Polynomial Representation of F 4 and a New Combinatorial Identity about Twenty-Four
aa r X i v : . [ m a t h . R T ] O c t Polynomial Representation of F and a NewCombinatorial Identity about Twenty-Four Xiaoping Xu
Institute of Mathematics, Academy of Mathematics & System SciencesChinese Academy of Sciences, Beijing 100190, P.R. China Abstract
Singular vectors of a representation of a finite-dimensional simple Lie algebra are weightvectors in the underlying module that are nullified by positive root vectors. In this article, weuse partial differential equations to find all the singular vectors of the polynomial representationof the simple Lie algebra of type F over its basic irreducible module. As applications, we obtain anew combinatorial identity about the number 24 and explicit generators of invariants. Moreover,we show that the number of irreducible submodules contained in the space of homogeneousharmonic polynomials with degree k ≥ ≥ [ | k/ | ] + [ | ( k − / | ] + 2. It has been known for may years that the representation theory of Lie algebra is closelyrelated to combinatorial identities. Macdonald [M] generalized the Weyl denominatoridentities for finite root systems to those for infinite affine root systems, which are nowknown as the Macdonald’s identities. Lepowsky and Garland [LG] gave a homologicproof of Macdonald’s identities. Kac (e.g., cf [Ka]) derived these identities from hisgeneralization of Weyl’s character formula for the integrable representations of affine Kac-Moody algebras, known as Weyl-Kac formula. Lepowsky and Wilson [LW1, LW2] founda representation theoretic proof of the Rogers-Ramanujan identities. There are a numberof the other works relating combinatorial identities to representations of Lie algebras. Research supported by China NSF 10871193
1e present here a consequence of the Macdonald’s identities taken from Kostant’swork [Ko1]. Let G be a finite-dimensional simple Lie algebra over the field C of com-plex numbers. Denote by Λ + the set of dominant weights of G and by V ( λ ) the finite-dimensional irreducible G -module with highest weight λ . It is known that the Casimiroperator takes a constant c ( λ ) on V ( λ ). Macdonald’s Theorem implies that there existsa map χ : Λ + → {− , , } such that the following identity holds:( ∞ Y n =1 (1 − q n )) dim G = X λ ∈ Λ + χ ( λ )(dim V ( λ )) q c ( λ ) . (1 . G .The number 24 is important in our life; for instance, we have 24 hours a day. Math-ematically, it is also a very special number. The minimal length of doubly-even self-dualbinary linear codes is 24. Indeed there is a unique such code of length 24 (cf. [P]), knownas the binary Golay code (cf. [Go]). The automorphism group of this code is a sporadicfinite simple group. The minimal dimension of even unimodular (self-dual) integral lin-ear lattices without elements of square length 2 is also 24. Again there exists a uniquesuch lattice of dimension 24 (cf. [C1]), known as Leech’s lattice (cf. [Le]). Conway [C2]found three sporadic finite simple groups from the automorphism group of Leech’s lat-tice. Griess [Gr] constructed the Monster, the largest sporadic finite simple group, as theautomorphism group of a commutative non-associative algebra related to Leech’s lattice.The Dedekind function η ( z ) = q / ∞ Y n =1 (1 − q n ) with q = e πzi (1 . / q / is crucial forthe modularity. Moreover, the Ramanujan series∆ ( z ) = ( η ( z )) = q ∞ Y n =1 (1 − q n ) = ∞ X n =1 τ ( n ) q n , (1 . τ ( n ) is called the τ - function of Ramanujan. The function τ ( n ) is multiplicative andhas nice congruence properties such as τ (7 m + 3) ≡ τ (23 + k ) ≡ k is any quadratic non-residue of 23. The theta series of an integral linear latticeis the generating function of counting the numbers of lattice points on spheres. Hecke[He] proved that the theta series of any even unimodular lattice must be a polynomialin the Essenstein series E ( z ) and the Ramanujan series ∆ ( z ). The factor (1 − q n ) is important to ∆ ( z ). Let λ r be the r th fundamental weight of G . In this article, we2btain the following identity(1 + q )(1 + q + q ) = (1 − q ) ∞ X m,n,k =0 (dim V ( mλ + ( n + k ) λ )) q m +2 n + k (1 . G is the simple Lie algebra of type F . In other words, the dimensions of the modules V ( kλ + lλ ) are linearly correlated by the binomial coefficients of 24. Numerically,dim V ( kλ + lλ ) = ( l + 1)( k + 3)( k + l + 4)39504568320000 (2 k + l + 7)(3 k + l + 10)(3 k + 2 l + 11) × ( Y r =1 ( k + r ))( Y s =2 ( k + l + s ))( Y q =5 (2 k + l + q )) . (1 . G F of type F can be realized as thefull derivation algebra of the unique exceptional finite-dimensional simple Jordan algebra,which is 27-dimensional (e.g., cf. [A]). The identity element spans a one-dimensionaltrivial module. The quotient space of the Jordan algebra over the trivial module formsa 26-dimensional irreducible G F -module, which is the unique G F -module of minimaldimension (So it is called the basic module of G F ). Singular (highest-weight) vectors ofa representation of G F are weight vectors in the underlying module that are nullified bypositive root vectors. In this article, we use partial partial differential equations to find allthe singular vectors in the polynomial algebra over the basic irreducible module of G F .Then the identity (1.4) is a consequence of the Weyl’s theorem of complete reducibility.Another corollary of our main theorem is that the algebra of polynomial invariants overthe basic module is generated by two explicit invariants. In addition, there is also a simpleapplication to harmonic analysis.Denote by E r,s the square matrix with 1 as its ( r, s )-entry and 0 as the others. Theorthogonal Lie algebra o ( n, C ) = X ≤ r j. (1 . i, i + j = { i, i + 1 , i + 2 , ..., i + j } (1 . i and positive integer j throughout this article.Since our proof of the main theorem heavily depends on precise explicit representationformulas, we present in Section 2 a construction of the basic representation of G F fromthe simple Lie algebra G E of type E . In this way, the reader has the whole picture ofour story, and it is easier for us to track errors. The proofs of our main theorem and itscorollaries are given in Section 3 4 Basic Representation of F E . As weall known, the Dynkin diagram of E is as follows: E : ❡ ❡ ❡ ❡ ❡ ❡ { α i | i ∈ , } be the simple positive roots corresponding to the vertices in thediagram, and let Φ E be the root system of E . Set Q E = X i =1 Z α i , (2 . E . Denote by ( · , · ) the symmetric Z -bilinear form on Q E suchthat Φ E = { α ∈ Q E | ( α, α ) = 2 } . (2 . E , we have the following automorphism of Q E : σ ( X i =1 k i α i ) = k α + k α + k α + k α + k α + k α (2 . P i =1 k i α i ∈ Q E . Define a map F : Q E × Q E → { , − } by F ( X i =1 k i α i , X j =1 l j α j ) = ( − P i =1 k i l i + k l + k l + k l + k l + k l , k i , l j ∈ Z . (2 . α, β, γ ∈ Q E , F ( α + β, γ ) = F ( α, γ ) F ( β, γ ) , F ( α, β + γ ) = F ( α, β ) F ( α, γ ) , (2 . F ( α, β ) F ( β, α ) − = ( − ( α,β ) , F ( α, α ) = ( − ( α,α ) / . (2 . F ( α, β ) = − F ( β, α ) if α, β, α + β ∈ Φ E . (2 . F ( σ ( α ) , σ ( β )) = F ( α, β ) for α, β ∈ Q E . (2 . H = M i =1 C α i . (2 . E is G E = H ⊕ M α ∈ Φ E C E α (2 . · , · ] determined by:[ H, H ] = 0 , [ h, E α ] = − [ E α , h ] = ( h, α ) E α , [ E α , E − α ] = − α, (2 . E α , E β ] = (cid:26) α + β Φ E ,F ( α, β ) E α + β if α + β ∈ Φ E . (2 . σ of the Lie algebra G E with order 2:ˆ σ ( X i =1 b i α i ) = X i =1 b i σ ( α i ) , b i ∈ C , (2 . σ ( E α ) = E σ ( α ) for α ∈ Φ E . (2 . F is F : ❡ ❡ i ❡ ❡ F of type F . In particular, we let { ¯ α , ¯ α , ¯ α , ¯ α } be the simple positive roots correspondingto the above Dynkin diagram of F , where ¯ α , ¯ α are long roots and ¯ α , ¯ α are short roots.The simple Lie algebra of type F is G F = { u ∈ G E | ˆ σ ( u ) = u } (2 . H F = C ( α + α ) + C ( α + α ) + C α + F α . (2 . V = { w ∈ G E | ˆ σ ( w ) = − w } . (2 . V forms the basic 26-dimensional G F -module with representation ad | G E .Next we want to find the explicit representation formulas of the root vectors of G F on V in terms of differential operators. Note the σ -invariant positive roots of Φ E are: α , α , α + α , X i =3 α i , X i =2 α i , α + X i =3 α i , α + X i =2 α i , X i =1 α i , (2 . α + X i =1 α i , X i =1 α i + X r =3 α r , α + X i =1 α i + X r =3 α r , α + X i =1 α i + X r =2 α r . (2 . σ : α , α , α + α , α + α , α + α + α , α + α + α , α + α + α + α , (2 . X i =1 α , X i =1 α i , α + X i =1 α i , α + α + X i =1 α i , α + α + X i =1 α i . (2 . x = E α + α + P i =1 α i − E α + α + P i =1 α i , x = E α + α + P i =1 α i − E α + α + P i =2 α i , (2 . x = E α + P i =1 α i − E α + P i =2 α i , x = E P i =1 α i − E P i =2 α i , (2 . x = E P i =1 α i − E α + P i =4 α i , x = E α + P i =3 α i − E P i =3 α i , (2 . x = E α + α + α − E α + α + α , x = E α + α + α − E P i =4 α i , x = E α + α − E α + α (2 . x = E α + α − E α + α , x = E α − E α , x = E α − E α , (2 . x = α − α , x = α − α , x = E − α − E − α , (2 . x = E − α − E − α , x = E − α − α − E − α − α , x = E − α − α − E − α − α , (2 . x = E − α − α − α − E − P i =4 α i , x = E − α − α − α − E − α − α − α , (2 . x = E − α − P i =3 α i − E − P i =3 α i , x = E − P i =1 α i − E − α − P i =4 α i , (2 . x = E − P i =1 α i − E − P i =2 α i , x = E − α − α − P i =1 α i − E − α − α − P i =2 α i , (2 . x = E − α − P i =1 α i − E − α − P i =2 α i , x = E − α − α − P i =1 α i − E − α − α − P i =1 α i . (2 . { x i | i ∈ , } forms a basis of V and we treat all x i as variables for technicalconvenience.Again denote by E ¯ α the root vectors of G F as follows: ε = ± E ε ¯ α = E εα , E ε ¯ α = E εα , E ε ¯ α = E εα + E εα , E ε ¯ α = E εα + E εα , (2 . E ε (¯ α +¯ α ) = E ε ( α + α ) , E ε (¯ α +¯ α ) = E ε ( α + α ) + E ε ( α + α ) , (2 . E ε (¯ α +¯ α ) = E ε ( α + α ) + E ε ( α + α ) , E ε (¯ α +¯ α +¯ α ) = E ε ( α + α + α ) + E ε ( α + α + α ) , (2 . E ε (¯ α +¯ α +¯ α ) = E ε ( α + α + α ) + E ε ( α + α + α ) , E ε (¯ α +2¯ α ) = E ε P i =3 α i , (2 . E ε (¯ α +¯ α +2¯ α ) = E ε P i =2 α i , E ε (¯ α +2¯ α +¯ α ) = E ε ( α + P i =3 α i ) + E ε P i =3 α i , (2 . E ε P i =1 ¯ α i = E ε ( P i =1 α i ) + E ε ( α + P i =4 α i ) , E ε (¯ α +2¯ α +2¯ α ) = E ε ( α + P i =2 α i ) , (2 . E ¯ α + ε P i =1 ¯ α i = E ε ( P i =1 α i ) + E ε ( P i =2 α i ) , E ε (¯ α +2¯ α +2¯ α ) = E ε ( α + P i =3 α i ) (2 . E ε (¯ α +2¯ α +2¯ α +¯ α ) = E ε ( α + P i =1 α i ) + E ε ( α + P i =2 α i ) , (2 . ε (¯ α +¯ α +2¯ α +2¯ α ) = E ε P i =1 α i , E ε (¯ α +2 P i =2 ¯ α i ) = E ε ( α + P i =1 α i ) , (2 . E ε (¯ α +2¯ α +3¯ α +¯ α ) = E ε ( α + α + P i =1 α i ) + E ε ( α + α + P i =2 α i ) , (2 . E ε (¯ α +2¯ α +3¯ α +2¯ α ) = E ε ( α + α + P i =1 α i ) + E ε ( α + α + P i =1 α i ) , (2 . E ε (¯ α +2¯ α +4¯ α +2¯ α ) = E ε ( P i =1 α i + P r =3 α r ) , (2 . E ε (¯ α +3¯ α +4¯ α +2¯ α ) = E ε ( α + P i =1 α i + P r =3 α r ) , (2 . E ε (2¯ α +3¯ α +4¯ α +2¯ α ) = E ε ( α + P i =1 α i + P r =2 α r ) . (2 . h = α , h = α , h = α + α , h = α + α . (2 . E ¯ α | V = x ∂ x + x ∂ x + x ∂ x − x ∂ x − x ∂ x − x ∂ x , (2 . E ¯ α | V = x ∂ x + x ∂ x + x ∂ x − x ∂ x − x ∂ x − x ∂ x , (2 . E ¯ α | V = − x ∂ x − x ∂ x − x ∂ x + x ∂ x + x ( ∂ x − ∂ x ) − x ∂ x − x ∂ x + x ∂ x + x ∂ x + x ∂ x , (2 . E ¯ α | V = − x ∂ x − x ∂ x − x ∂ x − x ∂ x + x ( ∂ x − ∂ x ) − x ∂ x + x ∂ x + x ∂ x + x ∂ x + x ∂ x , (2 . E ¯ α +¯ α | V = − x ∂ x + x ∂ x + x ∂ x − x ∂ x − x ∂ x + x ∂ x , (2 . E ¯ α +¯ α | V = x ∂ x + x ∂ x + x ∂ + x ∂ x + x ( ∂ x − ∂ x ) − x ∂ x − x ∂ x − x ∂ x − x ∂ x − x ∂ x , (2 . E ¯ α +¯ α | V = − x ∂ x + x ∂ x + x ∂ x − x ( ∂ x + ∂ x ) − x ∂ x + x ∂ x − ( x + x ) ∂ x − x ∂ x − x ∂ x + x ∂ x , (2 . E ¯ α +¯ α +¯ α | V = − x ∂ x + x ∂ x + x ∂ x + x ∂ x + x ( ∂ x − ∂ x ) − x ∂ x − x ∂ x − x ∂ x − x ∂ x + x ∂ x , (2 . E ¯ α +¯ α +¯ α | V = x ∂ x + x ∂ x − x ∂ x − x ( ∂ x + ∂ x ) − x ∂ x + x ∂ x − ( x + x ) ∂ x + x ∂ x − x ∂ x − x ∂ x , (2 . ¯ α +2¯ α | V = − x ∂ x + x ∂ x + x ∂ x − x ∂ x − x ∂ x + x ∂ x , (2 . E ¯ α +¯ α +2¯ α | V = x ∂ x + x ∂ x + x ∂ x − x ∂ x − x ∂ x − x ∂ x , (2 . E ¯ α +2¯ α +¯ α | V = − x ∂ x + x ∂ x + x ( ∂ x − ∂ x ) + x ∂ x + x ∂ x − x ∂ x − x ∂ x − x ∂ x − x ∂ x + x ∂ x , (2 . E ¯ α +¯ α +¯ α +¯ α | V = − x ∂ x − x ∂ x − x ∂ x − x ( ∂ x + ∂ x ) − x ∂ x + x ∂ x − ( x + x ) ∂ x + x ∂ x + x ∂ x + x ∂ x , (2 . E ¯ α +2¯ α +2¯ α | V = − x ∂ x + x ∂ x + x ∂ x − x ∂ x − x ∂ x + x ∂ x , (2 . E ¯ α +¯ α +2¯ α +¯ α | V = x ∂ x − x ∂ x + x ( ∂ x − ∂ x ) + x ∂ x + x ∂ x − x ∂ x − x ∂ x − x ∂ x + x ∂ x − x ∂ x , (2 . E ¯ α +2¯ α +2¯ α | V = x ∂ x + x ∂ x + x ∂ x − x ∂ x − x ∂ x − x ∂ x , (2 . E ¯ α +2¯ α +2¯ α +¯ α | V = − x ∂ x + x ∂ x + x ( ∂ x − ∂ x ) + x ∂ x + x ∂ x − x ∂ x − x ∂ x − x ∂ x − x ∂ x + x ∂ x , (2 . E ¯ α +¯ α +2¯ α +2¯ α | V = − x ∂ x + x ∂ x + x ∂ x − x ∂ x − x ∂ x + x ∂ x , (2 . E ¯ α +2¯ α +2¯ α +2¯ α | V = x ∂ x + x ∂ x + x ∂ x − x ∂ x − x ∂ x − x ∂ x , (2 . E ¯ α +2¯ α +3¯ α +¯ α | V = − x ∂ x − x ( ∂ x + ∂ x ) − x ∂ x + x ∂ x − x ∂ x + x ∂ x − x ∂ x + x ∂ x − ( x + x ) ∂ x + x ∂ x , (2 . E ¯ α +2¯ α +3¯ α +2¯ α | V = x ( ∂ x − ∂ x ) + x ∂ x − x ∂ x + x ∂ x + x ∂ x − x ∂ x − x ∂ x + x ∂ x − x ∂ x − x ∂ x , (2 . E ¯ α +2¯ α +4¯ α +2¯ α | V = x ∂ x − x ∂ x + x ∂ x − x ∂ x + x ∂ x − x ∂ x , (2 . E ¯ α +3¯ α +4¯ α +2¯ α | V = − x ∂ x + x ∂ x − x ∂ x + x ∂ x − x ∂ x + x ∂ x , (2 . E α +3¯ α +4¯ α +2¯ α | V = x ∂ x − x ∂ x + x ∂ x − x ∂ x + x ∂ x − x ∂ x , (2 . h | V = x ∂ x + x ∂ x − x ∂ x + x ∂ x − x ∂ x − x ∂ x + x ∂ x + x ∂ x − x ∂ x + x ∂ x − x ∂ x − x ∂ x , (2 . | V = x ∂ x − x ∂ x + x ∂ x + x ∂ x − x ∂ x − x ∂ x + x ∂ x + x ∂ x − x ∂ x − x ∂ x + x ∂ x − x ∂ x , (2 . h | V = x ∂ x − x ∂ x + x ∂ x − x ∂ x + x ∂ x − x ∂ x + x ∂ x +2 x ∂ x − x ∂ x + x ∂ x − x ∂ x − x ∂ x + x ∂ x − x ∂ x + x ∂ x − x ∂ x + x ∂ x − x ∂ x , (2 . h | V = x ∂ x − x ∂ x + x ∂ x − x ∂ x + x ∂ x − x ∂ x + x ∂ x − x ∂ x + 2 x ∂ x − x ∂ x + x ∂ x − x ∂ x + x ∂ x − x ∂ x + x ∂ x − x ∂ x + x ∂ x − x ∂ x . (2 . r = 27 − r for r ∈ , . (2 . τ on C [ x , ..., x ][ ∂ x , ..., ∂ x ] by τ ( x r ) = x ¯ r , τ ( ∂ x r ) = ∂ x ¯ r for r ∈ , \ { , } (2 . τ ( x ) = − x , τ ( x ) = − x , τ ( ∂ x ) = − ∂ x , τ ( ∂ x ) = − ∂ x . (2 . E − ¯ α | V = τ ( E ¯ α | V ) (2 . α ∈ Φ + F , the set of positive roots of G F . Thus we have given the explicit formulas forthe basic irreducible representation of F . F V is the irreducible module of the highest weight λ and x isa highest weight vector. In this section, we want to study the G F -module A = C [ x i | i ∈ ,
26] via the representation formulas (2.48)-(2.79).Suppose that η = 3 X r =1 x r x ¯ r + ax + bx x + cx (3 . F -invariant (cf . (2.76)), where a, b, c are constants to be determined. Then E ¯ α ( η ) = E ¯ α ( η ) = 0 naturally hold. Moreover,0 = E ¯ α ( η ) = 2( a − b ) x x + [ b − c − x x , (3 . a = b = 4 c + 3 , . (3 . E ¯ α ( η ) = 0 yield b = c = 4 a + 3. So we have the quadraticinvariant η = 3 X r =1 x r x ¯ r − x − x x − x . (3 . G F -invariant bilinear form on V .By (2.72)-(2.75), we try to find a quadratic singular vector of the form ζ = x ( a x + a x ) + a x x + a x x + a x x + a x x , (3 . a r are constants. Observe0 = E ¯ α ( ζ ) = ( a + a ) x x = ⇒ a = − a . (3 . E ¯ α ( ζ ) = ( a + a ) x x = ⇒ a = − a . (3 . E ¯ α ( ζ ) = ( a − a ) x x + ( a − a ) x x = ⇒ a = 2 a , a = a . (3 . E ¯ α ( ζ ) = ( a − a − a ) x x = ⇒ a = a − a . (3 . ζ = x (2 x + x ) − x x − x x + 3 x x − x x (3 . λ . So it generates an irreducible module that is isomorphic to the basic module V . Note E − ¯ α | V = − x ∂ x − x ∂ x − x ∂ x + x ∂ x + x ∂ x + x ∂ x , (3 . E − ¯ α | V = − x ∂ x − x ∂ x − x ∂ x + x ∂ x + x ∂ x + x ∂ x , (3 . E − ¯ α | V = x ∂ x + x ∂ x + x ∂ x − x ∂ x + x (2 ∂ x − ∂ x )+ x ∂ x + x ∂ x − x ∂ x − x ∂ x − x ∂ x , (3 . E − ¯ α | V = x ∂ x + x ∂ x + x ∂ x + x ∂ x + x (2 ∂ x − ∂ x )+ x ∂ x − x ∂ x − x ∂ x − x ∂ x − x ∂ x (3 . ζ compatible to { x i | i ∈ , } , we set ζ = E − ¯ α ( ζ ) = x ( − x + x ) + 3 x x − x x + 3 x x − x x , (3 . ζ = E − ¯ α ( ζ ) = − x ( x + 2 x ) + 3 x x + 3 x x + 3 x x − x x , (3 . ζ = − E − ¯ α ( ζ ) = − x ( x + 2 x ) − x x − x x + 3 x x − x x , (3 . ζ = E − ¯ α ( ζ ) = x ( − x + x ) + 3 x x − x x − x x + 3 x x , (3 . ζ = − E − ¯ α ( ζ ) = − x ( x + 2 x ) + 3 x x + 3 x x + 3 x x − x x , (3 . ζ = E − ¯ α ( ζ ) = x (2 x + x ) + 3 x x + 3 x x + 3 x x − x x , (3 . ζ = − E − ¯ α ( ζ ) = x ( − x + x ) − x x + 3 x x − x x + 3 x x , (3 . ζ = − E − ¯ α ( ζ ) = x (2 x + x ) − x x − x x + 3 x x − x x , (3 . ζ = − E − ¯ α ( ζ ) = x ( − x + x ) + 3 x x + 3 x x + 3 x x + 3 x x , (3 . ζ = − E − ¯ α ( ζ ) = x (2 x + x ) + 3 x x − x x − x x − x x , (3 . ζ = − E − ¯ α ( ζ ) = − x ( x + 2 x ) + 3 x x − x x − x x − x x , (3 . ζ = E − ¯ α ( ζ ) = − x ( x + 2 x ) − x x + 3 x x + 3 x x − x x +3 x x − x x + 3 x x − x x , (3 . ζ = E − ¯ α ( ζ ) = x (2 x + x ) − x x + 3 x x + 3 x x − x x +3 x x − x x − x x + 3 x x , (3 . ζ r = τ ( ζ ¯ r ) for r ∈ , , (3 . τ is an algebra automorphism determined by (2.77) and (2.78). The above con-struction shows that the map x r ζ r determine a module isomorphism from V to themodule generated by ζ . In particular, E ¯ α ( x i ) = ax j ⇔ E ¯ α ( ζ i ) = aζ j , a ∈ C , ¯ α ∈ Φ F . (3 . ϑ = ( x ζ − x ζ ) / x ( − x x + x x − x x + x x − x x )+ x ( x x + x x − x x + x x ) (3 . λ . Recall that the invariant η in (3.4) define an invariantbilinear form on V . Thus we have the following cubic invariant η = 3 X r =1 ( ζ r x ¯ r + x r ζ ¯ r ) − x ζ − x ζ − x ζ − x ζ . (3 . η = 9(1 + τ )[( x x + x x − x x + x x ) x + ( x x − x x + x x ) x +( x x − x x ) x + x ( x x + x x ) − x ( x x + x x + x x ) − x ( x x + x x )] + 2 x + 3 x x − x x − x + 3 x (2 x + x ) x +3 x ( x + 2 x ) x − x [ x x + x x + x x + x x − x x + x x − x x + x x − x x + x x ] − x [2 x x + 2 x x − x x +2 x x − x x − x x − x x − x x − x x + 2 x x ] , (3 . τ is an algebra automorphism defined in (2.77) and (2.78). Denote by N the set ofnonnegative integers. Now we are ready to prove our main theorem. Theorem 3.1 . Any polynomial f in A satisfying the system of partial differentialequations E ¯ α ( f ) = 0 for ¯ α ∈ Φ + F (3 . must be a polynomial in x , ζ , ϑ, η , η . In particular, the elements { x m ζ m ϑ m η m η m | m , m , m , m , m ∈ N } (3 . are linearly independent singular vectors and any singular vector is a linear combinationof those in (3.33) with the same weight. The weight of x m ζ m ϑ m η m η m is m λ + ( m + m ) λ .Proof. First we note x x = ζ − x x + 3 x x + 3 x x − x x + 3 x x , (3 . x x = ζ + x ( x − x ) + 3 x x − x x + 3 x x , (3 . x x = ζ + x ( x + 2 x ) − x x − x x + 3 x x , (3 . x x = 3 x x − ζ − x ( x + 2 x ) − x x − x x , (3 . x x = ζ + x ( x − x ) + 3 x x + 3 x x − x x , (3 . x x = ζ + x ( x + 2 x ) − x x − x x + 3 x x , (3 . x x = ζ + x ( x − x ) − x x + 3 x x − x x , (3 . x x = ζ + x ( x − x ) − x x − x x − x x , (3 . x x + 3 x x = η − X r =3 x r x ¯ r + x + x x + x , (3 . x x + x x − x x + x x ) + x ( x + 2 x )] x +3[3( x x + x x − x x + x x ) + x (2 x + x )] x = η − x ( x x − x x + x x ) − x ( x x − x x + x x ) − x x − τ )[( x x − x x ) x + x ( x x + x x ) − x ( x x + x x + x x ) − x ( x x + x x )] − x + 3 x x + 2 x + 3 x [ x x + x x + x x + x x − x x + x x − x x + x x − x x + x x ] − x [2 x x + 2 x x − x x + 2 x x − x x − x x − x x − x x − x x + 2 x x ] (3 . { x r | , , = r ∈ , } are rational functions in { x r , ζ s , η , η | r ∈ { , , , , } ; 7 , = s ∈ , } . (3 . f ∈ A is a solution of (3.32). Write f as a rational function f in thevariables of (3.44). In the following calculations, we will always use (3.28). By (2.71),0 = E α +3¯ α +4¯ α +2¯ α ( f ) = x ∂ x ( f ) . (3 . f is independent of x . Moreover, (2.70) gives0 = E ¯ α +3¯ α +4¯ α +2¯ α ( f ) = − x ∂ x ( f ) . (3 . f is independent of x . Furthermore, (2.69) yields0 = E ¯ α +2¯ α +4¯ α +2¯ α ( f ) = x ∂ x ( f ) . (3 . f is independent of x . Successively applying (2.68), (2.67), (2.66), (2.65) and(2.63) to f , we obtain that f is independent of x , x , x , x and x . Therefore, f isa rational function in { x r , ζ s , η , η | , = r ∈ ,
10; 7 , = s ∈ , } . (3 . E ¯ α +¯ α ( f ) = − x ∂ x ( f ) − ζ ∂ ζ ( f ) , (3 . E ¯ α +¯ α +¯ α ( f ) = x ∂ x ( f ) + ζ ∂ ζ ( f ) , (3 . E ¯ α +2¯ α +¯ α ( f ) = − x ∂ x ( f ) − ζ ∂ ζ ( f ) , (3 . E ¯ α +¯ α +¯ α +¯ α ( f ) = − x ∂ x ( f ) − ζ ∂ ζ ( f ) , (3 . E ¯ α +¯ α +2¯ α +¯ α ( f ) = x ∂ x ( f ) + ζ ∂ ζ ( f ) , (3 . E ¯ α +2¯ α +2¯ α +¯ α ( f ) = − x ∂ x ( f ) − ζ ∂ ζ ( f ) . (3 . η r = x ζ r − x r ζ , r ∈ , η = x ζ − x ζ , η = x ζ − x ζ . (3 . f can be written as a rational function f in { x r , ζ s , η q | r, s = 1 , q ∈ , } . (3 . f , we get0 = E ¯ α +2¯ α ( f ) = − ( x ζ − ζ x ) ∂ η ( f ) = − ϑ∂ η ( f ) , (3 . E ¯ α +¯ α +2¯ α ( f ) = 3 ϑ∂ η ( f ) , E ¯ α +2¯ α +2¯ α ( f ) = − ϑ∂ η ( f ) = 0 (3 . f is independent of η , η and η . Furthermore, we apply (2.50), (2.53)and (2.55) to f and obtain0 = E ¯ α ( f ) = − ϑ∂ η ( f ) , E ¯ α +¯ α ( f ) = 3 ϑ∂ η ( f ) , (3 . E ¯ α +¯ α +¯ α ( f ) = − ϑ∂ η ( f ) . (3 . f is a rational function in x , x , ζ , ζ , η , η . By (2.51),0 = E ¯ α ( f ) = − x ∂ x ( f ) − ζ ∂ ζ ( f ) . (3 . f can be written as a rational function f in x , ζ , ϑ, η , η . Since f = f is a polynomial in { x r | r ∈ , } , Expressions (3.29),(3.34), (3.42) and (3.43) imply that f must be a polynomial in x , ζ , ϑ, η , η . The otherstatements follow directly. ✷ Calculating the weights of the singular vectors in the above theorem, we have:
Corollary 3.2 . The space of polynomial G F -invariants over its basic module is ansubalgebra of A generated by η and η . 15et L ( m , m , m , m , m ) be the G F -submodule generated by x m ζ m ϑ m η m η m .Then L ( m , m , m , m , m ) is a finite-dimensional irreducible G F -submodule with thehighest weight m λ + ( m + m ) λ . Let A k be the subspace of polynomials in A withdegree k . Then A k is a finite-dimensional G F -module. By the Weyl’s theorem of completereducibility, A = ∞ M k =0 A k = ∞ M m ,m ,m ,m ,m =0 L ( m , m , m , m , m ) . (3 . d ( k, l ) the dimension of the highest weight irreducible module with the weight kλ + lλ . The above equation imply the following combinatorial identity:1(1 − t ) = 1(1 − t )(1 − t ) ∞ X k ,k ,k =0 d ( k , k + k ) t k +2 k + k . (3 . − t ) to the above equation, we obtain a new combinatorial identity abouttwenty-four: 1(1 − t ) = 1(1 + t )(1 + t + t ) ∞ X k ,k ,k =0 d ( k , k + k ) t k +2 k + k . (3 . Corollary 3.3 . The dimensions d ( p, l ) of the irreducible module with the weights kλ + lλ are linearly correlated by the following identity: (1 + t )(1 + t + t ) = (1 − t ) ∞ X k ,k ,k =0 d ( k , k + k ) t k +2 k + k . (3 . F from Euclidean space (e.g.,cf. [Hu]),( λ , ¯ α ) = ( λ , ¯ α ) = 1 , ( λ , ¯ α ) = ( λ , ¯ α ) = 1 / . (3 . δ = λ + λ + λ + λ . By the dimension formula of finite-dimensional irreduciblemodules of simple Lie algebras (e.g.,cf. [Hu]), d ( k, l ) = Q ¯ α ∈ Φ + F ( λ + δ, ¯ α ) Q ¯ α ∈ Φ + F ( δ, ¯ α ) = ( l + 1)( k + 3)( k + l + 4)39504568320000 (2 k + l + 7)(3 k + l + 10) × (3 k + 2 l + 11)( Y r =1 ( k + r ))( Y s =2 ( k + l + s ))( Y q =5 (2 k + l + q )) . (3 . η in (3.4). Dually we have G F invariant complexLaplace operator ∆ F = 3 X r =1 ∂ x r ∂x ¯ r − ∂ x − ∂ x ∂ x − ∂ x . (3 . k is H F k = { f ∈ A k | ∆ F ( f ) = 0 } . (3 . H F = V . Assume k ≥
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