aa r X i v : . [ m a t h . R T ] A p r REPRESENTATIONS OF LIE ALGEBRASARISING FROM POLYTOPES
R.M. Green
Department of MathematicsUniversity of ColoradoCampus Box 395Boulder, CO 80309-0395USA
E-mail: [email protected]
Abstract.
We present an extremely elementary construction of the simple Lie al-gebras over C in all of their minuscule representations, using the vertices of variouspolytopes. The construction itself requires no complicated combinatorics and essen-tially no Lie theory other than the definition of a Lie algebra; in fact, the Lie algebrasthemselves appear as by-products of the construction. Preliminary version, draft 2
Introduction
The simple Lie algebras over the complex numbers are objects of key importancein representation theory and mathematical physics. These algebras fall into fourinfinite families ( A n , B n , C n , D n ) and five exceptional types ( E , E , E , F and G ). The classical (i.e., non-exceptional) types of Lie algebras are easily definedin terms of Lie algebras of matrices; such representations are called the naturalrepresentations of the Lie algebras. However, it is not so easy to give similardescriptions of the exceptional algebras in a way that makes it easy to carry outcalculations with them. Another natural question is whether one can give easydescriptions of other representations of the classical Lie algebras, such as the spin Mathematics Subject Classification . 17B10, 52B20. Typeset by
AMS -TEX R.M. GREEN representations of algebras of types B n and D n , which are traditionally constructedin terms of Clifford algebras (see [ , § C . Two of these include Littelmann’s description of repre-sentations in terms of paths, and the crystal basis approach of Kashiwara and theKyoto school. Both these approaches are very versatile but can be combinatoriallycomplicated. Recent work of the author shows how to construct certain Lie algebrarepresentations using combinatorial structures called “full heaps”, whose theory isdeveloped in [ , ]. The approach of the present paper grew out of an attempt toexplain the full heap representations in as simple a way as possible, and it does notrequire any complicated combinatorial constructions.The polytopes we consider in this paper are convex subsets of R n whose vertices(i.e., 0-skeletons) have integer coordinates; such polytopes are sometimes called“lattice polytopes”. These include the hypercube, the hyperoctahedron (which isthe dual of the hypercube) and the polytopes known as 2 and 3 in Coxeter’snotation [ ]; the latter two polytopes have 27 and 56 vertices respectively. All thesepolytopes are highly symmetrical, and the symmetry groups have been known fora long time. The reason that these polytopes are relevant in Lie theory is that theset of weights for the minuscule representations of simple Lie algebras over C formthe vertices of one of the aforementioned polytopes. This is not obvious, but it isnot a complete surprise either: Manivel [ , Introduction] for example mentions inpassing that the weights of the 56-dimensional representation of e correspond tothe vertices of 3 .Our approach in this paper is to start with the vertices of the polytope and usethem to construct representations of Lie algebras without first constructing the Liealgebras themselves. All the minuscule representations of simple Lie algebras over C may be constructed in this way, and the construction is remarkably simple. In §
2, we introduce the notion of a “minuscule system”, which involves two subsetsof R n , denoted by Ψ and ∆. The set Ψ is said to be a minuscule system with EPRESENTATIONS OF LIE ALGEBRAS ARISING FROM POLYTOPES 3 respect to the simple system ∆ if two conditions are satisfied (see Definition 2.1).These conditions are very elementary and easy to check, and whenever they hold,the set ∆ defines a set of linear operators on a vector space with dimension | Ψ | (Definition 2.2). If one makes a judicious choice of Ψ and ∆, then these linearoperators turn out to be the representations of the Chevalley generators of a simpleLie algebra over C acting in one of its minuscule representations with respect to anobvious basis (the basis can be shown to be the crystal basis in the sense of [ ], byadapting the argument of [ , § α = − θ , where θ is the highest root. In the latter case,we obtain finite dimensional representations of certain derived affine Kac–Moodyalgebras. Formulating the results in terms of affine algebras can be more natural,as the affine algebras have a greater degree of symmetry. Another advantage isthat it is easier to see how the modules behave under restriction; for example, the56-dimensional module for the Lie algebra of type E , after inflation to a modulefor the derived affine algebra, can be restricted to a module for the Lie algebra oftype A which is the direct sum of two nonisomorphic 28-dimensional irreduciblesubmodules. Once this observation is made, our approach here to e is seen to bevery natural.The layout of this paper is as follows. In §
1, we recall some of the basic theory ofrepresentations of Lie algebras. Minuscule systems are defined in §
2, and developedin §
3. Our main result is Theorem 3.2, and the linear operators used in it are de-fined in Definition 2.2. Sections 4–6 are devoted to examples of minuscule systems. § e and e arising from the polytopes 2 and3 respectively. § R.M. GREEN including the spin representations of the Lie algebras of types B n and D n . Theminuscule representations of type A n − are obtained by restricting the spin repre-sentation in type B n to a subalgebra. § C n and D n . In §
7, we explore connections with algebraic geometry, and concludingremarks are given in §
1. Background on Lie algebras A Lie algebra is a vector space g over a field k equipped with a bilinear map[ , ] : g × g −→ g (the Lie bracket ) satisfying the conditions[ x, x ] = 0 , [[ x, y ] , z ] + [[ y, z ] , x ] + [[ z, x ] , y ] = 0 , for all x, y, z ∈ g . (These conditions are known respectively as antisymmetry andthe Jacobi identity .)If g and g are Lie algebras over a field k , then a homomorphism of Lie algebrasfrom g to g is a k -linear map φ : g −→ g such that φ ([ x, y ]) = [ φ ( x ) , φ ( y )] forall x, y ∈ g . An isomorphism of Lie algebras is a bijective homomorphism.If V is any vector space over k then the Lie algebra gl ( V ) is the k -vector spaceof all k -linear maps T : V −→ V , equipped with the Lie bracket satisfying[ T, U ] := T ◦ U − U ◦ T, where ◦ is composition of maps.A representation of a Lie algebra g over k is a homomorphism ρ : g −→ gl ( V ) forsome k -vector space V . In this case, we call V a (left) module for the Lie algebra g (or a g -module, for short) and we say that V affords ρ . If x ∈ g and v ∈ V , wewrite x.v to mean ρ ( x )( v ). The dimension of a module (or of the correspondingrepresentation) is the dimension of V . If ρ is the zero map, then the representation ρ and the module V are said to be trivial . EPRESENTATIONS OF LIE ALGEBRAS ARISING FROM POLYTOPES 5 A submodule of a g -module V is a k -subspace W of V such that x.w ∈ W forall x ∈ g and w ∈ W . If V has no submodules other than itself and the zerosubmodule, then V is said to be irreducible .If V and V are g -modules, then a k -linear map f : V −→ V is called a homo-morphism of g -modules if f ( x.v ) = x.f ( v ) for all x ∈ g and v ∈ V . An isomorphismof g -modules is an invertible homomorphism of g -modules.A subspace h of g is called a subalgebra of g if [ h , h ] ⊆ h . If, furthermore, we have[ g , h ] ⊆ h (respectively, [ h , g ] ⊆ h ) then h is said to be a left ideal (respectively, rightideal ) of g . If h is both a left ideal and a right ideal of g , then we call h a two-sidedideal (or “ideal” for short) of g . If g has no ideals other than itself and the zeroideal, then g is said to be simple . The derived algebra , g ′ , of g is the subalgebragenerated by all elements { [ x , x ] : x , x ∈ g } . It can be shown that g ′ is an idealof g . Definition 1.1.
Let A be an n × n matrix with integer entries. Following [ , § A = ( a ij ) a generalized Cartan matrix if it satisfies the following threeproperties:(i) a ii = 2 for all 1 ≤ i ≤ n ;(ii) a ij ≤ i = j ;(iii) a ij = 0 ⇒ a ji = 0.We call the matrix A symmetrizable if there exists an invertible matrix D and asymmetric matrix B such that A = DB .The next result is a well known presentation for the derived algebra of a Kac–Moody algebra corresponding to a symmetrizable Cartan matrix. R.M. GREEN
Theorem 1.2.
Let A be a symmetrizable generalized Cartan matrix. The derivedKac–Moody algebra g = g ′ ( A ) corresponding to A is the Lie algebra over C generatedby the elements { e i , f i , h i : i ∈ ∆ } subject to the defining relations [ h i , h j ] = 0 , [ h i , e j ] = A ij e j , [ h i , f j ] = − A ij f j , [ e i , f j ] = δ ij h i , [ e i , [ e i , · · · [ e i , | {z } − A ij times e j ] · · · ]] = 0 , [ f i , [ f i , · · · [ f i , | {z } − A ij times f j ] · · · ]] = 0 , where δ is the Kronecker delta.Proof. This is a special case of [ , Theorem 9.11]. (cid:3) Remark 1.3.
In this paper, we are mostly interested the case where A is of finitetype (as defined in [ , § g is simple.Suppose for the rest of § g is an algebra satisfying the hypotheses ofTheorem 1.2. Let h be the subalgebra of g spanned by the elements { h i : i ∈ ∆ } .Let h ∗ = Hom( h , C ) be the dual vector space of h , and let { ω i : i ∈ ∆ } be the basisof h ∗ dual to { h i : i ∈ ∆ } . Let V be a g -module. An element v ∈ V is called a weight vector of weight λ ∈ h ∗ if for all h ∈ h , we have h.v = λ ( h ) v . The weights ω i are known as fundamental weights . If the weight vector v is annihilated by theaction of all of the elements e i (respectively, all of the elements f i ), then we call v a highest weight vector (respectively, a lowest weight vector ).The following result is well known. EPRESENTATIONS OF LIE ALGEBRAS ARISING FROM POLYTOPES 7
Proposition 1.4. (i)
Let g be a simple Lie algebra over C . If λ is a nonnegative Z -linear combinationof the fundamental weights ω i then up to isomorphism there is a unique finitedimensional irreducible g -module L ( λ ) of the form g .v λ , where v λ is of weight λ and is the unique nonzero highest weight vector of L ( λ ) . The modules L ( λ ) arepairwise nonisomorphic and exhaust all finite dimensional irreducible modules of g . (ii) Suppose that V is a finite dimensional g -module containing a nonzero highestweight vector v λ of weight λ , and that dim( V ) = dim( L ( λ )) . Then V ∼ = L ( λ ) .Proof. Part (i) is a special case of [ , Theorem 10.21]. For part (ii), it follows fromthe proof of [ , Proposition 10.13] that any g -module g .v λ generated by a highestweight vector of weight λ is a quotient of the Verma module M ( λ ). The Vermamodule has a unique maximal submodule, J ( λ ) (see [ , Theorem 10.9]) and wehave L ( λ ) = M ( λ ) /J ( λ ) by definition. It follows that g .v λ has a quotient moduleisomorphic to L ( λ ). The assumption about dimensions allows this only if V ∼ = L ( λ )(and g .v λ = V ). (cid:3) If λ is a fundamental weight, the corresponding module L ( λ ) is called a funda-mental module . Certain of the fundamental modules for simple Lie algebras areknown as minuscule modules , for reasons we will not go into (although see [ ,2.11.15] for an explanation). The purpose of this paper is to provide a uniform andvery elementary construction of these modules. We now list the minuscule modules,their weights and their dimensions; more information on this may be found in [ , § ], andin some cases, this differs from Carter’s notation in [ ].For the simple Lie algebra of type A n , all the fundamental modules L ( ω ) , . . . , L ( ω n )are minuscule, and we have dim( L ( ω i )) = (cid:18) n + 1 i (cid:19) . R.M. GREEN
In this case, L ( ω ) is the natural module, and L ( ω i ) is the i -th exterior power of L ( ω ).For the simple Lie algebra of type B n (for n ≥ L ( ω n ), which has dimension 2 n .For the simple Lie algebra of type C n (for n ≥ L ( ω ), which has dimension 2 n .For the simple Lie algebra of type D n (for n ≥ L ( ω ), of dimension 2 n , and the two spinmodules L ( ω n − ) and L ( ω n ), each of which has dimension 2 n − .The simple Lie algebra of type E has two minuscule modules, L ( ω ) and L ( ω ),each of which has dimension 27.The simple Lie algebra of type E has one minuscule module, L ( ω ), which hasdimension 56.The simple Lie algebras of types E , F and G have no minuscule modules.
2. Minuscule systems
Definition 2.1.
Let Ψ and ∆ be subsets of vectors in R n for some n ∈ N , where R n is equipped with the usual scalar product and 0 ∆. We say that Ψ is a minuscule system with respect to the simple system ∆ if the following conditionsare satisfied for every v ∈ Ψ and a ∈ ∆.(i) We have 2 v . a = c a . a for some c = c ( v , a ) ∈ {− , , +1 } .(ii) Let c = c ( v , a ) be as in (i). Then we have v + a ∈ Ψ if and only if c = −
1, andwe have v − a ∈ Ψ if and only if c = 1. (In particular, if c = 0, then neithervector v ± a lies in Ψ.) Definition 2.2.
Let Ψ be a minuscule system with respect to the simple system ∆,and let k be a field. We define V Ψ to be the k -vector space with basis { b v : v ∈ Ψ } .For each a ∈ ∆, we define k -linear endomorphisms E a , F a , H a of V Ψ by specifying EPRESENTATIONS OF LIE ALGEBRAS ARISING FROM POLYTOPES 9 their effects on basis elements, as follows: E a ( b v ) = (cid:26) b v + a if v + a ∈ Ψ;0 otherwise; F a ( b v ) = (cid:26) b v − a if v − a ∈ Ψ;0 otherwise; H a ( b v ) = c ( v , a ) b v = 2 v . aa . a b v . Definition 2.3.
Let Ψ be a minuscule system with respect to the simple system∆. We define the generalized Cartan matrix , A , of ∆ to be the | ∆ | × | ∆ | matrixwhose ( a , b ) entry is given by A a , b = 2 a . ba . a . Although we have apparently given two meanings to the term “generalized Car-tan matrix” (the above meaning and Definition 1.1), they coincide in all the exam-ples of this paper. A formulation very similar to Definition 2.3 may be found in[ , §
3. Results on minuscule systems
The following lemma is the key ingredient for our main result.
Lemma 3.1.
Using the notation of Definition 2.2, we have the following identitiesin
End( V Ψ ) , where a , b ∈ ∆ : H a ◦ E b = E b ◦ H a + A a , b E b , (1) H a ◦ E a = E a = − E a ◦ H a , (2) H a ◦ F b = F b ◦ H a − A a , b F b , (3) H a ◦ F a = − F a = − F a ◦ H a , (4) E a ◦ F b = F b ◦ E a = 0 if A a , b < , (5) E a ◦ F b = F b ◦ E a if A a , b = 0 , (6) E a ◦ E a = 0 , (7) E a ◦ E b = E b ◦ E a if A a , b = 0 , (8) E a ◦ E b ◦ E a = 0 if A a , b = − , (9) F a ◦ F a = 0 , (10) F a ◦ F b = F b ◦ F a if A a , b = 0 , (11) F a ◦ F b ◦ F a = 0 if A a , b = − , (12) Proof.
We prove (1) by acting each side of the equation on a basis vector b v . If E b ( b v ) = 0, then both sides are trivial, so we may assume this is not the case,meaning that v + b ∈ Ψ. It follows that, in the notation of Definition 2.1, we have c ( v , b ) = − c ( v + b , b ) = 1. In turn, this means that H a ◦ E b ( b v ) = H a ( b v + b ) = 2 ( v + b ) . aa . a b v + b and that E b ◦ H a ( b v ) = 2 v . aa . a E b ( b v ) = 2 v . aa . a b v + b . EPRESENTATIONS OF LIE ALGEBRAS ARISING FROM POLYTOPES 11
Subtracting, we have( H a ◦ E b − E b ◦ H a )( b v ) == 2 b . aa . a b v + b = A a , b E b ( b v ) , which proves (1).If b = a , then the above argument shows that H a ◦ E a ( b v ) = H a ( b v + a ) = c ( v + a , a ) b v + a = E a ( b v ) . Part (2) follows from this and the fact that A a , a = 2.The proof of (3) (respectively, (4)) follows by adapting the argument used toprove (1) (respectively, (2)).We now prove that E a ◦ F b = 0 if A a , b <
0. By (1) and (2), we have E a F b = H a E a F b = E a H a F b + 2 E a F b = E a F b H a + (2 − A a , b ) E a F b . Rearranging, this gives E a F b H a = ( A a , b − E a F b . Suppose that E a F b = 0, and let b v be a basis element for which E a ◦ F b ( b v ) = 0.This implies that H a ( b v ) = ( A a , b − b v , but this is a contradiction to Definition2.1 (i), because c ( v , a ) = A a , b − ≤ −
2. This shows that E a ◦ F b = 0, and theproof that F b ◦ E a = 0 is very similar, proving (5).We next turn to (6). Let us first suppose that E a ◦ F b ( b v ) = 0 for some basiselement v . (This means that E a ◦ F b ( b v ) = b v − b + a and that v − b + a ∈ Ψ.) By(2) and (3), we have E a ◦ F b ( b v ) = − E a ◦ H a ◦ F b ( b v )= − E a ◦ F b ◦ H a ( b v ) . It follows that H a ( b v ) = − b v , and that c ( v , a ) = −
1. In turn, this implies that v + a ∈ Ψ and E a ( b v ) = b v + a = 0. Since v − b + a ∈ Ψ, we have F b ◦ E a ( b v ) = b v + a − b = E a ◦ F b ( b v ) . It follows that if E a ◦ F b = 0, then E a ◦ F b = F b ◦ E a . The converse statement alsofollows by a similar argument. This in turn implies that E a ◦ F b = 0 if and only if F b ◦ E a = 0, which completes the proof of (6).The proofs of (8) and (11) follow the same line of argument as the proof of (6).To prove (7), we show that E a ◦ E a ( b v ) = 0 for all basis elements b v . As before,we may reduce to the case where E a ( b v ) = 0, meaning that v + a ∈ Ψ, c ( v , a ) = − c ( v + a , a ) = 1. The latter fact implies that E a ( b v + a ) = 0, which completesthe proof. The proof of (10) follows the same argument.We now prove (9). As in the proof of (7), the proof reduces to showing that E a ◦ E b ◦ E a ( b v ) = 0in the case where c ( v , a ) = −
1. Using (1) and (2), we then have E a ◦ E b ◦ E a ( b v ) = − E a ◦ H a ◦ E b ◦ E a ( b v )= − E a ◦ E b ◦ ( H a ◦ E a ( b v )) + E a ◦ E b ◦ E a ( b v )= 0 , as required. The proof of (12) follows the same argument as the proof of (9). (cid:3) We are now ready to state our main result.
Theorem 3.2.
Let Ψ be a minuscule system with respect to the simple system ∆ , and let A be the generalized Cartan matrix of ∆ . Assume that A is a sym-metrizable generalized Cartan matrix in the sense of Definition 1.1, and let g bethe corresponding derived Kac–Moody algebra. Then the C -vector space V Ψ has thestructure of a g -module, where e i (respectively, f i , h i ) acts via the endomorphism E i (respectively, F i , H i ). EPRESENTATIONS OF LIE ALGEBRAS ARISING FROM POLYTOPES 13
Proof.
We need to show that the defining relations of Theorem 1.2 are satisfied.Since the operators H a are simultaneously diagonalizable with respect to thebasis { b v : v ∈ Ψ } , they commute, and so we have [ h i , h j ] = 0.Lemma 3.1 (1) establishes the relations between the h i and the e j , and Lemma3.1 (3) establishes the relations between the h i and the f j . Lemma 3.1 (5) and (6)prove that [ e i , f j ] = 0 if i = j .We now prove that [ e i , f i ] = h i , for which we need to show that E i ◦ F i − F i ◦ E i = H i . It is enough to evaluate each side of the equation on a basis element b v . If c ( v , i ) = 0then all terms act as zero. If c ( v , i ) = 1 then E i ◦ F i ( b v ) = b v , H i ( b v ) = b v , and F i ◦ E i ( b v ) = 0, thus satisfying the equation. The case c ( v , i ) = − e i , [ e i , · · · [ e i , | {z } − A ij times e j ] · · · ]] = 0is satisfied. If A ij = 0, this states that [ e i , e j ] = 0, which is immediate from Lemma3.1 (8). If A ij = −
1, this states that[ e i , [ e i , e j ]] = 0 , or in other words, E i ◦ E i ◦ E j − E i ◦ E j ◦ E i + E j ◦ E i ◦ E i = 0 , which is immediate from Lemma 3.1 (7) and (9). The only other possibility is that A ij ≤ −
2. In this case, every term of the corresponding identity in terms of E i and E j involves an E i ◦ E i , which is zero by Lemma 3.1 (7), and this completes theproof.A similar argument shows that the Serre relation involving the f i is also satis-fied. (cid:3) The following result provides some methods of constructing new minuscule sys-tems from known ones, and these will be useful in the sequel.
Proposition 3.3.
Let Ψ ⊂ R n be a minuscule system with respect to the simplesystem ∆ . Let Ψ ′ and ∆ ′ be nonempty subsets of Ψ and ∆ , respectively. (i) Suppose that for every v ∈ Ψ ′ and a ∈ ∆ ′ , the following conditions are satisfied. (a) If c ( v , a ) = − then v + a ∈ Ψ ′ . (b) If c ( v , a ) = 1 then v − a ∈ Ψ ′ .Then Ψ ′ is a minuscule system with respect to ∆ ′ . (ii) If Ψ ′ = Ψ and ∅ 6 = ∆ ′ ⊂ ∆ then Ψ ′ is a minuscule system with respect to ∆ ′ . (iii) Let n ∈ R n and l ∈ R . Suppose that the sets Ψ( n , l ) = { v ∈ Ψ : v . n = l } and ∆( n ) = { a ∈ ∆ : a . n = 0 } are nonempty. Then Ψ( n , l ) is a minuscule system with respect to the simplesystem ∆( n ) .Proof. Definition 2.1 applied to Ψ ′ and ∆ ′ follows immediately from the hypothesesof (i). Part (ii) is an immediate consequence of (i).Part (iii) follows from (i) and the observation that if v ∈ Ψ( n , l ) and a ∈ ∆( n )then ( v ± a ) . n = l ± l . (cid:3) Definition 3.4.
If Ψ ′ and ∆ ′ satisfy the hypotheses of Proposition 3.3 (i), we willcall the pair (Ψ ′ , ∆ ′ ) a minuscule subsystem of (Ψ , ∆).We now explain how minuscule systems associated with a Lie algebra also sup-port actions of the corresponding Weyl group. Definition 3.5.
Let n ∈ N and 0 = α ∈ V = R n . The reflection s α associated to α is the linear map s α : V −→ V given by s α ( v ) = v − v .αα.α α. If Ψ ⊂ R n is a minuscule system with respect to the simple system ∆, then wedefine the Weyl group W = W Ψ , ∆ of (Ψ , ∆) to be the group of automorphisms of R n generated by the set { s a : a ∈ ∆ } . EPRESENTATIONS OF LIE ALGEBRAS ARISING FROM POLYTOPES 15
It is not hard to check that this agrees with the usual notion of the Weyl groupassociated to a simple Lie algebra over C (see [ , (1.1.2), § , § V . Proposition 3.6. If Ψ is a minuscule system with respect to the simple system ∆ ,then W = W Ψ , ∆ acts on Ψ .Proof. It is enough to show that if a ∈ ∆ and v ∈ Ψ, then s a ( v ) ∈ Ψ. By thedefinitions of s a and c = c ( v , a ), we have s a ( v ) = v − c a , which lies in Ψ byDefinition 2.1. (cid:3) §
4. The Hesse polytope In §
4, we introduce some examples of minuscule systems related to the polytopeknown in Coxeter’s notation as 3 . This polytope does not have a consistent namein the literature; we will follow Conway and Sloane in calling 3 the Hesse polytope ,as this name does not appear to have any other connotations. The Hesse polytopehas 56 vertices, whose coordinates are given by the set Ψ E of Definition 4.1. Notethat we have multiplied Conway and Sloane’s coordinates for the vertices by 4, inorder to make them integers and to retain compatibility with du Val’s coordinates[ , § Schl¨afli polytope , which is called 2 in Coxeter’s notation, also plays a rolein the examples of this section involving the Lie algebra of type E . It has 27vertices, whose coordinates can be given by either of the sets Ψ( n , ±
8) appearingin Proposition 4.3. More details on the inclusion of the Schl¨afli polytope in theHesse polytope may be found in [ , § Definition 4.1.
Let ε , ε , . . . , ε ∈ R be such that ε i has a 1 in position i + 1,and zeros elsewhere. For 0 ≤ i, j ≤
7, define the vector v i,j = v { i,j } = v j,i ∈ R by v i,j := 4( ε i + ε j ) − X i =0 ε i ! . (For example, we have v , = (3 , , − , − , − , − , − , − E consist ofthe 56 vectors {± v i,j : 1 ≤ i < j ≤ } . It is convenient for later purposes to introduce the sets K = { , , , } and K = { , , , } . Lemma 4.2.
Let Ψ E be as in Definition 4.1, and let ∆ E (1)7 = { α , α , . . . , α } , where α i = 4( ε i − ε i +1 ) if ≤ i < , and α = ( − , − , − , − , , , , . Then Ψ E is a minuscule system with respect to the simple system ∆ E (1)7 .Proof. Suppose first that a = α i for some i <
7, and let v ∈ Ψ E . Write v = P j =0 λ j ε j . The proof is a case by case check according to the values of λ i and λ i +1 . There are three cases to check.The first possibility is that λ i = λ i +1 . This implies that v .α i = 0. The coef-ficients of ε i and of ε i +1 in v + a differ by 8, which means that v + a Ψ, anda similar argument shows that v − a Ψ. The conditions of Definition 2.1 aretherefore satisfied.The second possibility is that ( λ i , λ i +1 ) ∈ { ( − , , ( − , } , that is, λ i +1 = λ i + 4. This implies that, v . a = −
16 and a . a = 32. This satisfies Definition 2.1(i) with c = −
1. In this case, v − a Ψ, because the coefficients of ε i and ε i +1 in v − a do not lie in the set {± , ± } . However, the vector v + a is obtained from v by exchanging the coefficients of ε i and ε i +1 , which means that v + a ∈ Ψ. Thissatisfies Definition 2.1 (ii).The third possibility is that ( λ i , λ i +1 ) ∈ { (3 , − , (1 , − } , that is, λ i +1 = λ i − c = 1, v + a Ψ, and v − a ∈ Ψ, as required.It remains to show that Definition 2.1 is satisfied with a = α . To check this,we use the sets K , K of Definition 4.1. Let v = ± v i,j . As before, there are threecases to check.The first possibility is that { i, j } 6⊆ K l for some l ∈ { , } . (Informally, thismeans that the two occurrences of 3 (or −
3) in v do not occur in the same half of EPRESENTATIONS OF LIE ALGEBRAS ARISING FROM POLYTOPES 17 the vector.) This implies that v .α = 0. Furthermore, neither of the two vectors v ± α lies in Ψ, because in each of them, one of the basis vectors ε i appears withcoefficient ±
5. Definition 2.1 is therefore satisfied in this case.The second possibility is that either v = + v i,j with { i, j } ⊂ K , or that v = − v i,j with { i, j } ⊂ K . In each case, v .α = −
16 and v + α ∈ Ψ. However, ineach case, we have v − α Ψ, because two basis vectors appear in v − α withcoefficient ±
5. Since α .α = 32, Definition 2.1 is satisfied with c = − v = + v i,j with { i, j } ⊂ K , or that v = − v i,j with { i, j } ⊂ K . An analysis like that of the above paragraph shows that Definition2.1 is satisfied with c = 1, v − α ∈ Ψ, and v + α Ψ. This completes the proof. (cid:3)
Proposition 4.3.
Let
Ψ = Ψ E be as in Definition 4.1, and let ∆ = ∆ E (1)7 be asin Definition 4.2. (i) The -dimensional C -vector space V Ψ has the structure of a g -module, where g is the derived affine Kac–Moody algebra of type E (1)7 . (ii) Let Ψ ′ = Ψ and ∆ ′ = ∆ \{ α } . Then Ψ ′ is a minuscule system with respect tothe simple system ∆ ′ , and V Ψ ′ is a module for the simple Lie algebra e over C of type E . It is an irreducible module with highest weight vector − v , andlowest weight vector v , (as in Definition 4.1). (iii) Let n = v , . Then we have a disjoint union Ψ = Ψ( n ,
24) ˙ ∪ Ψ( n ,
8) ˙ ∪ Ψ( n , −
8) ˙ ∪ Ψ( n , . For l ∈ { , , − , − } , Ψ( n , l ) is a minuscule system with respect to the simplesystem ∆( n ) = ∆ \{ α , α } , and V Ψ( n ,l ) is a module for the simple Lie algebra e over C of type E . The two modules V Ψ( n , ± are trivial one-dimensional mod-ules for e , whereas the two modules V Ψ( n , ± are nonisomorphic -dimensionalirreducible modules. The module V Ψ( n , has highest weight v , and lowest weight v , . The module V Ψ( n , − has highest weight − v , and lowest weight − v , . (iv) For l ∈ { , , − , − } , Ψ( n , l ) is a minuscule system with respect to the simplesystem ∆( n ) ∪ { α } , where α = 4( ε − ε ) . This makes V Ψ( n ,l ) into a module for the derived affine Kac–Moody algebra g of type E (1)6 .Proof. By Lemma 4.2, Ψ is a minuscule system with respect to the simple system∆. One may check directly (using [ , § A is thesymmetrizable generalized Cartan matrix of type E (1)7 of [ ]. Theorem 3.2 thenestablishes (i).For (ii), we know that Ψ ′ is a minuscule system with respect to the simplesystem ∆ ′ by Proposition 3.3 (ii). The matrix A in this case is the symmetrizable(generalized) Cartan matrix of type E of [ ]. It follows from Theorem 3.2 that V Ψ ′ is a module for e . Direct checks show that − v , is annihilated by all theoperators E i , v , is annihilated by all the operators F i , and − v , is annihilatedby all the operators H i except H , in which case we have H ( − v , ) = − v , . Since V Ψ ′ has the same dimension as L ( ω ) and contains a highest weight vector of weight ω , the modules V Ψ ′ and L ( ω ) are isomorphic and irreducible by Proposition 1.4.We next establish the decomposition of Ψ described in (iii). We have Ψ( n ,
24) = { v , } and Ψ( n , −
24) = {− v , } . The set Ψ( n ,
8) consists of the vectors { v ,i : 1 ≤ i ≤ } ∪ { v i, : 1 ≤ i ≤ } ∪ {− v i,j : 1 ≤ i < j ≤ } , and we have Ψ( n , −
8) = − Ψ( n , n , l ) is a minuscule systemwith respect to ∆( n ), and Theorem 3.2 shows that the modules V Ψ( n ,l ) are modulesfor e (after the generalized Cartan matrix has been recognized as symmetrizableof type E ). The assertions about dimensions and weight vectors are easy to check.A quick calculation shows that, for H i . v , = (cid:26) v , if i = 1 , i ∈ { , , , , } ;in contrast, we have H i . ( − v , ) = (cid:26) − v , if i = 5 , i ∈ { , , , , } . EPRESENTATIONS OF LIE ALGEBRAS ARISING FROM POLYTOPES 19
This shows that V Ψ( n , (respectively, V Ψ( n , − has the same dimension as, and anonzero weight vector of the same weight as L ( ω ) (respectively, L ( ω )). Proposi-tion 1.4 now shows that the two modules V Ψ( n , ± are irreducible and nonisomor-phic.To prove (iv), we need to check that Definition 2.1 is satisfied with a = α . Thisfollows by imitating the case analysis for the case i < n .α = 0. (cid:3)
5. The hypercube In §
5, we consider examples relating to the polytope known as the the hypercube or measure polytope ; in Coxeter’s notation it is denoted γ n . The set Ψ defined inLemma 5.1 is our standard set of coordinates for the 2 n vertices of the hypercube.We will show how the hypercube may be used to construct the spin represen-tations of the simple Lie algebras of types B n and D n . By passing to appropriatesubsystems, we obtain all the fundamental representations of the simple Lie algebraof type A n as a by-product. Lemma 5.1.
Let n ≥ , let ε , . . . , ε n − ∈ R n be the usual basis for R n , and let Ψ be the set of n vectors of the form ( ± , ± , . . . , ± . Let ∆ = { α , α , . . . , α n } , where α = − ε + ε ) , α n = 4 ε n − , and α i = 4( ε i − − ε i ) for < i < n . Then Ψ is a minuscule system with respect to the simple system ∆ .Proof. We check that Definition 2.1 holds for each of the α i in turn. Let v = P n − j =0 λ j ε j ∈ Ψ.Suppose first that 0 < i < n . The proof is a case by case check according tothe values of λ i and λ i +1 . There are three cases to check, and we omit the detailsbecause the cases are almost identical to those in the first part of the argumentproving Lemma 4.2. Next, suppose that i = 0. There are three cases to check, according to thevalues of λ and λ . If λ = λ = +2, then v + α ∈ Ψ, v − α Ψ, and2 v .α = −
32 = − α .α , giving c = c ( v , α ) = − λ = λ = − v − α ∈ Ψ, v + α Ψ, and 2 v .α = 32 = α .α , giving c = 1 as required.If λ = λ , then neither vector v ± α lies in Ψ, and 2 v .α = 0, giving c = 0.Definition 2.1 is therefore satisfied in all three cases.Finally, suppose that i = n . There are two cases to check, according to thevalue of λ n − . If λ n − = +2 then v − α n ∈ Ψ and v + α n Ψ. We also have2 v .α n = 16 = α n .α n , giving c = c ( v , α n ) = 1, thus satisfying Definition 2.1. If λ n − = − v + α n ∈ Ψ and v − α n Ψ. We also have 2 v .α n = −
16 = − α n .α n ,giving c = −
1, thus satisfying Definition 2.1 and completing the proof. (cid:3)
We may now state an analogue of Proposition 4.3.
Proposition 5.2.
Maintain the notation of Definition 5.1. Let j = P n − j =0 ε j and S = { n − j : 0 ≤ j ≤ n } . (i) The n -dimensional C -vector space V Ψ has the structure of a g -module, where g is the derived affine Kac–Moody algebra of type B (1) n . (ii) Let Ψ ′ = Ψ and ∆ ′ = ∆ \{ α } . Then Ψ ′ is a minuscule system with respect tothe simple system ∆ ′ , and V Ψ ′ is a module for the simple Lie algebra b n over C of type B n . It is an irreducible module with highest weight vector j and lowestweight vector − j , and affords the spin representation of b n . (iii) We have a disjoint union
Ψ = n [ j =0 Ψ( j , n − j ) . For l ∈ S , Ψ( j , l ) is a minuscule system with respect to the simple system ∆( j ) = ∆ \{ α , α n } , EPRESENTATIONS OF LIE ALGEBRAS ARISING FROM POLYTOPES 21 and V Ψ( j ,l ) is a module for the simple Lie algebra a n − over C of type A n − .The two modules V Ψ( j , ± n ) are trivial one-dimensional modules for a n − , andthe other modules V Ψ( j ,l ) satisfy V Ψ( j , n − j ) ∼ = L ( ω n − j ) . The module V Ψ( j , n − j ) has highest weight − j + 4 n − j − X i =0 ε i ! and lowest weight j − j − X i =0 ε i ! . (iv) For l ∈ S , Ψ( j , l ) is a minuscule system with respect to the simple system ∆( j ) ∪{ α } , where α = 4( ε n − − ε ) . This makes V Ψ( j ,l ) into a module for the derivedaffine Kac–Moody algebra g of type A (1) n − .Proof. Using Lemma 5.1 in place of Lemma 4.2, the proof of (i) follows the sameargument as the proof of Proposition 4.3 (i).The proof of (ii) now follows by copying the argument of Proposition 4.3 (ii). Inthis case, the module turns out to be L ( ω n ).It is easily checked that Ψ( j , n − j ) consists precisely of the vectors in Ψ thathave j occurrences of −
2, from which the first assertion of (iii) follows. Proposition3.3 (iii) shows that Ψ( j , n − j ) is a minuscule system with respect to ∆( j ), andTheorem 3.2 shows that the modules V Ψ( j , n − j ) are modules for a n − (after thegeneralized Cartan matrix has been recognized as symmetrizable of type A n − ).The assertions about dimensions and weight vectors are easy to check. If j = ± n and v is the highest weight vector v of V Ψ( j , n − j ) , then we have H i . v = 0 unless i = n − j , in which case H i . v = v . The required isomorphism now follows fromProposition 1.4.To prove (iv), we may copy the argument of Proposition 4.3 (iv) to check thatDefinition 2.1 is satisfied with a = α . (Note that j .α = 0.) (cid:3) Lemma 5.3.
Let n ≥ , let ε , . . . , ε n − ∈ R n be the usual basis for R n , and let Ψ be as in Lemma 5.1. Let Ψ + D (respectively, Ψ − D ) be the subset of Ψ whose vectorscontain an even (respectively, odd) number of occurrences of − .Let ∆ D = { α , α , . . . , α n − , α ′ n } , where α ′ n = 4( ε n − + ε n − ) and the othervectors α i are as in Lemma 5.1.Then Ψ = Ψ + D ˙ ∪ Ψ − D is a minuscule system with respect to the simple system ∆ D , and both (Ψ + D , ∆ D ) and (Ψ − D , ∆ D ) are minuscule subsystems of (Ψ , ∆ D ) .Proof. Most of the work for checking that Ψ is a minuscule system with respect to∆ D is done in the proof of Lemma 5.1. The only extra criterion to check is thatDefinition 2.1 holds for a = α ′ n . This follows by making appropriate sign changesto the argument used to check Definition 2.1 for a = α as in the proof of Lemma5.1.Letting j be as in Proposition 5.2, we see that v ∈ Ψ lies in Ψ + D if the integer v . j is a multiple of 8, and v lies in Ψ − D otherwise. We observe that each a ∈ ∆ D hasthe property that a . j is a multiple of 8. We now apply Proposition 3.3 (i), whichproves that (Ψ ± D , ∆ D ) are minuscule subsystems. (cid:3) Proposition 5.4.
Maintain the notation of 5.1–5.3. (i)
Each of the n − -dimensional C -vector spaces V Ψ ± D has the structure of a g -module, where g is the derived affine Kac–Moody algebra of type D (1) n . (ii) Let Ψ ± = Ψ ± D and ∆ ± = ∆ D \{ α } . Then each of the two sets Ψ ± is a minusculesystem with respect to each of the simple systems ∆ ± respectively, and each ofthe two spaces V Ψ ± is a module for the simple Lie algebra d n over C of type D n .The modules are nonisomorphic and both irreducible, and they afford the twospin representations of d n . The highest weight vector of V Ψ + (respectively, V Ψ − )is j (respectively, j − ε n − ). The lowest weight vectors of V Ψ ± are − j and − j + 4 ε n − , where the assignment of vectors to modules depends on whether n is even or odd.Proof. Using Lemma 5.3 in place of Lemma 4.2, the proof of (i) follows the same
EPRESENTATIONS OF LIE ALGEBRAS ARISING FROM POLYTOPES 23 argument as the proof of Proposition 4.3 (i).The first assertion of (ii) follows by using Lemma 5.3 and copying the argumentof Proposition 4.3 (ii). The operators H i (for 1 ≤ i < n −
1) all act as zero on 2 j and 2 j − ε n − . The operator H n − (corresponding to α n − ) acts as zero on 2 j andacts as the identity on 2 j − ε n − . The operator H n (corresponding to α ′ n ) acts asthe identity on 2 j and as zero on 2 j − ε n − . The second assertion is then provedby adapting the corresponding argument in Proposition 4.3 (iii). (cid:3)
6. The hyperoctahedron In §
6, we consider examples relating to the polytope known as the the hyper-octahedron or cross polytope ; in Coxeter’s notation it is denoted β n . The set Ψdefined in Lemma 6.1 is our standard set of coordinates for the 2 n vertices of thehyperoctahedron.We will show how to use the hyperoctahedron to construct the remaining twotypes of minuscule representations, namely the natural representations for Lie al-gebras of types C n and D n . Lemma 6.1.
Let n ≥ , let ε , . . . , ε n − ∈ R n be the usual basis for R n , and let Ψ = {± ε i : 0 ≤ i ≤ n − } . Let ∆ D be as in Lemma 5.3. Then Ψ is a minuscule system with respect to ∆ D .Proof. We check Definition 2.1, treating each vector a ∈ ∆ D in turn. Suppose firstthat a = α i for some 1 ≤ i ≤ n −
1, and let v ∈ Ψ.Define ε j to be the unique basis element such that v = ± ε j . If j
6∈ { i − , i } then we have c = c ( v , a ) = 0 and neither vector v ± a lies in Ψ, satisfying Definition2.1 (ii). If v ∈ { ε i − , − ε i } then v − a ∈ Ψ, 2 v . a = 32 = a . a , giving c = 1 asrequired. The other possibility is that v ∈ {− ε i − , ε i } , in which case v + a ∈ Ψ,2 v . a = −
32 = − a . a , giving c = − a = α ′ n . In this case, if j
6∈ { n − , n − } then we have c = c ( v , a ) =0 and neither vector v ± a lies in Ψ. If v ∈ { ε n − , ε n − } then v − a ∈ Ψ, v + a Ψ, and 2 v . a = 32 = a . a , giving c = 1 as required. The other possibility is that v ∈ {− ε n − , − ε n − } , in which case v + a ∈ Ψ, v − a Ψ, and 2 v . a = −
32 = − a . a ,giving c = − a = α . In this case, if j
6∈ { , } then we have c = c ( v , a ) = 0and neither vector v ± a lies in Ψ. If v ∈ {− ε , − ε } then v − a ∈ Ψ, v + a Ψ,and 2 v . a = 32 = a . a , giving c = 1 as required. The other possibility is that v ∈ { ε i − , ε i } , in which case v + a ∈ Ψ, v − a Ψ, and 2 v . a = −
32 = − a . a ,giving c = − (cid:3) Proposition 6.2.
Maintain the notation of Lemma 6.1. (i)
The n -dimensional C -vector space V Ψ has the structure of a g -module, where g is the derived affine Kac–Moody algebra of type D (1) n . (ii) Let Ψ ′ = Ψ and ∆ ′ = ∆ \{ α } . Then Ψ ′ is a minuscule system with respect tothe simple system ∆ ′ , and V Ψ ′ is a module for the simple Lie algebra d n over C of type D n . It is an irreducible module with highest weight vector ε and lowestweight vector − ε , and affords the natural representation of d n .Proof. Using Lemma 6.1 in place of Lemma 4.2, the proof of (i) follows the sameargument as the proof of Proposition 4.3 (i).The first assertion of (ii) follows by using Lemma 6.1 and copying the argumentof Proposition 4.3 (ii). The operators H i (for 1 < i ≤ n , where H n corresponds to α ′ n ) all act as zero on 4 ε . The operator H acts as the identity on 4 ε . The secondassertion is then proved by adapting the corresponding argument in Proposition4.3 (ii). (cid:3) Lemma 6.3.
Let n ≥ , let Ψ be as in Lemma 6.1, and let ∆ C = { α , . . . , α n − } ∪ { α ′′ , α ′′ n } , where α i is as in Lemma 5.1 for ≤ i ≤ n − , α ′′ = − ε and α ′′ n = 8 ε n − . Then Ψ is a minuscule system with respect to ∆ C . EPRESENTATIONS OF LIE ALGEBRAS ARISING FROM POLYTOPES 25
Proof.
We check Definition 2.1, treating each vector a ∈ ∆ C in turn. The onlycases not already covered by Lemma 6.1 are the cases where a ∈ { α ′′ , α ′′ n } . Let v ∈ Ψ, and define ε j to be the unique basis element such that v = ± ε j .Suppose that a = α ′′ n . If j = n − c = c ( v , a ) = 0 and neithervector v ± a lies in Ψ, satisfying Definition 2.1. If v = ± ε n − then v ∓ a ∈ Ψ, v ± a Ψ, and 2 v . a = ±
64 = ± a . a , giving c = ± a = α ′′ . If j = 0 then we have c = c ( v , a ) = 0 andneither vector v ± a lies in Ψ, satisfying Definition 2.1. If v = ± ε then v ± a ∈ Ψ, v ∓ a Ψ, and 2 v . a = ∓
64 = ∓ a . a , giving c = ∓ (cid:3) Proposition 6.4.
Maintain the notation of Lemma 6.3. (i)
The n -dimensional C -vector space V Ψ has the structure of a g -module, where g is the derived affine Kac–Moody algebra of type C (1) n . (ii) Let Ψ ′ = Ψ and ∆ ′ = ∆ \{ α ′′ } . Then Ψ ′ is a minuscule system with respect tothe simple system ∆ ′ , and V Ψ ′ is a module for the simple Lie algebra c n over C of type C n . It is an irreducible module with highest weight vector ε and lowestweight vector − ε , and affords the natural representation of c n .Proof. The proof is the same as the proof of Proposition 6.2, using Lemma 6.3 inplace of Lemma 6.1. (cid:3)
7. Lines on Del Pezzo surfaces In §
7, we revisit the examples of § e and e . We will highlight the close link between the representation theory andthe combinatorial algebraic geometry associated with configurations of lines on DelPezzo surfaces. For more details on the latter, the reader is referred to [ , § V.4].
Lemma 7.1.
Let Ψ and ∆ be as in Proposition 4.3, and let K and K be asin Definition 4.1. The action of the generators { s a : a ∈ ∆ } of the Weyl group W = W Ψ , ∆ on Ψ are as follows. If a = α i with ≤ i ≤ , then s a ( ± v j,k ) = s a ( ± v s i ( j ) ,s i ( k ) ) , where s i is the simple transposition ( i, i + 1) . We have s α ( ± v i,j ) = ± v i,j unless { i, j } ⊂ K k for some k ∈ { , } , in which case we have s α ( ± v { i,j } ) = ∓ v K k \{ i,j } . The action of W on Ψ is transitive. (cid:3) Proof.
The formulae for the action of the s a are obtained by a routine case by casecheck.The action of the s α i for 0 ≤ i ≤ { + v i,j : 0 ≤ i The transformations induced by s α described in the preceding proofare sometimes known as bifid transformations (see Example 3 of [ , § Lemma 7.3. Let Ψ and ∆ be as in Proposition 4.3. The diagonal action of theWeyl group W on Ψ × Ψ has four orbits, each of which consists of a set { ( v , v ) : v , v ∈ Ψ and | v − v | = D } for some fixed number D . More explicitly, the orbits are as follows: (i) { ( v , v ) : v ∈ Ψ } , corresponding to D = 0 ; (ii) { ( ± v i,j , ± v i,k ) : |{ i, j, k }| = 3 }∪{ ( ± v i,j , ∓ v k,l ) : |{ i, j, k, l }| = 4 } , correspondingto D = √ , (iii) { ( ± v i,j , ∓ v i,k ) : |{ i, j, k }| = 3 }∪{ ( ± v i,j , ± v k,l ) : |{ i, j, k, l }| = 4 } , correspondingto D = √ , (iv) { ( v , − v ) : v ∈ Ψ } , corresponding to D = √ .Proof. The assertions about D are easy to check. This other assertions, which arealso not difficult to prove, are a restatement of [ , (4.1)]. (cid:3) Proposition 7.4. The elements of Ψ are in natural bijection with the linesof the Del Pezzo surface of degree ; more precisely, if v , v ∈ Ψ are distinct points EPRESENTATIONS OF LIE ALGEBRAS ARISING FROM POLYTOPES 27 with | v − v | = √ D , then D − is the intersection number of the lines corre-sponding to v and v . In particular, pairs of points at distance √ correspond toskew lines on the Del Pezzo surface.The elements of Ψ( n , (defined in Proposition 4.3 (iii)) are { v ,i : 1 ≤ i ≤ } ∪ {− v i,j : 1 ≤ i < j ≤ } ∪ { v i, : 1 ≤ i ≤ } . These are in natural bijection with the lines of the Del Pezzo surface of degree : in Hartshorne’s notation [ , Theorem V.4.9] , we identify E i with v ,i , F ij with − v i,j and G i with v i, . The intersection number is defined as in the case of the lines.Proof. The assertions about the Del Pezzo surface of degree 2 are proved in [ ,p28], where it is shown that | v − v | = d ( x + 1) , where v and v are two distinct points of Ψ, d is the minimal nontrivial distancebetween two points, and x is the intersection number of the pair of lines corre-sponding to v and v . (The precise link with the polytopes 2 and 3 is givenon [ , p33].)It is easily checked that the 27 elements of Ψ( n , 8) are as listed. By the resultmentioned above, the only possible intersection numbers for two distinct lines onthe Del Pezzo surface of degree 3 are 0 (meaning the lines are skew) and 1 (meaningthe lines are incident). Since no two elements of Ψ( n , 8) are at distance √ 96, itremains to check that two distinct points of Ψ( n , 8) are at distance √ 32 if andonly if the corresponding lines are skew, and this follows from the rules given in [ ,Remark V.4.10.1]. (cid:3) Note that, because Ψ( n , − 8) = − Ψ( n , e are interchangeable in this context.The next result explains how to recover the root system of type E from the set E of directed edges of the polytope 3 . Proposition 7.5. Maintain the notation of Lemma 7.3. Let E = { ( v , v ) : v , v ∈ Ψ and | v − v | = √ } has size . The vectors E ′ = { v − v : ( v , v ) ∈ E } form a root system oftype E , and each of the roots occurs with multiplicity in E .Proof. If v = v , then one checks directly that there are 27 vectors v such that( v , v ) ∈ E . The fact (Lemma 7.1) that W acts transitively on Ψ implies byLemma 7.1 that E has size | Ψ | × 27 = 1512.For the second assertion, note that v , − v , = α . The (additive) action of W on Ψ induces an action on E ′ , and Lemma 7.3 (ii) shows that W acts transitivelyon E ′ . Note that if A ij = A ji = − 1, then s i s j ( α i ) = α j . This implies that allthe roots α i are conjugate under the action of the Weyl group, and then [ , § W.α consists precisely of the root system of type E . Bytransitivity of the action of W on the root system, each root in E ′ occurs withthe same multiplicity, and by [ , Appendix] there are 126 roots of type E . Since1512 / 126 = 12, the proof is completed. (cid:3) Proposition 7.6. Let Ψ ′ and ∆ ′ be as in Proposition 4.3, and let V Ψ ′ be thecorresponding -dimensional representation of the Lie algebra e . If v ∈ Ψ ′ and x ∈ e , then we have x. v = X u ∈ Ψ ′′ λ u u , where Ψ ′′ = { v } ∪ { u : | v − u | = √ } . In other words, if λ u = 0 , then either u = v or the lines on the Del Pezzo surface of degree corresponding to u and v are skew.A similar result holds for either of the -dimensional representations of e andthe Del Pezzo surface of degree .Proof. By [ , § e = h ⊕ M α ∈ Φ g α , EPRESENTATIONS OF LIE ALGEBRAS ARISING FROM POLYTOPES 29 where h is a 7-dimensional Cartan subalgebra, Φ is the root system for e , and thesubspaces g α are one-dimensional. We identify e with the algebra of operators onthe 56-dimensional module V as described in Proposition 4.3 (ii).With these identifications, if α = α i for i = 0, then g α (respectively, g − α isspanned by the Lie algebra element E i = E α i (respectively, F i = F α i ). The Cartansubalgebra h has as a basis the operators H i = H α i for 1 ≤ i ≤ e in which (a) the subspace g α for α a positive root is spanned by a vector of the form[ · · · [[ E i , E i ] E i ] · · · E i m ]where α = P mj =1 α i j and (b) the subspace g α for α a negative root is spanned bya vector of the form [ · · · [[ F i , F i ] F i ] · · · F i m ]where − α = P mj =1 α i j . (See [ , Proposition 5.4 (ii), (iv)] or [ , (7.8.5)] for moredetails.)It follows that if b v is a basis element of V , α ∈ Φ and g α ∈ g α , then g α .b v = λb v + α for some scalar λ (meaning that g α .b v = 0 if v + α Ψ). If λ = 0, thenProposition 7.5 shows that the distance from v to v + α is √ 32. By Proposition7.4, we see that v and v + α correspond to skew lines. It follows easily from thedefinition of the H i that if h ∈ H then h.b v = λb v for some scalar λ .Combining these observations proves the assertions about the 56-dimensionalrepresentation. The argument can be easily adapted to work for the 27-dimensionalrepresentation, because the root system of type E embeds naturally into the rootsystem of type E . (cid:3) 8. Concluding remarks In the various constructions presented above for irreducible modules for simpleLie algebras, we did not provide self-contained proofs that the modules constructed were irreducible. However, this was done only to save space, and it is not hard togive an elementary field-independent proof that these modules are irreducible.One application of the polytope approach to minuscule representations is thatone can describe the crystal graph of each of the irreducible modules that arisesfrom the construction directly in terms of the polytope. To do this, one starts withthe vertices of Ψ, and for each element a ∈ ∆ corresponding to a simple root of thesimple Lie algebra, one connects two vertices v and v of Ψ by an edge labelled a if v − v = a . It is not hard to show that this produces a realization of the crystalgraph, with the extra property that two edges are parallel if and only if they havethe same label.In the cases where the pair (Ψ , ∆) corresponds to a representation of a simpleLie algebra, the elements of Ψ may be partially ordered by stipulating that v ≤ v if v − v is a positive linear combination of elements of ∆; this corresponds to theusual partial order on the weights of a representation. The resulting partial order onΨ makes Ψ into a distributive lattice under the operations of greatest lower boundand least upper bound. (This is not a priori obvious, but follows, for example, fromthe full heaps approach; see [ , Corollary 2.2].) It would be interesting to knowwhether there is an easy way to define the meet and join operations directly fromthe data (Ψ , ∆).It may be tempting to think that one can describe a basis for each of the simpleLie algebras described in this paper by including operators E a and F a for every positive root a . However, such an algebra of operators would not be closed underthe Lie bracket (except in trivial cases) and what is needed instead is to modifythe definition of these new operators to introduce sign changes in certain places.We do not know if there is an easy way to keep track of these signs using theformalism developed in this paper, although there is a good way to do it in the fullheaps approach, using the notion of the “parity” of a heap; see [ , Definition 4.3,Definition 6.3] for details. EFERENCES 31 References [ ] S.C. Billey and V. Lakshmibai, Singular Loci of Schubert Varieties , Progr. Math. 182,Birkh¨auser, Boston, 2000.[ ] R.W. 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