Subsystems of a Root System (E 8 )

Transcription

1 Subsystems of a Root System (E 8 ) Toshio OSHIMA Graduate School of Mathematical Sciences University of Tokyo Geometry and Representations in Lie Theory August, 007 Tambara Institute of Mathematical Sciences

2 . A classification of root systems Definition. G : Affine Dynkin diagram A diagram consists of vertices and lines/arrows connecting vertices. Each arrow has a stem with multiple lines. Every vertex has a value of a positive real number. The minimal real number in the connected component of G equals. Fix any vertex P and suppose P has a value m. Let L,...,L p be lines/arrows connecting P to vertices Q,...,Q p with values m,...,m p, respectively. Then m = k m + + k p m p. () Here k j L j is an arrow with k j -tuple lines pointing toward P. Theorem. The connected affine Dynkin diagrams is G( Ψ), G( Ψ ) or G( BC n ) with the fundamental systems Ψ of an irreducible root system (by adding one root in the closure of the negative Weyl chamber)

3 à n B n à B = C C n Dn D 4 Ẽ 6 3 Ẽ Ẽ F G 3 B n C n F 4 3 G BC n BC C = B

4 Ã n B n (n ) : D n B n BC n (n 3) : D n+ C n C n (n 3) : D 4 G : Ẽ 6 F 4 : Ẽ 7 F 4 : D4 BC : 3 Ẽ 6 G : Ẽ 8 : R = R () (R = A n, B n, C n, D n, E 6, E 7, E 8, F 4, G ), B n = D () n+, C n = A () n, BC n = A () n, F 4 = E () 6, G = D (3)

5 Allow lines/arrows connecting the same vertices Allow infinite number of vertices A + 3 A D B C

6 E 8 α α 3 α 4 α 5 α 6 α 7 α 8 α α The numbers indicate the linear relation of the roots. {α,α 3,α 4,...,α 8,α 0 }:TypeD 8 α = ɛ + ɛ, α j = ɛ j ɛ j (3 j 8), α 0 = ɛ 7 ɛ 8, This linear relation α = (α 0 +3α +4α 3 +6α 4 +5α 5 +4α 6 +3α 7 +α 8 ) = (ɛ + ɛ 8 ) (ɛ + ɛ 3 + ɛ 4 + ɛ 5 + ɛ 6 + ɛ 7 ), = { ±(ɛ i ɛ j ), ±(ɛ i + ɛ j ), 8 k= ( )ν(k) ɛ k ; i<j 8, 8 k= ν(k) is even}, # = 40.

7 . Subsystems of a root system : a (reduced) root system of rank n (a finite subset of R n \{0}) ( i) s α () = ( α ) sα (x) :=x (α x) α, x Rn) (α α) ii) (α β) Z ( α, β ) (α α) iii) α Rα = Rn iv) Rα = {±α} ( α ) F W F := s α ; α F O(n) W := W : the Weyl group of F := W F F F : a subsystem of F = F Ξ, :root systems Hom(Ξ, ) := { ι :Ξ ; (ι(α) ι(β)) (ι(α) (ι(α)) =(α β) ( α, β Ξ) } (α α) Hom(Ξ, ) := W \ Hom(Ξ, ) Aut() := Hom(, ), Out() := Hom(, ) Ξ (isomorphic) def ι Hom(Ξ, ) such that ι(ξ) =

8 Ξ, Ξ : subsystems of Ξ Ξ : (equivalent by ) w W such that w(ξ) = Ξ Ξ w Ξ : (weakly equivalent) g Aut() such that g(ξ) = Ξ Q. For given root systems and, Hom(, )? Q. Ξ Ξ Ξ Ξ? If not, how to distinguish them? Q3. #{F ; F Ξ}? Q4. Does Aut(Ξ) come from W? (ex. orthogonal systems) Q5. σ Aut(Ξ) : stabilizes each irreducible component of Ξ σ is realized by an element of W? Q6. Ξ = σ(ξ) with σ Out() Ξ Ξ? Q7. #{Θ Ψ; Θ Ξ}? (Ψ: a fundamental system of ) Q8. Hom(Ξ, ), Out()\Hom(Ξ, ), Hom(Ξ, )/Out(Ξ), Hom(Ξ, )/Out (Ξ) Q9. Closures of Ξ ( -, S-, L-, fundamental)?

9 Aut (Ξ) := Aut(Ξ ) Aut(Ξ m) Aut(Ξ) (Ξ =Ξ + +Ξ m) Ξ := {α ; α Ξ} Ξ: -closed def Ξ=(Ξ ) Ξ: L-closed def α Ξ Rα =Ξ Ξ: S-closed def α Ξ, β Ξ, α + β α + β Ξ def Ξ:fundamental a fundamental system of containing Ξ -closed fundamental L-closed S-closed Q0. Out (Ξ) := {w Ξ ; w W with w(ξ) = Ξ} / W Ξ? Dynkin, Semisimple subalgebras of semisimple Lie algebras, 957. Aslaksen and Lang, Extending π-systems to bases of root systems, Hom(Ξ, ) Lemma. Hom ( Ξ +Ξ, ) ῑ Hom(Ξ,) (ῑ, Hom ( Ξ,ι(Ξ ) )) Lemma Study ι Hom(Ξ, ) and ι(ξ) for irreducible Ξ and.

10 Theorem. i) :classical type or Ξ A m with m. When Ξ D 4 or (Ξ, ) (D 4,D 4 ), Hom(Ξ, ) {Imbeddings ῑ of G(Φ) to G( Ψ) or G( Ψ ) such that ῑ(β 0 )=α 0 or α 0} () β 0 : any root in Φ such that the right hand side of () is not empty and if such β 0 doesn t exit, Hom(Ξ, ) =. When Ξ=D 4, # ( Hom(D 4, )/Out(D 4 ) ) (3) and Hom(D 4, )/Out(D 4 ) is given by the above ῑ #Hom(D 4,B n )=#Hom(D 4,C n )=#Hom(D 4,D n+ )=3 (n 4) In general ι(ξ) α Ψ; α ῑ(φ) (4)

11 ii) :exceptional type and Ξ=R m (= B m,c m,d m,e m,f 4 or G ) m m R 0 :=, 3, 4, 6, 4, (R = B, C, D, E, F, G, respectively) m R : maximal m such that G(R m) G( Ψ) or G( Ψ ) G(R m R ) G(Φ R ) with ΦR Ψ or Ψ When (R m, ) (D 4,F 4 ), (m R =0if it doesn t exist) # ( Hom(R m, )/Out(R m ) ), () 0 (m>m R ), #Hom(R m, ) = #Out(R m R ) (m = m R ), (5) (m R 0 m<m R ), R m =(R m R m R )+ (Φ R ) Ψ (or Ψ ) (m R 0 m m R ) through G(R m ) G(R m R ) G(Φ R ) G( Ψ) (or G( Ψ )) and Rm R m R is given by i) or (4). #Hom(D 4,F 4 )= D 4 F4 L and D 4 F 4 \ F4 L.

12 Examples i) #Hom(A,A n )=and A A n A n 3 (n ). α α α3 α n α n α α α3 α n α n α 0 α 0 ii) #Hom(A,E 6 )=and A E 6 A, #Hom(3A,E 6 )=8. #Hom(A 4,E 6 )=, # ( Out(E 6 )\Hom(A 4,E 6 ) ) =and A 4 E 6 A. α α3 α4 α5 α6 α α 0 α α3 α4 α5 α6 α α 0 α α3 α4 α5 α6 α α 0 iii) #Hom(A 7,E 8 )=# ( Out(E 8 )\Hom(A 7,E 8 )/Out(A 7 ) ) =. α α3 α4 α5 α6 α7 α8 α 0 α α α3 α4 α5 α6 α7 α8 α 0 α A 7 E 8 = A 7 E 8 A

13 \ Ξ A A A 3 A 4 A 5 A 6 A 7 A 8 E 6 () () #Hom : E 7 () 0 E 8 () iv) #Hom(A 4 + A,E 8 )=and # ( Hom(A 4 + A,E 8 )/Out(A 4 + A ) ) =. (A 4 + A ) E 8 A. α α3 α4 α5 α6 α7 α8 α 0 α α3 α4 α or α α3 α4 α α α α3 α4 α5 α6 α7 α8 α 0 α α α3 α4 α5 α6 α or α α3 α4 α5 α6 α v) #Hom(D 4,E n )= α α3 α4 α5 α6 α 0 α α 0 E 6 = D 4 α 0 α α3 α4 α5 α6 α7 α 0 α D 4 E 7 3A α α3 α4 α5 α6 α7 α8 α 0 α α 0 D 4 E 8 D 4

14 vi) Hom(mA,E 8 ), Hom(mA,E 7 ) α α3 α4 α5 α6 α7 α8 α 0 α3 α4 α5 α6 α7 ɛ 7 ɛ 8 α α ɛ 7 ɛ 8 E 8 E 7 α3 α4 α5 α7 α ɛ 3 ɛ 4 ɛ 7 ɛ 8 ɛ 5 ɛ 6 E 7 D 6 D 4 + A 4A ɛ 6 ɛ 5 3A A A D 4 3A ( k 3) (k =4) 5 (k =5) #Hom(kA,E 8 )=#Hom((k )A,E 7 )= 5 (k =6) 30 (7 k 8) 0 (k>8) Out D8 (8A ) Out E8 (8A ) Out(8A ) S 8, #Out E8 (8A )= 8! 30 = 344

15 4. Duality (Ξ, Ξ ):a dual pair in def Ξ, Ξ, Ξ =Ξ, Ξ =Ξ def A dual pair (Ξ, Ξ ) of is special Out(Ξ ) {w Ξ ; w W,w(Ξ )=Ξ }/W Ξ Out(Ξ ) Theorem 3. A dual pair (Ξ, Ξ ) in is special #Hom(Ξ, ) = and Out(Ξ ) Out(Ξ ) #Hom(Ξ, ) = #Hom(Ξ, ) = Example. (, Ξ, Ξ ):(Ξ, Ξ ) are special dual pairs in. (D m+n,d m,d n ) (m, n, m 4,n 4), (E 6,A 3, A ), (E 7,A 5,A ), (E 7,A 3 + A,A 3 ), (E 7, 3A,D 4 ), (E 8,E 6,A ), (E 8,A 5,A + A ), (E 8,A 4,A 4 ), (E 8,D 6, A ), (E 8,D 5,A 3 ), (E 8,D 4,D 4 ), (E 8,D 4 + A, 3A ), (E 8, A, A ), (E 8,A 3 + A,A 3 + A ), (E 8, 4A, 4A ), (F 4,A,A ). is E 8 all the dual pairs are special!

16 Result. UsingTheoremand3,wegetTables which answer Q Q0. A classification of subsystems of a root system, 47pp. or math.rt/06904 Ξ \ E 6 E 7 E 8 F 4 G equivalent (isomorphic) 0 (0) 46 (40) 76 (7) 36 () 6(4) S-closed L-closed ( -closed) 6 (7) 3 (3) 40 (8) (9) 3(3) Ξ = (rank Ξ = rank ) 0 (3) 9 (7) 33 (3) 0 (6) 4(4) maximal (S-closed) 3(3) 4(4) 5(5) 3(3) 3() dual pairs (special) 3() 6(3) () 5(4) ()