A CLASSIFICATION OF SUBSYSTEMS OF A ROOT SYSTEM

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1 A CLASSIFICATION OF SUBSYSTEMS OF A ROOT SYSTEM TOSHIO OSHIMA Abstract. We classify isomorphic classes of the homomorphisms of a root system Ξ to a root system Σ which do not change Cartan integers. We examine several types of isomorphic classes defined by the Weyl group of Σ, that of Ξ and the automorphisms of Σ or Ξ etc. We also distinguish the subsystem generated by a subset of a fundamental system. We introduce the concept of the dual pair for root systems which helps to study the action of the outer automorphism of Ξ on the homomorphisms. Contents. Introduction. Notation 3 3. A theorem 7 4. Lemmas 3 5. Proof of the main Lemma 5 6. Dual pairs and closures 9 7. Making tables 8. Some remarks 6 9. List of irreducible root systems Tables 38 References 48. Introduction Root systems were introduced by W. Killing and E. Cartan for the study of semisimple Lie algebras and now they are basic in several fields of mathematics. In this note a subsystem of a root system means a subset of a root system which is stable under the reflections with respect to the roots in the subset. The purpose of this note is to study subsystems of a root system. It is not difficult to classify the subsystems if the root system is of the classical type but we do it in a unified way. The method used here will be useful in particular when the root system is of the exceptional type. Let Ξ and Ξ be subsystems of a root system Σ. We define that Ξ is equivalent to Ξ by Σ and we write Ξ Ξ if w(ξ) = Ξ with an element w of the Weyl group Σ W Σ of Σ. By the classification in this note we will get complete answers to the following fundamental questions (cf. Remark 0. for the answers). Q. What kinds of subsystems of Σ exist as abstract root systems? Q. Suppose Ξ is isomorphic to Ξ as abstract root systems, which is denoted by Ξ Ξ. How do we know Ξ Ξ? Σ Q3. How many subsystems of Σ exist which are equivalent to Ξ? 000 Mathematics Subject Classification. 7B0. Key words and phrases. root system, Weyl group.

2 TOSHIO OSHIMA Q4. Does the outer automorphism of Ξ come from W Σ? Q5. Suppose σ is an outer automorphism of Ξ which stabilizes every irreducible component of Ξ. Is σ realized by an element of W Σ? Q6. Suppose that Ξ is transformed to Ξ by an outer automorphism of Σ. Is Ξ Ξ valid? Σ Q7. Is Ξ equivalent to a subsystem Θ generated by a subset Θ of a fundamental system Ψ of Σ? How many elements exist in {Θ Ψ; Θ Ξ}? Σ For example, Q4 may be interesting if Ξ has irreducible components which are mutually isomorphic to each other. An orthogonal system is its typical example (cf. Remark 8.). The first question of Q7 is studied by [] and the answer is given there when Ξ is irreducible (cf. Remark 8.3 iii)). To answer these questions we will study subsystems as follows. Let Ξ and Σ be reduced root systems and let Hom(Ξ, Σ) denote the set of maps of Ξ to Σ which keep the Cartan integers (α β) (β β) invariant for the roots α and β. Since the map is injective and its image is a root system, the image is a subsystem of Σ isomorphic to Ξ. Let W Ξ and W Σ denote the Weyl groups of Ξ and Σ respectively and put Aut(Ξ) = Hom(Ξ, Ξ) and Aut(Σ) = Hom(Σ, Σ). We will first study the most refined classification, that is, W Σ \Hom(Ξ, Σ) after the review of the standard materials for rootsystems in. In 3 we will give Theorem 3.5 which reduces the structure of W Σ \Hom(Ξ, Σ) to a simple graphic combinatorics in the extended Dynkin diagrams. It is a generalization of the fact that an element of W Σ \Aut(Σ) corresponds to a graph automorphism of the Dynkin diagram associated to Σ (cf. Example 3.6) and will be proved in 5 after the preparation in 4. In 6 we define the dual pair of subsystems, which helps us to study the action of Aut(Ξ) on Hom(Ξ, Σ). In 0 we have the table of all the non-empty Hom(Ξ, Σ) with irreducible Σ. The table gives the numbers of the elements of the cosets W Σ \Hom(Ξ, Σ), Aut(Σ)\Hom(Ξ, Σ), W Σ \Hom(Ξ, Σ)/Aut(Ξ), W Σ \Hom(Ξ, Σ)/Aut (Ξ) and the number of the subsystems generated by subsets of a fundamental system of Σ which correspond to a coset. Here Aut (Ξ) is the subgroup of Aut(Ξ) defined by the direct product of the automorphisms of the irreducible components of Ξ. The table also determines certain closures of Ξ (cf. Definition 6.3, 6.6). In many cases # ( W Σ \Hom(Ξ, Σ)/Aut(Ξ) ) =, which is equivalent to say that the subsystems of Σ which are isomorphic to Ξ form a single W Σ -orbit. We will also distinguish the orbits when the number is larger than one. In 8 we give some remarks obtained by our study. For example, Q4 will be examined for the orthogonal systems of the root systems of type E 7 and E 8. In 9 we give the extended Dynkin diagrams and roots of the irreducible root systems following the notation in [3], which is for the reader s convenience and will be constantly used in this note. A proof of the classification of the root systems is also given for the completeness (cf. Proposition 9.3 and Remark 9.4 iv)). Dynkin [4] classified regular subalgebras of a simple Lie algebra in his study of semisimple subalgebras. The classification is given by Table 9 and Table in [4]. In Table, A 6 + A and the second one of A 7 + A should be replaced by E 6 + A and E 7 + A, respectively. These tables describe the classification of Aut(Σ)\Hom(Ξ, Σ)/Aut(Ξ) for S-closed subsystems (cf. Definition 6.6) in our table in 0 (cf. Remark 0.7 ii)) and were obtained from Dynkin diagrams given by successive procedures removing roots from extended Dynkin diagrams. The procedure

3 A CLASSIFICATION OF SUBSYSTEMS OF A ROOT SYSTEM 3 is the way to classify maximal S-closed subsystems used by [] (cf. Remark 8.4). The maximal S-closed subsystems are also classified by [8]. Our classification based on Theorem 3.5 gives a more refined classification of W Σ \Hom(Ξ, Σ). In fact we give a simple algorithm to give the complete representatives of the coset W Σ \Hom(Ξ, Σ). The author would like to thank E. Opdam for pointing out (8.8) and related errors in the table in 0.. Notation In this section we review the root systems and fix the notation related to them. All the materials in this section are elementary and found in [3]. Fix a standard inner product ( )ofr n and an orthonormal basis {ɛ,...,ɛ n } of R n.forα R n \{0} the reflection s α with respect to α is defined by (.) and we put α = (α α). s α : R n R n x s α (x) :=x (α x) (α α) α Definition.. A reduced root system of rank n is a finite subset Σ of R n \{0} which satisfies (.) R n = α Σ Rα, (.3) s α (Σ) = Σ ( α Σ), (.4) (α β) (α α) Z ( α, β Σ), (.5) Rα Σ={±α} ( α Σ). In general the rank of a root system Σ is denoted by rank Σ. Remark.. i) In this note any non-reduced root system, which doesn t satisfy (.5), doesn t appear except in 9 and hereafter for simplicity a root system always means a reduced root system. ii) We use the notation N for the set {0,,,...} of non-negative integers. Definition.3. Let Σ be a root system of rank n. Afundamental system ΨofΣ is a finite subset {α,...,α n } of Σ which satisfies (.6) R n = Rα + Rα + + Rα n, n (.7) α = m j (α)α j Σ ( m (α),...,...,m n (α) ) N n or N n. j= The fundamental system Ψ exists for any root system Σ and the root α Σis positive (with respect to Ψ) if m j (α) 0forj =,...,n, which is denoted by α>0. Definition.4. Let Θ be a finite subset of Σ and put (.8) W Θ := s α ; α Θ = the group generated by {s α ; α Θ}, (.9) W := W Σ = W Ψ, (.0) Θ := W Θ Θ, (.) Θ := {α Σ; (α β) =0 ( β Θ)}. The group W is called the Weyl group of Σ. A subset Ξ of Σ is called a subsystem of Σifs α (Ξ) = Ξ for any α Ξ. Then Ξ is a root system with rank Ξ = dim α Ξ Rα.

4 4 TOSHIO OSHIMA We put α = {α} for α Σ. Note that Θ and Θ are subsystems of Σ and (.) rank Θ +rankθ rank Σ. Definition.5. Amapι of a root system Ξ to a root system Σ is a homomorphism if ι keeps the Cartan integers: (.3) (ι(α) ι(β)) (ι(α) ι(α)) =(α β) ( α, β Ξ). (α α) In this case ι is injective and ι(ξ) is a subsystem of Σ. The set of all homomorphisms of Ξ to Σ is denoted by Hom(Ξ, Σ) and define (.4) Aut(Σ) := Hom(Σ, Σ). Note that W Σ and W Ξ naturally act on Hom(Ξ, Σ) and (.5) ι s α = s ι(α) ι (ι Hom(Ξ, Σ), α Ξ). Two homomorphisms ι and ι of Ξ to Σ are W Σ -equivalent if (.6) ι = w ι for a suitable w W Σ and we define (.7) Hom(Ξ, Σ) := W Σ \Hom(Ξ, Σ) W Σ \Hom(Ξ, Σ)/W Ξ, (.8) Out(Σ) := W Σ \Aut(Σ) = Hom(Σ, Σ) Aut(Σ)/W Σ {g Aut(Σ) ; g(ψ) = Ψ}. The root system Ξ is isomorphic to Σ, which is denoted by Ξ Σ, if there exists a surjective homomorphism of Ξ onto Σ. Suppose Σ and Σ are subsystems of Σ such that Σ = Σ Σ and Σ Σ. Then we say that Σ is a direct sum of Σ and Σ, which is denoted by Σ = Σ +Σ. Arootsystemisirreducible if if has no non-trivial direct sum decomposition. Note that every root system is decomposed into a direct sum of irreducible root systems and (.9) Aut(Σ) {g O(n); g(σ) = Σ} if Σ is an irreducible root system of rank n. HereO(n) is the orthogonal group of R n with respect to ( ). For root systems Σ and Σ there exists a root system Σ = Σ +Σ such that Σ j Σ j for j = and. This root system Σ is determined modulo isomorphisms and hence we simply write Σ = Σ +Σ.WhenΣ =Σ, we sometimes write Σ in place of Σ +Σ. For any two elements α and α in Ψ, there exists an isomorphism ι of α, α to one of the following four root systems with the fundamental system {β,β } such that ι(α) =β and ι(α )=β : A + A =A :(β,β )=(ɛ,ɛ ) (β β ) (β β) =0, (β β ) (β β ) =0 β β A :(β,β )=(ɛ ɛ,ɛ ɛ 3 ) (β β ) (β β) =, (β β ) (β β ) = β β B :(β,β )=(ɛ ɛ,ɛ ) (β β ) (β β) =, (β β ) (β β ) = β β G :(β,β )=( ɛ + ɛ + ɛ 3,ɛ ɛ ) (β β ) (β β) =, (β β ) (β β ) = 3 β β The Dynkin diagram G(Ψ) of a root system Σ with the fundamental system Ψ is the graph which consists of vertices expressed by circles and edges expressed by some lines or arrows such that the vertices are associated to the elements of Ψ. The lines or arrows connecting two vertices represent the isomorphic classes of the corresponding two roots in Ψ according to the above Dynkin diagram of rank.

5 A CLASSIFICATION OF SUBSYSTEMS OF A ROOT SYSTEM 5 Here the number of lines which link β to β in the diagram equals The arrow points toward a shorter root. (β β ) min{(β β),(β β )}. Definition.6. Arootα of an irreducible root system Σ is called maximal and denoted by α max if every number m j (α)forj =,...,nin Definition.3 is maximal among the roots of Σ. It is known that the maximal root uniquely exists. Let Ψ = {α,...,α n } be a fundamental system of Σ. Define (.0) (.) α 0 := α max, Ψ :=Ψ {α 0 }. The extended Dynkin diagram of Σ in this note is the graph G( Ψ) associated to Ψ which is defined in the same way as G(Ψ) associated to Ψ. We call Ψ theextended fundamental system of Σ. A subdiagram of G( Ψ) is the Dynkin diagram G(Θ) associated to a certain subset Θ Ψ. In 9 the extended Dynkin diagrams of all the irreducible root systems are listed, which are based on the notation in [3]. The vertex expressed by a circled circle in the diagram corresponds to the special root α 0. If the vertex and the lines starting from it are removed from the diagram, we get the corresponding Dynkin diagram of the irreducible root system. The numbers below vertices α j in the diagram in 9 are the numbers m j (α max ) given by (.7). We define m 0 (α max )=andthen (.) α j Ψ m j (α max )α j =0. Remark.7. i) There is a bijection between the isomorphic classes of root systems and the Dynkin diagrams. The irreducible decomposition of a root system Σ corresponds to the decomposition of its Dynkin diagram G(Ψ) into the connected components G(Ψ j ). It also induces the decomposition of the fundamental system Ψ = Ψ Ψ m such that Σ= Ψ + + Ψ m is the decomposition into irreducible root systems. Then we call each Ψ j an irreducible component of Ψ. The irreducible root systems are classified as follows (cf. 9): (.3) A n (n ), B n (n ), C n (n 3), D n (n 4), E 6, E 7, E 8, F 4,G. We will also use this notation A n,... for a root system or a fundamental system. For example, A +B 3 means a root system isomorphic to the direct sum of the root system of type A and two copies of the root system of type B 3 or it means its fundamental system. ii) Out(Σ) is naturally isomorphic to the group of graph automorphisms of the Dynkin diagram associated to Σ. If Σ is irreducible, it also corresponds to the graph automorphisms of the extended Dynkin diagram which fix the vertex corresponding to α 0. Here we give the list of irreducible root systems Σ with non-trivial Out(Σ): (.4) Out(A n ) Z/Z (n ), Out(D 4 ) S 3 (= the symmetric group of degree 3), Out(D n ) Z/Z (n 5), Out(E 6 ) Z/Z.

6 6 TOSHIO OSHIMA iii) The graph automorphism σ of the extended Dynkin diagram G( Ψ) with the following property corresponds to a transformation by an element of W Σ. A rotation of G( Ψ) (Σ = A n,e 6 ), Any automorphism (Σ = B n, C n, E 7 ), (.5) σ ( (α 0,α,α n,α n ) ) =(α,α 0,α n,α n ) (Σ = D n ), σ ( (α 0,α,α n,α n ) ) =(α n,α n,α,α 0 ) (Σ = D n, n :even 4), σ ( (α 0,α,α n,α n ) ) =(α n,α n,α 0,α ) (Σ = D n, n : odd). When Σ is irreducible, we have the bijection: (.6) {w W Σ ; w( Ψ) = Ψ} {α j Ψ; m j (α max )=} σ σ(α 0 ) To classify subsystems contained in a root system we prepare more definitions. Definition.8. We put (.7) Aut (Ξ) := Aut(Ξ ) Aut(Ξ m ) Aut(Ξ), (.8) Out (Ξ) := Aut (Ξ)/W Ξ for a root system Ξ with an irreducible decomposition Ξ = Ξ + +Ξ m. Definition.9. Let Ξ, Ξ and Θ be subsystems of Σ. (.9) Ξ Ξ w W Θ such that Ξ = w(ξ), Θ (.30) Ξ w Θ Ξ g Aut(Θ) such that Ξ = g(ξ). If Ξ Ξ (resp. Ξ w Ξ ), we say that Ξ is equivalent (resp. weakly equivalent) to Θ Θ Ξ by Θ. Since Aut(Ξ) {ι Hom(Ξ, Σ) ; ι(ξ) = Ξ}, wehave {Ξ Σ; s α (Ξ )=Ξ ( α Ξ )andξ Ξ}/ Σ (.3) (.3) (.33) W Σ \Hom(Ξ, Σ)/Aut(Ξ) Hom(Ξ, Σ)/Out(Ξ), {Ξ Σ; s α (Ξ )=Ξ ( α Ξ )andξ Ξ}/ w Σ Aut(Σ)\Hom(Ξ, Σ)/Aut(Ξ) Out(Σ)\Hom(Ξ, Σ)/Out(Ξ), W Σ \Hom(Ξ, Σ)/Aut (Ξ) Hom(Ξ, Σ)/Out (Ξ). Definition.0 (fundamental subsystems). A subsystem Ξ of Σ is called fundamental if there exists Θ Ψ such that Ξ Σ Θ. Remark.. Suppose Σ is of type A n. Then it is clear that (.34) any subsystem of Σ is fundamental, ( (.35) Ξ Σ Ξ Ξ Ξ ) for subsystems Ξ and Ξ of Σ. Our aim in this note is to clarify the structure of Hom(Ξ, Σ), Out(Σ)\Hom(Ξ, Σ), Hom(Ξ, Σ)/Out(Ξ), Hom(Ξ, Σ)/Out (Ξ), Out(Σ)\Hom(Ξ, Σ)/Out(Ξ) and fundamental subsystems of Σ. For this purpose we prepare the following definition.

7 A CLASSIFICATION OF SUBSYSTEMS OF A ROOT SYSTEM 7 Definition.. i) A root α Ψ(resp. Ψ) is an end root of Ψ (resp. Ψ) if (.36) #{β Ψ(resp. Ψ) ; (β,α) < 0}. Arootα Ψ(resp. Ψ) is called a branching root of Ψ (resp. Ψ) if (.37) #{β Ψ(resp. Ψ) ; (β,α) < 0} 3. The corresponding vertex in the (extended) Dynkin diagram is also called an end vertex or a branching vertex, respectively. ii) When Σ is irreducible, we put (.38) Σ L := {α Σ; α = α max } and denote its fundamental system by Ψ L.ThenΣ L is a subsystem of Σ and A L n = A n,bn L = D n (n ), Cn L = na (n 3), Dn L = D n (n 4), (.39) E6 L = E 6,E7 L = E 7, E8 L = E 8, F4 L = D 4,G L = A. A root system whose Dynkin diagram contains no arrow is called simply laced. 3. Atheorem In this section we will give a simple procedure to clarify the set Hom(Ξ, Σ) := W Σ \Hom(Ξ, Σ) for root systems Ξ and Σ. Remark 3.. i) Note that (3.) (3.) Hom ( Ξ, Σ +Σ ) Hom ( Ξ +Ξ, Σ ) Ξ Ξ: component ῑ Hom(Ξ,Σ) ( Hom ( Ξ, Σ ), Hom ( (Ξ ), Σ ) ), ( ῑ, Hom ( Ξ,ι(Ξ ) )). Here ῑ means a class of ι Hom(Ξ, Σ) in Hom(Ξ, Σ) and the component Ξ of Ξ is the subsystem of Ξ such that Ξ = Ξ +(Ξ ). The empty set and Ξ are also components of Ξ. The identification (3.) follows from (3.3) {w W Σ ; w ι(ξ) = id} = W ι(ξ) W Σ for any ι Hom(Ξ, Σ) (cf. [3]). ii) The identifications (3.) and (3.) assure that we may assume Ξ and Σ are irreducible. In fact, the study of the structure of Hom(Ξ, Σ) is reduced to the study of ῑ Hom(Ξ, Σ) and ι(ξ) for irreducible Ξ and Σ. iii) We may moreover assume ι(ξ) Σ L by considering the dual root systems Ξ := { α (α α) ; α Ξ} and Σ := { α (α α) ; α Σ} in place of Ξ and Σ, respectively. Definition 3.. When G(Φ) is isomorphic to a subdiagram G(Θ) of G( Ψ) with a map ῑ :Φ Θ Ψ, it is clear that ῑ defines an element of Hom(Ξ, Σ). In this case we say that ῑ is an imbedding of G(Φ) into G( Ψ). Recalling Definition.4,.6 and., we now state a main lemma in this note, which will be proved in 5 by using lemmas in 4. Lemma 3.3. Let Ξ and Σ be irreducible root systems and let Φ and Ψ be their fundamental systems, respectively. Denoting (3.4) Hom (Ξ, Σ) := { ι Hom(Ξ, Σ) ; ι(ξ) Σ L }, (3.5) Hom (Ξ, Σ) := W Σ \Hom (Ξ, Σ), we have the following claims according to the type of Ξ:

8 8 TOSHIO OSHIMA ) Ξ is of type A m. Hom (Ξ, Σ) { Imbeddings ῑ of G(Φ) into G( Ψ) } with the end vertex α 0. Let ῑ be this graph imbedding corresponding to ι Hom(Ξ, Σ). Then (3.6) ι(ξ) α Ψ; α ῑ(φ). In the case #Hom (Ξ, Σ) >, we have #Hom (Ξ, Σ) = 3 if (Ξ, Σ) is of type (A 3,D 4 ) and if otherwise. Moreover for ῑ, ῑ Hom (Ξ, Σ) ῑ and ῑ are conjugate under an element of Out(Σ) or Out(Ξ) (3.7) ι(ξ) ι (Ξ). ) Ξ is of type D m (m 4). Let Φ m = {β 0,...,β m } be a fundamental system of Ξ with the Dynkin diagram β 0 β β β m 3 β m and m Σ denote the maximal integer m such that there β m is an imbedding ῑ m of G(Φ m ) into G( Ψ L ). We put m Σ =0if such an imbedding doesn t exist. Then 0 (Σ is of type A n, C n, G ), rank Σ (Σ is of type B n, D n, E 8, F 4 ), m Σ = 5 (Σ is of type E 6 ), 6 (Σ is of type E 7 ) and Hom (D m, Σ) (4 )m m Σ # ( Hom (Ξ, Σ)/Out(Ξ) ) =. Σ is of type E 6, E 7 or E 8. { (m = m Σ ), #Hom(D m, Σ) = (4 m<m Σ ), { ι(ξ) A (n =7), D mσ m + (n =6, 8). Σ is of type D n, B n or F 4 (m n). 6 (Σ : D 4 (m = n =4)), #Hom 3 (Σ : B n and D n (m =4<n)), (Ξ, Σ) = (Σ : D n (4 <m= n)), (Σ : F 4 (4 = m), B n (4 <m n), D n (4 <m<n)), D n m (ι Hom(D m,d n )), ι(ξ) B n m (ι Hom(D m,b n )), (ι Hom(D 4,F 4 )). 3) Ξ is of type B m (m ). Σ is of type B n with m n, Hom(Ξ, Σ) #Hom(Ξ, Σ) = and Σ is of type C n with m =, Σ is of type F 4 with m 4, ι(ξ) T n T n m (T = B, C, F, F = B, F = A and ι(b 3 ) F 4 Σ L ).

9 A CLASSIFICATION OF SUBSYSTEMS OF A ROOT SYSTEM 9 4) Ξ is of type C m (m 3). { Σ is of type C n with m n, Hom(Ξ, Σ) #Hom(Ξ, Σ) = and Σ is of type F 4 with m 4, ι(ξ) T n T n m (T = C, F, F = C,F = A and ι(c 3 ) F 4 Σ L ). 5) Ξ is of type E m (m =6,7and8). Hom(Ξ, Σ) { Imbeddings ῑ of G(Φ) into G( Ψ) }/, ι(ξ) α Ψ; α ῑ(φ). Here / is interpreted that all the imbeddings of G(Φ) are considered to be isomorphic except for (Ξ, Σ) (E 6,E 6 ).Namely#Hom(Ξ, Σ) if (Ξ, Σ) (E 6,E 6 ). 6) Ξ is of type G or F 4. Hom(Ξ, Σ) #Hom(Ξ, Σ) = and Ξ Σ. Remark 3.4. i) In the proof of Lemma 3.3 ) we will have { ῑmσ ι(ξ) (Φ mσ ) Ψ (m Σ m m Σ ), ῑmσ (Φ mσ ) Ψ, ῑ mσ (β m ),...ῑ mσ (β mσ ) (4 m m Σ ) for the imbedding ῑ mσ with ῑ(β mσ )=α 0 if Σ is of type D n, E 6, E 7 or E 8. Let Θ m be a subset of Ψ such that Θ m D m. If Σ is of type B n or D n,we may assume that ι m Hom(D m, Σ) satisfies ι m (Ξ) = Θ m and then (3.8) ι m (Φ m ) = Θ m Ψ. Suppose Σ is of type E 6, E 7 or E 8.Let α max be the maximal root of Θ mσ.put α 0 = α max and Θ mσ =Θ mσ { α 0 }. We may assume ι m Hom(D m, Σ) satisfies ι m (Ξ) = Θ m and Θ m Θ mσ.then (3.9) ι m (Φ m ) = Θ m Θ mσ, Θ m Σ Ψ. Note that G( Θ mσ ) is the extended Dynkin diagram of Θ mσ D mσ.seeexample 3.6 viii) and ix). ii) Using a graph automorphism of G( Ψ) corresponding to a suitable element of W Σ,wemayreplaceα 0 by another element α j of Ψ with m j (α max )=in Theorem 3.5 and in the remark above (cf. Remark.7 ii)). iii) The image ι(ξ) corresponding to the graph automorphism ῑ in Lemma 3.3 is obtained by Proposition 4.4. Lemma 3.3 can be summarized in the following form. Theorem 3.5. Let Σ and Ξ be irreducible root systems and let Ψ and Φ be their fundamental systems, respectively. Retain the notation given in Definition.4.6 and.. If Σ is not simply laced, we denote the maximal root in Σ\Σ L by α max and the Dynkin diagram of Ψ by G( Ψ ).Hereweputα 0 = α max and Ψ =Ψ {α 0}. i) Suppose Σ isoftheclassicaltypeorξ A m with m. When Ξ D 4 or (Σ, Ξ) (D 4,D 4 ), Hom(Ξ, Σ) {Imbeddings ῑ of G(Φ) to G( Ψ) or G( Ψ ) (3.0) such that β 0 corresponds to α 0 or α 0 by ῑ} for a suitable root β 0 Φ. Here we delete G( Ψ ) and α 0 in the above if Σ is simply laced. Moreover β 0 is any root in Φ such that the right hand side of (3.0) is not empty and if such β 0 doesn t exit, Hom(Ξ, Σ) =. When Ξ D 4, (3.) # ( Hom(Ξ, Σ)/Out(Ξ) )

10 0 TOSHIO OSHIMA and the representative of Hom(Ξ, Σ)/Out(Ξ) is given by the above imbedding ῑ and (3.) #Hom(D 4,B n )=#Hom(D 4,C n )=#Hom(D 4,D n+ )=3 (n 4). For ι Hom(Ξ, Σ) corresponding to this imbedding ῑ of G(Φ) we have (3.3) ι(ξ) = α Ψ; α ῑ(φ). Moreover for ῑ, ῑ Hom(Ξ, Σ) ῑ and ῑ are conjugate under an element of Out(Σ) or Out(Ξ) (3.4) ι(ξ) Σ L ι (Ξ) Σ L and ι(ξ) ι (Ξ). ii) Suppose Σ is of the exceptional type and Ξ=R m with R = B, C, D, E, F and G. Put m R 0 =, 3, 4, 6, 4 and according to R = B, C, D, E, F and G, respectively, and moreover suppose m m R 0. Let mr Σ be the maximal number m such that the Dynkin diagram G(R m ) of the root system R m is a subdiagram of G( Ψ) or G( Ψ ). Thus for a subset Φ R Σ of Ψ or Ψ we identify G(R m R Σ ) with the subdiagram G(Φ R Σ ).PutmR Σ =0ifsuchanumberm with m mr 0 does not exists. When (Σ,R m ) (F 4,D 4 ), we have (3.) and 0 (m>m R Σ ), (3.5) #Hom(R m, Σ) = #Out(R m R Σ ) (m = m R Σ ), (m R 0 m<m R Σ ), (3.6) R m Σ=(R m R m R Σ )+ (Φ R Σ ) Ψ (or Ψ ) (m R 0 m mr Σ ) through the natural map G(R m ) G(R m R Σ ) G(Φ R Σ ) G( Ψ) (or G( Ψ )) and Rm R m is given by i) or Lemma 3.3 5). The coset Hom(D 4,F R Σ 4 ) consists of the two elements corresponding to the identifications D 4 F4 L and D 4 F 4 \ F4 L. Proof. When Σ is of type R with R = B n, C n, F 4 or G, G( Ψ ) is the affine Dynkin diagram R given by Proposition 9.3. This theorem follows from Lemma 3.3, Remark 3. iv), Remark 9.4 iii) and Remark 4.. Example 3.6. ( Hom(Ξ, Σ) and Ξ ) i) #Hom(A,A n )=anda A n A n 3 (n ). Two elements of Hom(A,A n ) are defined by (α,α ) (α 0,α )and(α,α ) (α 0,α n ), respectively. They are isomorphic to each other under Out(A ). Note that the rotation of the extended Dynkin diagram corresponds to an element of W An. α α α3 α n α n α α α3 α n α n α 0 α 0 ii) #Hom(A 3,D 4 )=3,# ( Out(D 4 )\Hom(A 3,D 4 ) ) =anda 3 D 4 =. The group Out(D 4 ) S 3 corresponds to that of the graph automorphisms of the extended Dynkin diagram which fix α 0. α α 4 α α 3 α 0 α α 4 α α 3 α 0 α α 4 α α 3 α 0 iii) #Hom(A 3,D n )=# ( Out(D n )\Hom(A 3,D n )/Out(A 3 ) ) =forn>4. α n α α α3 α4 α5 α n α 0 α n α n α α α3 α4 α5 α n α 0 α n

11 A CLASSIFICATION OF SUBSYSTEMS OF A ROOT SYSTEM A 3 D n D n 3 or D n 4 according to the imbeddings A 3 D n. iv) #Hom(A,E 6 )=anda E 6 A.Then3A E 6 and #Hom(3A,E 6 )= #Hom(A, A )=8(cf. 8..5). α α3 α4 α5 α6 α α 0 v) #Hom(A 4,E 6 )=,# ( Out(E 6 )\Hom(A 4,E 6 ) ) =anda 4 E 6 A. α α3 α4 α5 α6 α α 0 α α3 α4 α5 α6 α α 0 vi) #Hom(A 5,E 7 )=# ( Out(E 7 )\Hom(A 5,E 7 )/Out(A 5 ) ) =. α 0 α α3 α4 α5 α6 α7 α α 0 α α3 α4 α5 α6 α7 α A 5 E 7 A or A according to the imbeddings A 5 E 7. vii) #Hom(4A,D 4 )=6and# ( Out(D 4 )\Hom(4A,D 4 ) ) =. α α 4 α α 3 α 0 α 0 = {±α, ±α 3, ±α 4 } viii) #Hom(D 4,E n ) = (cf. Remark 3.4 i)) α α 3 α 4 α 5 α 6 α 0 α α 0 E 6 = D 4 α 0 α α3 α4 α5 α 6 α 7 α 0 α D 4 E 7 3A α α 3 α4 α5 α 6 α 7 α8 α 0 α α 0 D 4 E 8 D 4 ix) #Hom(D 5,D 9 )=#Hom(D 5,B 9 ) = (cf. Remark 3.4 i)). α α α3 α4 α5 α6 α7 α8 α 0 α 9 D 5 D 9 D 4 α α α3 α4 α5 α6 α7 α8 α 9 α 0 D 5 B 9 B 4 x) #Hom(A,F 4 )=,(A L ) F 4 A S,(AS ) F 4 A L with AS = A \ A L. α 0 α α α3 α4 α α α3 α4 α 0 xi) #Hom(C 3,F 4 )=,G(C 4 ) G( F 4), G(C 3 ) G( C 4 )andc 3 F 4 = A L. α α α3 α4 α 0 α 0 C 4 = {α,α 3,α 4,α 0 },C 3 = {α 4,α 0, α 0}. xii) #Hom(A 4 + A,E 8 )=and# ( Hom(A 4 + A,E 8 )/Out(A 4 + A ) ) =. Putting (Ξ, Ξ, Σ) = (A 4,A,E 8 )(resp.(a,a 4,E 8 )) in the identification (3.), we have the first (resp. second) line of diagrams below. These two reductions lead

12 TOSHIO OSHIMA to the same result. In particular (A 4 + A ) E 8 A. Note that (A 4,A 4 )and (A,E 6 ) are special dual pairs in E 8 (cf. Definition 6.3). α α3 α4 α5 α6 α7 α8 α 0 α α α3 α4 α5 α6 α7 α8 α 0 α α α3 α4 α α α3 α4 α5 α6 α or or α α3 α4 α α α3 α4 α5 α6 α Corollary 3.7. i) Suppose Σ is not of type A. Let G({α 0,α j,...,α jm }) be a maximal subdiagram of G( Ψ) isomorphic to G(A m ) such that α 0 and α jm are the end vertices of the subdiagram and α jν are not the branching vertex of G( Ψ) for ν =,...,m. Then #Hom (A k, Σ) = (k =,...,m), { ( #Hom =0 αjm is not a branching vertex of G( (A m+, Σ) ( > αjm is a branching vertex of G( Ψ) ) with (Σ = B n, n 3), (Σ = B, C n ), (Σ = D n, n 4), m = 3 (Σ = E 6 ), 4 (Σ = E 7 ), 6 (Σ = E 8 ), 3 (Σ = F 4 ), (Σ = G ). Here α jm is the branching vertex if Σ=B n (n 3), D n (n 4), E 6, E 7 or E 8. ii) We consider the following procedure for a Dynkin diagram X: If X is connected, we replace it by the subdiagram X of the extended Dynkin diagram X of X where the vertices of X correspond to the roots orthogonal to the maximal root of X. If an irreducible component of X has no root with the length of the maximal root, we remove the component. If X is not connected, we choose one of the connected component of X and change the component by the above procedure. Then Hom (ra, Σ) corresponds to the totality of r steps of the above procedures starting from G(Ψ). The existence of these steps implies Hom (ra, Σ) and in this case #Hom (ra, Σ) = if and only if any non-connected Dynkin diagram does not appear except for the final step. In particular, we have the following: Let r(σ) be the maximal integer r satisfying Hom (ra, Σ). Then (3.7) r(σ) = + r(σ j ). j Here {Σ j } is the set of irreducible components of α 0 such that Σ j ΣL and r(a n )=+r(a n )=[ n+ ] (n ), r(a )=, r(a 0 )=0, r(b n )=+r(b n )=[ n ] (n 4), r(b 3)=r(B )=, r(c n )=+r(c n )=n (n 3), r(c )=, r(d n )=+r(d n )=[ n ] (n 4), r(d 3)=r(D )=, r(e 6 )=+r(a 5 )=4, r(e 7 )=+r(d 6 )=7, r(e 8 )=+r(e 7 )=8, r(f 4 )=+r(c 3 )=4, r(g )=.

13 A CLASSIFICATION OF SUBSYSTEMS OF A ROOT SYSTEM 3 Remark 3.8. i) If Σ is of type A, D or E, thenhom(ra, Σ) is figured as follows according to the procedures in Corollary 3.7 ii) and the notation in 9. A n A n A n 4 A n 6 D n D n + A ɛ + +ɛ 4 ɛ 5+ɛ 6+ɛ 7 ɛ 8 α 0 ɛ 4 ɛ E 6 A 5 A 3 A ɛ 3 ɛ ɛ 7 ɛ 8 E 8 E 7 ɛ3 ɛ 4 ɛ 7 ɛ 8 ɛ 5 ɛ 6 E 7 D 6 D 4 + A 4A ɛ 6 ɛ 5 D 4 3A 3A A A There appear the subsystems 3A of E 7 twice in the above. They are distinguished by the structure of (3A ) E 7 but they are in the same W E8 -orbit under the above inclusion E 7 E 8 (cf. 7., 7.3 and 8..3). For example, it follows from the procedures shown above that #Hom(5A,E 7 )=#Hom(4A,D 6 )=#Hom(3A,D 4 + A ) (3.8) =#Hom(A,D 4 )+#Hom(A, 4A ) =#Hom(A, 3A )+4#Hom(A, 3A )=3+4 3=5. ii) For an irreducible root system Σ, we can easily calculate #Hom(Ξ, Σ) and Ξ Σ for any root system Ξ in virtue of Theorem 3.5 together with Remark 3. (cf. Example 3.4 x)). The complete list for non-trivial Hom(Ξ, Σ) is given in 0. More refined structures related to the actions of Out(Σ) and Out(Ξ) etc. are also given in 0, which will be studied in later sections. 4. Lemmas In this section we prepare some lemmas to prove Lemma 3.3 and we always assume that Ψ is a fundamental system of an irreducible root system Σ and Ψ is the corresponding extended fundamental system. Firstnotethatforα Ψ Σ L we have { (4.) (α β) (α α) {0, } ( β Ψ \{α} and β>0), {0, } ( β Ψ \{α} and β<0). Here we put Ψ \{α} =Ψifα/ Ψ. Lemma 4.. If a subset Θ of Ψ contains α 0 and the diagram G(Θ) is connected, (4.) Θ = α Ψ; α Θ. Proof. Note that (α i α j ) 0for0 i<j n. We will prove the lemma by the induction on #Θ. We may put Θ = {α 0,α,...,α m } and we may assume Θ := {α 0,α,...,α m } is empty or forms a connected subdiagram. Let α = n j= m j(α)α j Θ with m j (α) 0. Then the induction hypothesis for Θ implies m j (α) =0forj m and 0=(α m α) = n j=m+ m j(α)(α m α j ). Hence (α m α j ) 0meansm j (α) =0. Remark 4.. In Lemma 4. we may replace α 0 by any element α 0 satisfying (α 0 α) 0 for all α Ψ. Then the Dynkin diagram G( Ψ )of Ψ =Ψ {α 0 } is an affine Dynkin diagram in Proposition 9.3.

14 4 TOSHIO OSHIMA Lemma 4.3. Fix Θ Ψ and m Z #Θ \{0}. Define the map p Θ : Σ Z #Θ β = α m i Ψ i(β)α i ( m i (β) ) α i Θ Then p Θ (m) ΣL is empty or a single W Ψ\Θ -orbit. Moreover p Θ (m) \ ΣL is also empty or a single W Ψ\Θ -orbit. Proof. Fix 0 m =(m i ) α Θ in the image of p Θ. Let g be the complex simple Lie algebra with the root system Σ and let X α g be a root vector for α Σ. We denote by g Ψ\Θ the semisimple Lie algebra generated by {X α ; α Ψ \ Θ}. Then the space V m := α p Θ (m) CX α g is a g Ψ\Θ -stable subset under the adjoint representation of g, which is an irreducible representation of g Ψ\Θ as is shown in [6, Proposition.39 ii)]. Let π Θ be the orthogonal projection of α Ψ Rα onto α Ψ\Θ Rα with respect to ( ). Put v m = α m i Θ iα i π Θ ( α m i Θ iα i ). Then π Θ (α) =α v m, (π Θ (α) π Θ (α)) = (α α) (v m v m ) ( α p Θ (m)). The set of the weights of the irreducible representation (g Ψ\Θ,V m )isπ Θ (p Θ (m)) and the set of the weights with the longest length is π Θ (p Θ (m) ΣL ). Hence p Θ (m) ΣL is a single W Ψ\Θ -orbit. When p Θ (m) ΣL, we have the last statement in the lemma by combining the above argument with [6, Proposition.37 ii)]. Proposition 4.4. For a proper subset Θ of the extended fundamental system Ψ of Σ we have { ( ) p Ψ\Θ 0 (α 0 / Θ), (4.3) Θ = p Ψ\Θ( {0, ±pψ\θ (α 0 )} ) (α 0 Θ) under the notation in Lemma 4.3. Proof. Note that Θ p ) Ψ\Θ( 0. We assume α0 Ψ because the claim is clear when α 0 / Θ. Then (4.) implies Θ p Ψ\Θ( {0, ±pψ\θ (α 0 )} ).LetΘ 0 be the irreducible component of Θ containing α 0.Since Θ is W Θ\{α0}-invariant, Lemma 4.3 implies that ( ) ( 0 = p {±pψ\θ (α 0 )} ) or p Θ \p Ψ\Θ Ψ\Θ Ψ\Θ( {±pψ\θ (α 0 )} ) Σ L. Hence the proposition is clear if Σ is simply laced or if Θ 0 is of type B n or C n. It is also easy to check p Ψ\Θ( pψ\θ (α 0 ) ) Σ L in any other case when (Ψ, Θ 0 )= (B n,a m ), (B n,d m ), (C n,a )or(f 4,A k )withm n and k 3. Lemma 4.5 (roots orthogonal to the end root). Suppose α is an end root of Ψ with α Σ L. Then the set (4.4) Q = { α = α + m (α)α + m 3 (α)α m n (α)α n Σ L ;(α α )=0 } is empty if Ψ is of type A and it is a single W Ψ α -orbit if otherwise. Proof. We may assume #Ψ >. Then there is a unique β Ψwith(α β) < 0. We may assume β = α and we have Q = { α = α +α + m 3 (α)α m n (α)α n Σ L}. Then Q = if and only if Ψ is of type A. IfΨisnotoftypeA, Lemma 4.3 assures that Q is a single W Ψ\{α,α }-orbit. Note that Ψ α =Ψ\{α,α }..

15 A CLASSIFICATION OF SUBSYSTEMS OF A ROOT SYSTEM 5 Lemma 4.6 (special imbeddings of A and A 3 ). Let Ψ Ψ. If Ψ Ψ,we assume that we can choose α Ψ Σ L with α / Ψ. If Ψ =Ψ,weputα = α 0. Define Q := { β Ψ Σ L ;(β α ) < 0 }, Q := { (β,β ) (Ψ Σ L ) ( Ψ Σ L ); (β,α )=(β α ) < 0 and (β β )=0 }, Θ:= { α Ψ ;(α α ) < 0 }, Θ L := Θ Σ L. Then Θ L is the set of complete representatives of Q /W Ψ \Θ. Moreover if Ψ Ψ, Q /#W Ψ \Θ =#Θ L( #Θ L ) + { α Θ L ; G(Ψ α) is not of type A or not an end root of G(Ψ α) }. Here Ψ α is the irreducible component of Ψ containing α Θ. Proof. Let β Q. It follows from (4.) that there exists α m Ψ satisfying (4.5) β = α m + m j (β)α j, (4.6) α j Ψ \Θ (α m β) < 0. If α m / Σ L,Ψ α m is of type A or C and therefore β of the form (4.5) does not belong to Σ L. Hence α m Θ L and α m W Ψ \Θβ by Lemma 4.3. Let α m, α m Θ L with m m.wehaveα m W Ψ \Θα m and therefore Θ L is the set of complete representatives of Q /W Ψ \Θ. Let (β,β) Q. We may assume β = α k Θ L by the argument above and β is of the form (4.5) with α m Θ L. If k m, we may similarly assume β = α m and (α k,α m ) Q. α k α α m α α m α α m k=m k=m α p α q Suppose k = m. Ifα m is the end root of Ψ αm, it follows from Lemma 4.5 that Ψ α m is not of type A and (α k,β) corresponds to a unique element of Q /W Ψ \Θ. If α m is not the end root of Ψ α m,ψ α m is of type A and it is easy to see that (α k,β) also corresponds to a unique class in Q /W ψ \Θ. In fact, we may put {α Ψ αm ;(α α m ) < 0} = {α p,α q } and β = α m + α p + α q + m j (β)β Σ. α j Ψ \{α m,α p,α q} Note that the roots β with this expression are in a single W Ψ \{α m,α p,α q}-orbit. Thus we have the lemma. 5. Proof of the main Lemma Retain the notation in Lemma 3.3 to prove it. ) Let Ξ be of type A m+ with the fundamental system Φ = {β 0,...,β m } and the Dynkin diagram β 0 β β m β m. Firstnotethatῑ naturally corresponds to an element of Hom (Ξ, Σ) and then (3.6) follows from Lemma 4.. We will prove the lemma by the induction on m. Let ι Hom (Ξ, Σ). Since {α Σ; α = α max } = W Σ α max, the lemma is clear when m = 0. Suppose m. By the induction hypothesis we may assume

16 6 TOSHIO OSHIMA that there exists a unique sequence (α 0,...,α m )ofelementof Ψ and an element w W Σ such that w ι(β j )=α j for j =0,...,m. w ι : β 0 β α 0 β m β m α αm Put α m = w ι(β m )and Ψ = {α Ψ; (α α j )=0 (j =0,...,m )}, Θ={α Ψ ;(α α m ) < 0}. Since (α m α j ) = 0 for j =0,...,m, α m Ψ. Applying Lemma 4.6 to α := α m,wehaveα m Θ Σ L and w W Ψ \Θ such that w (α m)=α m. Hence w w ι corresponds to a required imbedding of G(Φ) into G(Ψ). The uniqueness of α m Θ Σ L is proved as follows. Suppose there exists w W Σ such that wα j = α j for j =0,...,m and wα m Θ Σ L. Then w W Ψ \Θ and Lemma 4.6 assures wα m = α m. Thus we have proved the first claim and then Lemma 4. assures (3.6). The last claim is easily obtained by applying the claims we have proved to the extended Dynkin diagrams in 9. ) Let Ξ is of type D m with m 4. We may assume that Σ is of type B n, D n, E n or F n. Let ι Hom (Ξ, Σ). Lemma 3.3 ) assures that there exists a unique sequence α 0,α j,...,α jm 3 in Ψ and an element w W Σ such that (5.) w ι(β ν )=α jν (ν =0,...,m 3) with j 0 =0. Putting Ψ = { α Ψ, (α α jν )=0 (ν =0,...,m 4) }, Θ= { α Ψ ;(α, α jm 3 )) < 0 }, α = α jm 3, (β,β )= ( w ι(β m ),w ι(β m ) ), β m 4 β m 3 β m β m we have β, β Ψ and we can apply Lemma 4.6 as in the case when Ξ is of type A. Thus #W Σ \ { ι Hom (Ξ, Σ) ; w W Σ such that (5.) is satisfied. } = ( #(Θ Σ L ) )( #(Θ Σ L ) ) +# { α Θ Σ L : the irreducible component of Ψ containing α is not of type A or α is not an end vertex of the component }. Hence Hom (D m, Σ) = if Σ is of type A n, C n or G or m>rank Σ. Moreover we have #Hom(D m, Σ) shown in the following table under the notation in 0.

17 A CLASSIFICATION OF SUBSYSTEMS OF A ROOT SYSTEM 7 Σ Ξ Ψ # D 4 D 4 {α,α 3,α 4 } 3A 6 D 5 D 4 Ψ \{α } A + A 3 ( α 3 : not an end root) 3 D n (n 6) D 4 Ψ \{α } A + D n 3 B 4 D 4 Ψ \{α } A + B 3 F 4 D 4 Ψ \{α } C 3 D n (4<m=n) D m {α n,α n } A D n (4<m=n ) D m {α n,α n,α n } A 3 ( α n : not an end root) D n (4<m n ) D m {α m,...,α n } D n m+ B n (4<m n) D m {α m,...,α n } B n m+ E 6 D 4 Ψ \{α } A 5 ( α 4 : not an end root) D 5 {α,α 3,α 5,α 6 } A E 7 D 4 Ψ \{α } D 6 D 5 {α,α 4,...,α 7 } A 5 ( α 4 : not an end root) D 6 {α,α 5,α 6,α 7 } A + A 3 ( α 5 :anendroot) E 8 D 4 Ψ \{α 8 } E 7 D 5 {α,...,α 6 } E 6 D 6 {α,...,α 5 } D 5 D 7 {α,α,α 3,α 4 } A 4 ( α 4 : not an end root) D 8 {α,α,α 3 } A + A Here m Σ is the rank of the maximal subdiagram of type D m containedinthe extended Dynkin diagram of Σ L and then { n (Σ is of type B n or D n ), (5.) m Σ = 5, 6, 8 (Σ is of type E 6, E 7 or E 8, respectively). (5.3) α n α α α3 α4 α n α 0 α α3 α4 α5 α6 α α 0 α n α 0 α α3 α4 α5 α6 α7 α α α α3 α4 α n α n α 0 α α3 α4 α5 α6 α7 α8 α 0 α Fix ι Hom(D 4,F 4 ). Since Hom(D 4,F 4 )isasinglew F4 -orbit, for any g Aut(D 4 )thereexistsw g W F4 with ι g = w g ι. Herew g is uniquely determined by g because rank F 4 =rankd 4. Hence we have Aut(D 4 ) W F4 Aut(B 4 )=W B4 W D4, (5.4) Out(D 4 ) S 3, W B4 /W D4 Z/Z. Let ι : D 4 D n ( B n ) be the natural imbedding given by the realization in 9 and let g Aut(D 4 ) be a non-trivial rotation of G(D 4 ). Then it is easy to see ι g w ι for any w W Bn. Hence if n 4, we have (5.5) # ( Hom(D 4,D n )/Out(D 4 ) ) =# ( Hom(D 4,B n )/Out(D 4 ) ) = because # ( Hom(D 4,D n ) ) =# ( Hom(D 4,B n ) ) = 3 and moreover we have (5.6) D m D n D n m, D m B n B n m for m =4. HereD 0 = D = B 0 =, D A, B A S and A S is the root space of type A such that A S (B n) L =. Note that (5.7) Aut(D n ) W Bn and Out(D n ):=Aut(D n )/W Dn Z/Z (n 5)

18 8 TOSHIO OSHIMA under the natural imbedding D n B n of root spaces. Thus we have (5.8) # ( Hom(D m, Σ)/Out(D m ) ) =0orifm 4 and Σ is irreducible and therefore {ι(d m ) ; ι Hom(D m, Σ)} is a single W Σ -orbit if it is non-empty. Thus we have (5.6) for 4 m n since it does not depend on the imbedding of D m. Let n {6, 7, 8} and put m = m En. There exists ι Hom(D m,e n ) such that ι corresponds to the imbedding of Φ mσ to Ψ withι(β 0 ) = α 0. Then we have D 5 E 6 =, D 6 E 7 A and D 8 E 8 = from Lemma 4.. Moreover there exists ι Hom(D m,e n ) such that ι (D m )= { ι(β 0 ),...,ι(β m 3 ),ι(β m 3 )+ι(β m )+ι(β m ) } and it is clear that D m E n D m E n. Let 4 k 6. Then D k E 8 D k D 8 D 8 k and we can conclude D k E 8 D 8 k because rank(d k E 8) 8 k and there is no root system containing D 8 k as a proper subsystem such that its roots have the same length and its rank is not larger than 8 k. Since D 6 E 7 A and D 4 D 6 A, D 4 E 7 3A and we have D 4 E 7 3A by the same argument as above. Thus we have obtained the claims in the lemma and therefore Remark 3.4 i) is also clear. 3) Suppose Ξ is of type B m with m. Note that for any β Ξ \ Ξ L,thereexistsβ,β Ξ L such that β = (β + β ) and (β β )=0. Henceι Hom(Ξ, Σ) is determined by ι Ξ L. Note that Ξ L is of type D m with D A and D 3 A 3. Then Hom(Ξ, Σ) means Σ is of type B n (n m) orf 4 if m>. If m>orifσisoftypef 4,#Hom(Ξ L, Σ) = and therefore #Hom(Ξ, Σ) =. If m =andσ=b n or C n, it is easy to see that { ι Hom(A, Σ) ; ( ι(β )+ι(β ) ) Σ } is a single W Σ -orbit and we have also #Hom(Ξ, Σ) =. Here Ξ L = β + β A. 4) When Σ is of type C n,wehavethelemmafromthecase3)byconsidering the dual root systems Ξ and Σ. 5) We first examine Hom(E 6,E 8 )andhom(e 7,E 8 ) under the notation in 9. Since #Hom(A 5,E 8 )=#Hom(A 6,E 8 ) =, we may assume E o 6 Ψ A5 = {α 0 = ɛ 7 ɛ 8,α 8 = ɛ 7 ɛ 6,α 7 = ɛ 6 ɛ 5,α 6 = ɛ 5 ɛ 4,α 5 = ɛ 4 ɛ 3 } for the imbedding E 6 E6 o E 8.Let α Φ \ Ψ A5.Wehave { 8 α = c j ɛ j E6 o 0 (j =0, 8, 6, 5), E 8 : α, α j = (j =7). Thus j= α = c ɛ + c ɛ + c(ɛ 3 + ɛ 4 + ɛ 5 )+(c )(ɛ 6 + ɛ 7 ɛ 8 ). Since α is a root of E 8,wehavec = and hence α = α ± := (±(ɛ + ɛ )+ɛ 3 + ɛ 4 + ɛ 5 ɛ 6 ɛ 7 + ɛ 8 ).

19 A CLASSIFICATION OF SUBSYSTEMS OF A ROOT SYSTEM 9 Since α = ɛ + ɛ is orthogonal to α j (j =0, 5, 6, 7, 8) and s α α + = α,wehave #Hom(E 6,E 8 )=and (E o 6 ) E 8 Ψ A 5 α + E 8 =( α,α 3 + α ) α + = α,α 3 A. α α3 α4 α5 α6 α7 α8 α 0 Let E 7 E o 7 E 8. Then we may moreover assume α 4 = ɛ 3 ɛ E 7 and the condition α α 4 implies α = α +. Hence #Hom(E 7,E 8 )=and α (E o 7 ) E 8 α,α 3 α 4 = α A. Now we examine Hom(E 6,E 7 ). Since A 5 E 6 E6 o E 7, the argument in ) assures that we may assume E6 o Ψ A 5 := Ψ A4 {α = ɛ + ɛ } or E6 o Ψ A5 := Ψ A4 {α 5 = ɛ 4 ɛ 3 } Ψ A4 := {α 0 = ɛ 8 ɛ 7,α = (ɛ ɛ ɛ 3 ɛ 4 ɛ 5 ɛ 6 ɛ 7 + ɛ 8 ), α 3 = ɛ ɛ,α 4 = ɛ 3 ɛ }. Then there exists α = 8 j= c jɛ j E o 6 E 7 such that ( α α j )=0 (j =0,, 4), ( α α 3 )=, ( α α )( α α 5 )=0. α α 0 α α3 α4 α5 α6 α7 α α Then the condition ( α α 0 ) = 0 implies c 7 = c 8 =0and α =(c +)ɛ + c(ɛ + ɛ 3 )+c 4 ɛ 4 + c 5 ɛ 5 + c 6 ɛ 6, c c 4 c 5 c 6 =0, (c +)(c c 4 )=0. Hence c =0,E6 0 Ψ A5 and α = ɛ + ɛ 5 or ɛ + ɛ 6.Since ΨA 5 E 7 = α 7 = ɛ 6 ɛ 5 and s ɛ6 ɛ 5 (ɛ + ɛ 5 )=ɛ + ɛ 6,wehave#Hom(Ξ, Σ) = and (E o 6 ) E 7 =. If # Hom(Ξ, Σ) =, any element of Hom(Ξ, Σ) is isomorphic to the imbedding ι corresponding to the graphic imbedding ῑ given in the claim. Since ι(ξ) Ψ ῑ(ξ) and ι(ξ) Ψ ῑ(ξ),wehaveι(ξ) = Ψ ῑ(ξ). 6) If Ξ is of type G or F 4, the lemma is clear and thus we have completed the proofofthelemma. 6. Dual pairs and closures Definition 6.. For a subsystem Ξ of a root system Σ and a subgroup G of Aut(Σ) we put (6.) N G (Ξ) := {g G ; g(ξ) = Ξ}, Z G (Ξ) := {g G ; g Ξ = id}, (6.) Aut Σ (Ξ) := N WΣ (Ξ)/Z WΣ (Ξ) Aut(Ξ), (6.3) Out Σ (Ξ) := Aut Σ (Ξ)/W Ξ N WΣ (Ξ)/(W Ξ W Ξ ) Out(Ξ). Note that the isomorphism in (6.3) follows from the equality Z WΣ (Ξ) = W Ξ. Proposition 6.. Let Ξ be a subsystem of Σ. Put Ξ =Ξ. Then there is a homomorphism (6.4) ϖ :Out Σ (Ξ ) Out Σ (Ξ ) Out Σ (Ξ )

20 0 TOSHIO OSHIMA and (6.5) (6.6) (6.7) ϖ is bijective if Ξ =Ξ, Out Σ (Ξ ) Out(Ξ ) if # ( Hom(Ξ, Σ)/Out(Ξ ) ) =#Hom(Ξ, Σ), Out Σ (Ξ ) Out(Ξ ) if # ( Hom(Ξ, Σ)/Out(Ξ ) ) =#Hom(Ξ, Σ). Proof. Since N WΣ (Ξ ) N WΣ (Ξ )andξ Ξ, (6.4) is well-defined and (6.5) is clear. Suppose # ( Hom(Ξ, Σ)/Out(Ξ ) ) =#Hom(Ξ, Σ). Then for any g Aut(Ξ )thereexistsw W Σ with w Ξ = g Ξ and (6.6) is clear. We similarly have (6.7). The isomorphism in (6.4) follows from (6.5) and the relation (Ξ ) =Ξ. Definition 6.3 (dual pairs). Apair(Ξ, Ξ ) of subsystems of a root system Σ is called a dual pair in Σ if (6.8) Ξ =Ξ and Ξ =Ξ. If (Ξ, Ξ ) is a dual pair, the map ϖ in Proposition 6. is an isomorphism. The dual pair is called special if the map ϖ is the isomorphism (6.9) ϖ : Out(Σ ) Out(Σ ). ForasubsystemΞofΣ,its -closure Ξ is defined by Ξ:=(Ξ ). Then (Ξ, Ξ ) is a dual pair if and only if Ξ is -closed (i.e. (Ξ ) = Ξ) and hence (Ξ, Ξ )is always a dual pair. We say that Ξ is -dense in Σ if Ξ =. Corollary 6.4. Let (Ξ, Ξ ) be a dual pair in Σ. Then # ( Hom(Ξ, Σ)/Out(Ξ ) ) < #Hom(Ξ, Σ) (6.0) Out(Ξ ) Out(Ξ ) or # ( Hom(Ξ, Σ)/Out(Ξ ) ) < #Hom(Ξ, Σ). Suppose #Hom(Ξ, Σ) =. Letι Hom(Ξ, Σ). Then we have (6.) (Ξ, Ξ ) is a special dual pair # Out(Ξ ) = # Out(Ξ ), (6.) w W Σ such that ι(ξ )=w(ξ ) ι(ξ ) Ξ. Proof. Note that (6.0) is the direct consequence of Proposition 6.. Suppose #Hom(Ξ, Σ) =. Then Proposition 6. implies Out(Ξ ) Out Σ (Ξ ) Out(Ξ ) and (6.) is clear. Then if ι(ξ ) Ξ,thereexistsw W Σ with ι(ξ )=w(ξ ) and therefore ι(ξ )=w(ξ ), which implies the claim. Example 6.5. i) The followings are examples of the triplets (Σ, Ξ, Ξ ) such that (Ξ, Ξ ) are special dual pairs in Σ. (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 ). In these examples except for (D 4, A, A )and(e 8, 4A, 4A ), #Hom(Ξ, Σ) = and the triplet is uniquely determined by the data (Σ, Ξ, Ξ )uptotheautomorphisms defined by W Σ. If the imbedding 4A E 8 satisfies (4A ) 4A,wehave a special dual pair (4A, 4A )ine 8, which is also uniquely defined. The imbedding A D 4 is unique up to Aut(D 4 ).

21 A CLASSIFICATION OF SUBSYSTEMS OF A ROOT SYSTEM ii) The isomorphism ϖ Out(A ) defined by the dual pair (A, A )ine 8 satisfies (6.3) ϖ ( Out(A ) Out(A ) ) Out(A ) Out(A ) because # ( Hom(4A,E 8 )/ Out (4A ) ) =#Hom(A,E 8 )=. See iii) It happens that any dual pair of E 8 is special. But for example, if (Σ, Ξ, Ξ ) is (E 6,A 5,A )or(e 7,D 6,A ), (Ξ, Ξ ) is a dual pair in Σ satisfying Out(Ξ ) Out(Ξ )and#hom(ξ, Σ) =, which implies #Hom(Ξ, Σ) >. Definition 6.6 (S-closure and L-closure). Let Ξ be a subsystem of Σ. Then Ξ is S-closed if and only if (6.4) α, β Ξ and α + β Σ α + β Ξ and L-closed if and only if (6.5) β Σ α Ξ Rα β Ξ. The smallest S-closed (resp. L-closed) subsystem of Σ containing Ξ is called the S-closure (resp. L-closure) of Ξ. Remark 6.7. i) We have the following relation for a subsystem Ξ of Σ: (6.6) -closed L-closed S-closed. ii) Let g be a complex semisimple Lie algebra with the root system Σ and let X α be root vectors corresponding to α Σ. Then the root system of the semisimple Lie algebra g Ξ generated by {X α ; α Ξ} is the S-closure of Ξ. Let Ξ and Ξ be S-closed subsystems of Σ. Then (6.7) [g Ξ, g Ξ ]=0 Ξ Ξ. Hence if (Ξ, Ξ ) is a dual pair with rank Ξ +rankξ = rank Σ, the dual pair of root systems gives a dual pair in semisimple Lie algebras (cf. [7]). iii) Suppose Σ is irreducible and there exist α, β Σwithα + β Σ \ Ξ. Then α, β, α + β is of type B or of type G, which implies that Σ is not simply laced. For example, D n C n is not S-closed and the S-closure of D n equals C n (n ). iv) Let Ξ be an L-closed subsystem of Σ. Then for any subsystem Ξ of Σ (6.8) W Ξ W Ξ = W Ξ Ξ. This is proved as follows. Choose a generic element v of the orthogonal complement of α Ξ Rα in α Σ Rα so that {α Σ; (α v) =0} =Ξ. SinceW Ξ = {w W Σ ; wv = v}, W Ξ W Ξ = {w W Ξ ; wv = v} = W {α Ξ ;(α v)=0} = W Ξ Ξ. v) Put Ξ = {±ɛ ± ɛ, ±ɛ 3 ± ɛ 4 } 4A and Ξ = {±ɛ ± ɛ 3, ±ɛ ± ɛ 4 } 4A. Then the subsystems Ξ and Ξ of D 4 under the notation in 9 do not satisfy (6.8). When Σ = B n, C n, F 4 or G and Ξ = Σ L and Ξ =Σ\ Ξ, (6.8) is not valid. 7. Making tables We are ready to answer the questions in the introduction by completing the tables in 0. In this section we do it when the root system Σ is of the exceptional type. Following the argument in 3, we easily get all Ξ satisfying Hom(Ξ, Σ) together with #Hom(Ξ, Σ) and Ξ by Theorem 3.5. In fact, we start from the irreducible Ξ and then examine other Ξ by using (3.) in a suitable lexicographic order (as in the tables) to avoid confusion (cf. Example 3.6 xii)). As a result we finally get (Ξ ) and the dual pairs. Moreover (6.) tells us whether the dual pair is special or not. We will calculate #{Θ Φ; Θ Ξ} in 7.5.

22 TOSHIO OSHIMA Now we prepare the lemma to examine the action of W Σ on the imbeddings of a root system Ξ into Σ. Lemma 7.. Let Ξ and Ξ be subsystems of Σ with Ξ Ξ.Then (7.) (7.) (7.3) (7.4) # ( Hom(Ξ, Σ)/Out(Ξ ) ) =# ( Hom(Ξ, Ξ )/Out(Ξ ) ) = # ( Hom(Ξ +Ξ, Σ)/Out(Ξ +Ξ ) ) =, # ( Hom(Ξ, Σ)/Out (Ξ ) ) =# ( Hom(Ξ, Ξ )/Out (Ξ ) ) = # ( Hom(Ξ +Ξ, Σ)/Out (Ξ +Ξ ) ) =, { # ( Hom(Ξ, Σ)/Out(Ξ ) ) =# ( Out(Ξ )\Hom(Ξ, Ξ )/Out (Ξ ) ) =, Out (Ξ ) Out(Ξ ) and (Ξ, Ξ ) is a special dual pair # ( Hom(Ξ +Ξ, Σ)/Out (Ξ +Ξ ) ) =, # ( Hom(Ξ, Σ)/Out(Ξ ) ) =# ( Hom(Ξ, Ξ )/Out(Ξ ) ) = and ι(ξ ) Ξ ( ι Hom(Ξ, Σ)) # ( Hom(Ξ, Σ)/Out(Ξ ) ) =. Proof. The claims (7.) and (7.) are clear because for ι Hom(Ξ +Ξ, Σ) the assumptions assure that there exists w W Σ such that ι(ξ )=w(ξ ) and hence we may assume ι(ξ )=Ξ in Hom(Ξ +Ξ, Σ). Under the assumption in (7.3) there exists w W Σ such that w ι stabilizes every irreducible component of Ξ and therefore it also stabilizes Ξ and we have (7.3). The claim (7.4) is also clear because for ι Hom(Ξ, Σ), w W Σ such that w ι(ξ ) =Ξ, which implies w ι(ξ ) Ξ. 7.. Type E 6. The automorphism group of G( Ψ) is of order 6, which is generated by a rotation and a reflection. Since the rotation has order 3, it corresponds to an element of W E6 and the reflection corresponds to a non-trivial element of Out(E 6 ). α 4 α 6 α 3 α 5 α α α 0 The set Hom(A 4,E 6 ) has two elements which are shown in Example 3.6 v). It also shows that Out(E 6 ) non-trivially acts on this set. If A 4 is imbedded to E 6 given as in the above imbedding G(A 4 ) G(Ẽ6) with the starting vertex {α 0 }, the non-trivial action by Out(A 4 ) changes the starting vertex as is shown above. Then by an element of W A5 with A 5 = α 0,α,α,α 5,α 6 the imbedding is transformed as is shown by the second arrow. Then the result corresponds to a reflection, which implies that Out(A 4 ) also acts non-trivially on Hom(A 4,E 6 ) and hence # ( Hom(A 4,E 6 )/Out(A 4 ) ) =. The same argument works for Ξ = A 5, A + A and A 3 + A. Similarly (α,α 0,α 5,α 6 ) is transformed to (α 6,α 5,α 0,α )byanelementofw A5 and furthermore to (α 0,α,α 4,α 3 ) by a rotation. Hence a non-trivial element of Out(A ) for A = α 0,α induces the transposition of two irreducible components of A A + A, which implies # ( Hom(A,E 6 )/Out (A ) ) =. From our construction of the representatives of Hom(Ξ,E 6 )itisobvioustohave # ( Hom(Ξ,E 6 )/Out (Ξ) ) =forξ=d 5 (cf. (5.3)) and E 6 and we can easily calculate # ( Out(E 6 )\Hom(Ξ,E 6 ) ).PutΣ=E 6 and let (Ξ, Ξ )beanyoneofthe pairs (A + A,A ), (A,A ), (A,A ), (A,A 3 ), (A 4,A )and(a 5,A ). Then applying (7.) to Σ and (Ξ, Ξ ), we have # ( Hom(Ξ +Ξ, Σ)/Out (Ξ +Ξ ) ) =.

23 A CLASSIFICATION OF SUBSYSTEMS OF A ROOT SYSTEM Type E 7. Note that G(Ẽ7) has an automorphism of order and it corresponds to an element of W E7 because W E7 =Aut(E 7 ). Let Σ = E 7 and let (Ξ, Ξ )beanyoneof(a,d 6 ), (A,A ), (A,A + A ), (A,A 3 ), (A,A 3 + A ), (A 3,A 3 ) and (A 3,A 3 + A ). We have # ( Hom(Ξ + Ξ, Σ)/Out (Ξ +Ξ ) ) = by (7.). Here we note that A D 6, A A 5 and A 3 A 3 + A. We can apply (7.3) to (Ξ, Ξ ) = (D 4,kA ) with k 3 and we have the same conclusion. Applying (7.) to (A 3, 3A ), we have # ( Hom(A 3 +3A,E 7 )/Out(A 3 +3A ) ) =. The subsystems Ξ of E 7 which are isomorphic to 3A and satisfy Ξ 4A are mutually equivalent by Σ. Hence Ξ 4A also have this property. Namely # ( W E7 \{ι Hom(4A,E 7 ); (ι(4a ) ) = ι(4a )}/ Aut(4A ) ) =. Put (A ) o = α 0.WehaveG ( ) (A ) o as is given in the following first diagram. Put (A ) o = α 0,α p. Then the extended Dynkin diagrams of the components of (A ) o = α,α 3,α 4,α 5,α 7 D 4 + A are also given by the following second diagram. These diagrams correspond to the last figure in Remark 3.8 i). Here α p := (α + α α 7 )+(α 4 + α 5 + α 6 )=ɛ 5 + ɛ 6, α q := α + α 3 +α 4 + α 5 = ɛ 3 + ɛ 4, α 0 α α3 α4 α5 α6 α7 α α p α q α 0 α α3 α5 α6 α 7 α α p α 0 α α α 3 α q 5 α 6 α 7 α α p α q α 0 α α3 α5 α6 α7 α α p In the above diagrams the vertices expressed by asterisks are considered to be removed and the diagrams are (extended) Dynkin diagrams for other roots. There are two equivalence classes in the imbeddings of 3A to E 7, whose representatives are (3A ) = α 0,α p,α q, (3A ) = α 0,α p,α 7, which satisfy (3A ) = α,α 3,α 5,α 7 4A, (3A ) = α,α 3,α 4,α 5 D 4. Thus the image of the imbedding of 4A to E 7 is equivalent to one of the following subsystems (4A ) j of E 7 : (4A ) = α 0,α p,α q,α 7, (4A ) = α,α 3,α 5, (4A ) = α 0,α p,α q,α 5, (4A ) = α,α 3,α 7, (4A ) 3 = α 0,α p,α q,α, (4A ) 3 = α 3,α 5,α 7, (4A ) 4 = α 0,α p,α q,α 3, (4A ) 4 = α,α 5,α 7. In view of Remark 3.4 ii) the above procedures for 3A E 6 can be also explained by the following isomorphic ones. α 0 α α3 α4 α 5 α 6 α 7 α r α α α0 α s 3 α 4 α 5 α 6 α 7 α r α α 0 α s α3 α4 α5 α6 α7 α r α α α 0 α s 3 α 4 α5 α6 α7 α r α

24 4 TOSHIO OSHIMA Here α r = α p and α s = α q. In fact α p, α r α,α 3,α 4,α 5,α 7 =(D 4 + A ) A.Asm 7 (α p ) < 0andm 7 (α r ) > 0, we have α r = α p. Similarly we have α s = α q from α q, α s α 0,α,α 3,α 5,α 7,α p A. This is also easily verified by the Dynkin diagrams with the coefficients m j ( α 0 )in 9. Note that α,α 3,α 5 α,α 3,α 7 α,α 4,α 7 α 3,α 5,α 7. α 5,α 6,α 7 α,α,α 3,α 4 α,α 3,,α 7 Thus we can conclude (7.5) ( (4A ) j ) { α 3,α 5,α 7 = α 0,α,α r,α s 4A (j =,, 3), = α 0,α,α 3,α r,α s D 4 (j =4). Since (4A ) j for j =,, 3 are equivalent to each other by E 8, so are the subsystems (4A ) j = ((4A ) j ) for j =,, 3. Moreover we have (7.6) α 0,α p,α q,α 3 α 0,α,α q,α 7 α 0,α 3,α 5,α 7. α,α 3,...,α 7 α,α 3,α 4,α 5 Put P Ξ = {Θ Ψ; Θ Ξ}. It is easy to see that if Θ P 3A satisfies Θ {α,α 3 }, Θ (3A ).MoreoverifΘ P 3A satisfies Θ {α,α 3 } =, E7 then Θ = {α,α 5,α 7 }. We will have #B 3A =in 7.5. Applying (7.4) to (Ξ, Ξ )=(A, 5A )and(a, 6A )withσ=e 7,wehave # ( Hom(Ξ, Σ)/Out(Ξ ) ) =, respectively. Similarly applying (7.) to (Ξ, Ξ )= (5A,A )and(5a, A ), we have # ( Hom(Ξ +Ξ, Σ)/Out(Ξ +Ξ ) ) =, respectively Type E 8. Applying (7.4) to (3A, 5A ) and then (7.) to (5A,A ), (5A, A ) and (5A, 3A ), respectively, we have # ( Hom(kA,E 8 )/Out(kA ) ) =fork = 5, 6, 7 and 8. See 8..3 to get further results on Hom(kA,E 8 )with k 8. If (Ξ, Ξ ) is any one of the pairs (A,E 6 ), (A 4,A 4 ), (D 4,D 4 ), (D 5,A 3 )and (D 6,A ), we have Hom(Ξ, Ξ ) # ( Hom(Ξ +Ξ,E 8 )/Out (Ξ +Ξ ) ) = by applying (7.3). Hence if Ξ contains A, A 4, D 4, D 5 or D 6 as an irreducible component, the value of the column indicated by # Ξ equals one. Moreover (7.) can be applied to (Ξ, Ξ ) = (A 3, 3A ), (A 3, 4A )and(a 5, A ). The number # ( Hom(Ξ,E 8 )/Out (Ξ) ) for Ξ = A 3 +3A, A 3 +4A and A 5 +A is easily obtained from (A 3 + A ) A 3 + A and A 5 A + A. Put (A 7 ) o = α,α 3,...,α 8 (A 8 ) o = (A 7 ) o,α 0, (D 6 ) o = {±ɛ i ± ɛ j ;3 i<j 8} = {α,α 3 }, P Ξ = {Θ {α,...,α 8 } ; Θ Ξ}. Then we note the following for Θ,Θ P Ξ. Θ Θ if Θ j (A 8 ) o for j =,andθ Θ. Σ If Θ {α,α 3 },thenθ =Θ (D 6 ) o. Using these facts, we can easily examine P Ξ. For example, any Θ P 4A satisfies Θ (4A ) o := α,α 3,α 6,α 8.Here(4A ) o = α 6,α 8 D 6 4A. E Type F 4 and G. It is easy to examine the cases when Σ = F 4 and G by using Theorem 3.5 together with Remark 3., (.39) and (5.4).