ c 1,..., c n Ω W (Φ) W (Φ) A n 1, 1,..., 1 n + 1 (n + 1)! S n+1 B n 1, 2,..., 2 2 2 n n! C 2 S n C n 2,..., 2, 1 2 2 n n! C 2 S n D n 1, 2,..., 2, 1, 1 4 2 n 1 n! C n 1 2 : S n E 6 1, 2, 2, 3, 2, 1 3 2 7 3 4 5 S 4 (3) : 2 E 7 2, 2, 3, 4, 3, 2, 1 2 2 10 3 4 5 7 C 2 S 6 (2) E 8 2, 3, 4, 6, 5, 4, 3, 2 1 2 14 3 5 5 2 7 2.O + 8 (2).2 F 4 2, 3, 4, 2 1 2 7 3 2 1152 G 2 3, 2 1 2 2 3 D 12