An Extension of the Classical Gauss Series-product Identity by Fermionic Construction of \hat{sl}_n
aa r X i v : . [ m a t h . R T ] D ec An Extension of the Classical GaussSeries-product Identity by FermionicConstruction of c sl n Tomislav ˇSiki´cDepartment of MathematicsUniversity of ZagrebBijeniˇcka 3010000 Zagreb, CroatiaE-mail: [email protected].
Abstract
The main result of this paper is two infinity classes of series-productidentities which is based on classical Gauss identity and two differentinterpretations of character formula for irreducible highest weight modulesof affine Lie algebras.
It is well known that celebrated Macdonald identities (and especially the Jacobitriple product identity) are nothing else than the denominator identity for affineKac-Moody Lie algebras. Moreover, some specializations of the denominator1dentity give interesting series-product identities. For instance, the followingclassical identities ϕ ( q ) = X n ∈ Z ( − n q (3 n + n ) / ( Euler ) ,ϕ ( q ) = X n ∈ Z (4 n + 1) q n + n ( Jacobi ) ,ϕ ( q ) ϕ ( q ) = X n ∈ Z ( − n q n ( Gauss ) , and in particular ϕ ( q ) ϕ ( q ) = X n ∈ Z q n + n ( Gauss ) , (1.1)where ϕ ( q ) = Q j ≥ (1 − q j ) is Euler’s product function, can be also expressedfollowing the same approach (see [1], Exercise 12.4, pp. 241). It is also quiteinteresting that the classical Gauss identity (1.1) arises from the first concretecomputations of characters of nontrivial modules for affine Lie algebras (see [2]).As Victor Kac writes in his book [1] (pp. 216) the basic idea of this approachis very simple: ”one gets an interesting identity by computing the characterof integrable representation in two different ways and equating the results. Inparticular, Macdonald identities are deduced via trivial representation.”Following Kac’s inventive consideration the central object of the observation inthis article will be a character formula of an irreducible highest weight module L (Λ) of the affine Lie algebra b sl n , where Λ can be any fundamental weight, notjust a trivial representation. The above observation results with two infinityclasses of series-product identities. If we denote by m the arbitrary positiveinteger and by κ the following polynomial of the several variables κ ( k , ..., k m − ) = k + k + · · · + k m − − k k − k k − · · · − k m − k m − than the first class of series-product identities looks like ϕ ( q m − ) m − ϕ ( q m ) ϕ ( q ) ϕ ( q m ) = X k ,...,k m − ∈ Z q (4 m − κ ( k ,...,k m − )+ lin ( k ,...,k m − ) , (1.2)2here lin ( k , ..., k m − ) = (2 m − k − k −· · ·− k m − +(4 m − k m − k m +1 −· · ·− k m − . The another class ϕ ( q m ) m ϕ ( q ) ϕ ( q ) ϕ ( q ) = X k ,...,k m − ∈ Z q mκ ( k ,...,k m − )+ f lin ( k ,...,k m − ) (1.3)holds for f lin ( k , ..., k m − ) = − k −· · ·− k m − +(3 m − k m − k m +1 −· · ·− k m − +(3 m − k m − . Since the identity (1.1) will be essentially involved in the proof of mentionedclasses (1.2) and (1.3) we can interpret it as an extension of this classical Gaussidentity.As we say above, we looked at the mentioned object from two different pointsof view. One point of view is based on the character formula ch L (Λ) = e | Λ | δ P γ ∈ Q +Λ e Λ + γ − | γ | δ Q j ≥ (1 − e − jδ ) mult jδ . (1.4)for any dominant integral weight Λ in the special case of the affine Lie algebrasof type A (1) l , D (1) l , E (1) l (see [1] or [3] and [4]).Another point of view is based on a boson-fermionic realization of L (Λ) foraffine Lie algebra b gl n (see [5]), parameterized by a partition of a number n = { n , n , · · · , n r } . The corresponding character formula for the affine Lie algebra b sl n in the original notation [5] is T race L (Λ k ) ( q ) = q const ϕ ( q ) Q ri =1 ϕ ( q /n i ) X k + k + ··· + k r = k q ( k n + k n + ··· + k rnr ) . (1.5)The connection between two different points of view is made by a particularspecialization F s : C [[ e α , e α , e α , ..., e α l ]] → C [[ q ]] of type s = ( s , s , s , ..., s l )which satisfies the following formula F s ( ch L (Λ k )) = q const T race L (Λ k ) ( q N ) . (1.6)3he n -tuple s and positive integer N are generated (we shall provide detailslater) from the same partition n .In this article we would like to emphasize the importance of (1.6). Observethat both sides of (1.6) contain infinite sums (denoted by P left and P right )dependent on significantly different sizes of the set of indices. On the left handside we have n − r −
1. Veryoften we have the case (which is of particular interest to us) when n ≫ r . In this particular case we obtain a significant reduction of the sum P left by thesum P right modulo some fraction of Euler’s product functions ϕ ( q ). Hypothet-ically, if we recognize, in particulary cases, that the sum P right is a one sideof the some well known series-product identity we can express the sum P left ,which depends on arbitrary indices using the fraction of Euler’s product func-tions ϕ ( q ).Moreover, we would like to argue that our approach provides an algebraicmethod to reveal numerous important series-product identities; studied mainlyby number-theorists. This is one of the main points of our article. We illustratethe power of our method by constructing two infinite classes of series-productidentities; both are based on the classical Gauss series-product identity (1.1).Finally, let us also mention that these two classes have only one commonelement which was the starting point of our research. We shall conclude thisintroduction by providing details on this particular example. This exampleprovides a review of above observations and in the same time is a motivationfor the construction of the mentioned two series-product identity classes.We also believe that this example will guide our reader through the main bodyof the article.Let g be a simple Lie algebra of the type A , i.e., g = sl . For the corre-sponding affine Lie algebra ˆ g = b sl , the partition 4 = { , } and the fundamental4eight Λ the character formula (1.4) looks like ch L (Λ ) = e | Λ | δ P k ,k ,k ∈ Z e Λ +( k +1 / α +( k +2 / α +( k +3 / α − | γ | δ Q n ≥ (1 − e − jδ ) where12 | γ | = ( k + 14 ) + ( k + 24 ) + ( k + 34 ) − ( k + 14 )( k + 24 ) − ( k + 24 )( k + 34 )and δ = α + α + α + α is the corresponding imaginary root.For above settings ( b sl , { , } , Λ ) the trace formula (1.5) has the followingform T race L (Λ ) ( q ) = q const ϕ ( q ) ϕ ( q / ) ϕ ( q / ) X k + k =3 q ( k + k ) . Now using the substitution of variables k = 3 − k and classical Gauss identity(1.1) in the sum X k + k =3 q ( k + k ) we have T race L (Λ ) ( q ) = q const ϕ ( q / ) ϕ ( q / ) . As outlined above, we now apply a particular specialization, in this case thespecialization F s : C [[ e α , e α , e α , e α ]] → C [[ q ]] defined by parameters s = ( s , s , s , s ) = (2 , − , , { , } . Now, using (1.6) for N = 3we obtain ϕ ( q ) ϕ ( q ) ϕ ( q ) = X k ,k ,k ∈ Z q k + k + k − k k − k k )+ k − k +2 k , which seems to be a new series-product identity extended from the classicalGauss identity (1.1). 5 THE NOTATION AND BASIC SETTINGS
Let g = sl n be a simple Lie algebra defined for the Dynkin diagram A l , where l = n −
1. Denote by h the corresponding Cartan subalgebra and by∆ = ( α , ..., α l )the basis of the root system R ( ⊂ h ∗ ). Besides, by θ we denote the highest rootof the root system R . It is well known that R = {± ( ε i − ε j ) | ≤ i < j ≤ l + 1 } α = ε − ε , α = ε − ε , ..., α l − = ε l − − ε l , α l = ε l − ε l +1 (2.1) θ = α + α + α + · · · + α l = ε − ε l +1 . Let g = h ⊕ M α ∈R g α be a root space decomposition of the simple Lie algebra. Denote by x α ∈ g α the root vector which satisfies [ x α , x − α ] = α ∨ for the coroot α ∨ .Let ˆ g = g ⊗ C [ t, t − ] ⊕ ( C c + C d )and write x ( i ) = x ⊗ t i for x ∈ g and i ∈ Z . Then ˆ g = b sl n is an affine Lie algebrawith [ x ( i ) , y ( i )] = [ x, y ]( i + j ) + iδ i + j, ( x | y ) , where ( x | y ) is the Killing form for the simple Lie algebra g , c being a centralelement c = l X i =0 α ∨ i d a scaling element [ d, x ( i )] = ix ( i ) . The Cartan subalgebra of ˆ g is given byˆ h = h ⊕ ( C c + C d ) . The corresponding Dynkin diagram is of type A (1) l and related numerical labelsare a A (1) l = (1 , , , ..., , , . (2.2)We denote by δ the linear functional on CSA ˆ h defined by δ | h ⊕ C c = 0 h δ, d i = ( δ | d ) = 1 . The affine Lie algebra root system ˆ R ( ⊂ ˆ h ∗ ) is composed of the real and imagi-nary rootsˆ R = ˆ R Re ∪ ˆ R Im = { α + nδ | α ∈ R , n ∈ Z } ∪ { nδ | n ∈ Z \ { }} . If we denote by α the following root α = − θ + δ (2.3)then ˆ∆ = ( α , α , ..., α l )form the basis of the root system ˆ R and Q = P li =0 Z α i is the correspondingroot lattice.The imaginary root δ , spanned in above basis, due to (2.2) and (2.3), looks like δ = l X i =0 a i α i = l X i =0 α i . (2.4)It is well known that all affine Lie algebras are a Kac-Moody algebras g ( A ) for ageneralized Cartan matrix A of the corank one. Since the affine Lie algebra b sl n is7lso the Kac-Moody algebra then for every Λ ∈ h ∗ the irreducible highest-weightmodule L (Λ) is uniquely defined . The numberΛ( c ) = h Λ , c i (2.5)is called the level of the weight Λ or of the module L (Λ).Denote by P (Λ) the set of all weights of the module L (Λ) and by mult λ themultiplicity of λ ∈ P (Λ). The set P = { λ ∈ ˆ h ∗ | λ ( α ∨ i ) ∈ Z , i = 0 , , ..., n − } is called the weight lattice and weights from P are called integral weights. In-tegral weights from P + = { λ ∈ P | λ ( α ∨ i ) ≥ , i = 0 , , ..., n − } are called dominant. The weight lattice contains the root lattice Q and it isclear that P (Λ) ⊂ P if Λ ∈ P . Besides, the irreducible highest-weight g ( A )-module is integrable ifand only if Λ ∈ P + . The fundamental weights Λ i for i = 0 , , ..., n − i ( α ∨ j ) = δ ij , j = 0 , , ..., n − and Λ i ( d ) = 0 . (2.6)It is obvious that fundamental weights are always dominant. For a subset S of ˆ h ∗ denote by S the orthogonal projection of S on h ∗ by theextension of the Killing form from the simple g to the affine Lie algebra ˆ g . Thenwe have the following useful formula for λ ∈ ˆ h ∗ λ = λ + λ ( c )Λ + ( λ | Λ ) δ is the fundamental weight.Especially for the b sl n we have that Q = P li =1 Z α i andΛ i = Λ + Λ i where Λ = 0 and Λ , ..., Λ n − are the fundamental weights of simple Lie algebra sl n .Since Λ i ∈ P + then for all fundamental weights the irreducible highest-weight b sl n -module L (Λ i ) is integrable.Besides, from (2.5) and (2.6) follow that all b sl n -module L (Λ i ) are level onemodules. Using the notation P for the set of all level one dominant integralweights we can write Λ i ∈ P ∀ i = 0 , , ..., n − . (3.1)Moreover, when Λ ∈ P and the type of Dynkin diagram is equal to A (1) l , D (1) l or E (1) l the following formula P (Λ) = { Λ + 12 | Λ | δ + α − ( 12 | α | + s ) δ | α ∈ Λ +
Q, s ∈ Z + } (3.2)explicitly describes the weights system P (Λ). This result is proved in [3] or [6].A weight λ ∈ P (Λ) is called maximal if λ + δ / ∈ P (Λ). Denote by max (Λ) theset of all maximal weights of L (Λ). For λ ∈ max (Λ) the series a Λ λ = + ∞ X n =0 mult L (Λ) ( λ − nδ ) e − nδ is well defined. Using the result (3.2), the theory of the series a Λ λ (which hasbeen started by [2] and [3]) and the work [4] (or [1], Ch.12) the character formulaof L (Λ) can be written as ch L (Λ) = e | Λ | δ a Λ Λ X γ ∈ Q +Λ e Λ + γ − | γ | δ = e | Λ | δ P γ ∈ Q +Λ e Λ + γ − | γ | δ Q n ≥ (1 − e − nδ ) mult nδ Since the Dynkin diagram of affine Lie algebra b sl n is equal to A (1) l and allfundamental weights Λ k are level one integral dominant weights (see 3.1) then9t is obvious that the above character formula is appropriate for b sl n . Finally, wefinish this exposition by the character formula ch L (Λ k ) = e | Λ k | δ P γ ∈ Q +Λ k e Λ + γ − | γ | δ Q n ≥ (1 − e − nδ ) mult nδ (3.3)for the irreducible integrable highest weight b sl n -module L (Λ k ) where k = 0 , , ..., l . Many of the vertex operator constructions of integrable highest weight represen-tations are based on an inequivalent Heisenberg subalgebras. The inequivalentHeisenberg subalgebras, as conjugacy classes in S n , are parametrized by parti-tions of n . Denote by n = { n , n , ..., n r } a partition of n where n ≤ n ≤ ... ≤ n r . Moreover, the standard canonical base { E ij | i, j = 1 , , ..., n } for gl n is alsoparametrized by the partition n (see [5]). The associated partition of n × n matices is then given schematically by: B n × n B n × n · · · B s n × n r B n × n B n × n · · · B s n × n r ... ... . . . ... B s n r × n B s n r × n · · · B ss n r × n r where B ij is a block of size n i × n j . With this blockform in mind the standardcanonical base is remodeled with the set of matrices { E ijpq | p = 1 , ..., n i , q = 1 , ..., n j , i, j = 1 , , ..., s } where E ijpq = E n + ··· + n i − + p,n + ··· + n j − + q . (4.1)Using just mentioned notation, in [5] authors give an explicit vertex operatorconstructions of level one irreducible integrable highest weight representation10f b gl n for all inequivalent Heisenberg subalgebras (i.e. for all partitions). Theconstruction uses multicomponent fermionic fields and yields a correspondencebetween bosons (elements of Heisenberg subalgebra) and fermions. The men-tioned construction in addition to [7] results with explicit ”q-dimension” traceformula for b gl n -module T race Λ ∞ k C ∞ q D = q | H n | P k + k + ··· + k r = k q ( k n + k n + ··· + k rnr ) Q ri =1 Q j ≥ (1 − q jni ) , where H n is element of standard Cartan subalgebra h which satisfies the follow-ing commutation relations ad H n ( E ijkl ) = [ H n , E ijkl ] = ( ln j − kn i + 12 n i − n j ) E ijkl . (4.2)Finally, after restriction to b sl n case (also [5]), the corresponding irreducibleintegrable highest weight module L (Λ k ) have the following trace formula T race L (Λ k ) ( q ) = q | H n | Y j ≥ (1 − q j ) P k + k + ··· + k r = k q ( k n + k n + ··· + k rnr ) Q ri =1 Q j ≥ (1 − q jni ) . (4.3) Let s = ( s , s , ..., s n ) be a sequence of integers. Then the sequence s (undersome assumptions) defines a homomorphism F s : C [[ e − α , e − α , e − α , ..., e − α n ]] → C [[ q ]]by F s ( e − α i ) = q s i ( i = 0 , , ..., n ) . This homomorphism is called the specialization of type s .Let N ′ be the least common multiple of n , n ,..., n r , then the integer N is definedby: N = N ′ if N ′ ( n i + n j ) ∈ Z ∀ i, j N ′ if N ′ ( n i + n j ) / ∈ Z f or a pair ( i, j ) . (5.1)11ollowing the boson-fermionic construction for b gl n from the paper [5] we canconclude that the connection between the mentioned two different points of viewis made by particular specialization of type s = ( s , s , s , ..., s n ) where s wasparametrized with the partition n = { n , n , ..., n r } and integer N . In fact wehave the following proposition. Proposition 5.1
Let n = { n , ..., n r } be a partition of n . Let N be the corre-sponding integer defined by (5.1). For affine Lie algebra b sl n and for all funda-mental weights Λ k k = 0 , , ...n − the next equation F s ( P γ ∈ Q +Λ e Λ + γ − | γ | δ Q j ≥ (1 − e − jδ ) mult jδ )= q const Y j ≥ (1 − q jN ) P k + k + ··· + k r = k q N ( k n + k n + ··· + k rnr ) Q ri =1 Q j ≥ (1 − q jNni ) (5.2) holds for s = N ( n + n r n n r , n , ..., n , n + n n n − , n , ..., n , (5.3) n + n n n − , ..., n r − , ..., n r − , n r − + n r n r − n r − , n r , ..., n r ) . Proof.
Since the root subspaces ˆ g α , for α ∈ ˆ R Re , are one-dimensional wehave unique 1 − α ←→ ˆ x − θ ⊗ tα i ←→ ˆ x α i ⊗ i = 1 , ..., l . Hence, we have correspondence, based on (2.1), between the base ∆ and theelement of standard canonical base of gl n { E ij | i, j = 1 , ..., n } α i ←→ E i,i +1 i = 1 , ..., l . (5.4)From (2.3) and (2.4) it is evident α ←→ E n, ⊗ t . (5.5)12he commutation relations (4.2) for adH n and E ijkl as indexed in (4.1) expressdegrees of eigenvectors E ijkl by degE ijkl = N ( ln j − kn i + 12 n i − n j ) modN . (5.6)From (5.6) and from the exposition of [5] which lead to trace formula (4.3)follows that eigenvalues for the adjoint action of adH n are pointers for the rightspecialization s . More precisely, using 1 − s consists of the eigenvalues for eigenvectors { α ∨ , α ∨ , ..., α ∨ n − } = { E n, ⊗ t, E , ⊗ , E , ⊗ , ..., E n − ,n ⊗ } . as shown in s = ( degE n, + N, degE , , degE , , ..., degE n − ,n ) . (5.7)Due to remodeling (4.1) of the standard canonical base by { E pqij } we concludethat sequence s (5.7) is equal to (5.3), i.e. the equation (5.2) holds. Remark 5.2
In the paper [8] V.G.Kac showed that the automorphisms σ ˜s ( e i ) = e πiN · s i e i i = 0 , , ..., l exhaust all N -th order automorphisms of g . By { e i | i = 0 , , ..., l } are markedgenerators of g and s = ( s , s , ..., s l ) is a sequence of nonnegative relatively prime integers. The parameters s i arecalled the Kac parameters . Many of the vertex operator constructions ofintegrable highest weight representations and the corresponding gradations andspecializations do not provide Kac parameters (see, in particular [5] and [7]).Particulary, the sequence s (5.3) and the associated specialization are deter-mined by relatively prime integers, but all integers N ( n i + n i +1 n i n i +1 − are negative.So, the specialization (5.2) is not parametrized by Kac parameters. n the paper [9] it is given the exact algorithm for finding the Kac parameters s Kac of the sequence s (5.3) and equation (5.2) holds for specialization by Kacparameters, too. GAU SS ( n = 1 + (4 m − , Λ m ) Introduce the Euler product ϕ ( q ) = ∞ Y n =1 (1 − q n ) . Denote by κ the following polynomial of the several variables κ ( k , ..., k l ) = k + k + · · · + k l − k k − k k − · · · − k l − k l where l = n −
1. It is interesting to notice that above polynomial can beinterpreted by the Killing form in such a way that κ ( k , ..., k l ) = 12 ( k α ∨ + · · · k l α ∨ l | k α ∨ + · · · k l α ∨ l ) . Theorem 6.1
Let n = 4 m for an arbitrary positive integer m . Then the fol-lowing series-product identity X k ,...,k m − ∈ Z q (4 m − κ ( k ,...,k m − )+ lin ( k ,...,k m − ) = ϕ ( q m − ) m − ϕ ( q m ) ϕ ( q ) ϕ ( q m ) (6.1) holds for lin ( k , ..., k m − ) = (2 m − k − k − · · · − k m − + (4 m − k m − k m +1 − · · · − k m − . (6.2) Proof.
First of all, the proof is based on the following setting:ˆ g = b sl m n = 4 m = { , m − } (6.3)Λ = Λ m = Λ + Λ m . m is the corresponding fundamental weight of simple Lie algebra sl m we can write (see [10], pp. 69):Λ m = 14 m [ mα + 2 mα + · · · + (3 m − · mα m − + 3 m · mα m +3 m · ( m − α m +1 + 3 m · ( m − α m +2 (6.4)+ · · · + 3 m · α m − + 3 m · α m − ] . The numerator of the formula (3.3) looks like e Λ + | Λ m | δ X γ ∈ Q +Λ m e γ − | γ | δ where γ = k α + k α + · · · + k m − α m − + Λ m . (6.5)Using (6.4) the vector γ is written down by the base ∆ γ = ( k + 14 ) α + ( k + 24 ) α + · · · + ( k m + 3 m α m +( k m +1 + 3( m − α m +1 + ( k m +2 + 3( m − α m +2 + · · · + ( k m − + 3 ·
24 ) α m − + ( k m − + 3 ·
14 ) α m − . For the partition 4 m = { , m − } it is obvious (see 5.1) that the number N isequal 4 m −
1. Besides, the mentioned partition implicates that the blockformof 4 m × m matrices looks like B
11 1 × B
12 1 × m − B
21 4 m − × B
22 4 m − × m − Following (4.2), (5.6) and (5.7) we can conclude that the specialization F s ,defined by s = ( degE m − , + N , degE , , degE , , ..., degE m − , m − )= (2 m, − m + 1 , , ..., ,
15s the specialization for the connection between two standpoints. More explicitlythe specialization is given by e − α ←→ q m e − α ←→ q − m +1 e − α ←→ q ... ... e − α m − ←→ q e − δ ←→ q m − . After calculation | γ | = ( γ | γ ) = (6.5) = 2 κ ( k , ..., k m − ) + 2 k m + 9 m mult nδ always equals dim h = 4 m − F s ( P γ ∈ Q +Λ e Λ + γ − | γ | δ Q n ≥ (1 − e − nδ ) mult nδ )= q const P k ,...,k m − q (4 m − κ ( k ,...,k m − )+ lin ( k ,...,k m − ) [ ϕ ( q m − )] m − (6.6)where lin ( k , ..., k m − ) = (2 m − k − k −· · ·− k m − +(4 m − k m − k m +1 −· · ·− k m − . The right hand side of the formula (5.2) for the mentioned settings (6.3) lookslike q const Y j ≥ (1 − q (4 m − j ) P k + k =3 m q m − ( k + k m − ) Q j ≥ (1 − q (4 m − j ) Q j ≥ (1 − q (4 m − j m − ) == q const P k + k =3 m q [(4 m − k + k ] ϕ ( q ) . After the substitution k = 3 m − k the calculations X k + k =3 m q [(4 m − k + k ] = X k ∈ Z q [(4 m − k +(3 m − k ) )] q m X k ∈ Z q m (4 k − k ) = q m X k ∈ Z q m [2( k − +( k − − = q m − m X k ∈ Z q m (2 k + k ) ( Gauss q m − m · ϕ ( q m ) ϕ ( q m )implicate that the right hand side of the formula (5.2) has the form q const P k + k =3 m q [(4 m − k + k ] ϕ ( q ) = q const ϕ ( q m ) ϕ ( q ) ϕ ( q m ) . (6.7)Now, from (6.6) and (6.7) it is obvious that the series-product identity (6.1)holds for (6.2). GAU SS ( n = m + 3 m, Λ m − ) Denote again by κ the following polynomial of several variables κ ( k , ..., k l ) = k + k + · · · + k l − k k − k k − · · · − k l − k l . Theorem 7.1
Let n = 4 m for an arbitrary positive integer m . Then the fol-lowing series-product identity X k ,...,k m − ∈ Z q mκ ( k ,...,k m − )+ f lin ( k ,...,k m − ) = ϕ ( q m ) m ϕ ( q ) ϕ ( q ) ϕ ( q ) (7.1) holds for f lin ( k , ..., k m − ) = − k − · · · − k m − + (3 m − k m − k m +1 − · · · − k m − + (3 m − k m − . (7.2) Proof.
First of all, the proof is based on the following setting:17 g = b sl m n = 4 m = { m, m } (7.3)Λ = Λ m − = Λ + Λ m − . Now, it is well known (see [10]) thatΛ m − = 14 m [ α + 2 α + 3 α + · · · + (4 m − α m − + (4 m − α m − ] . (7.4)The numerator of the formula (3.3) looks like e Λ + | Λ m − | δ X γ ∈ Q +Λ m − e γ − | γ | δ where γ = k α + k α + · · · + k m − α m − + Λ m − . (7.5)Using (7.4) the vector γ is written down by the base ∆ γ = ( k + 14 m ) α + ( k + 24 m ) α + · · · + ( k m − + 4 m − m ) α m − Now, the number N is equal 3 m when the partition is 4 m = { m, m } (see 5.1).This partition implicates the blockform of 4 m × m matrices B m × m B m × m B
21 3 m × m B
22 3 m × m Following again (4.2), (5.6) and (5.7) we can concluded that the demandedspecialization F s is defined by s = ( degE m, + N, degE , , ..., degE m − ,m , degE m, , degE , , ..., degE m − , m )= (2 , , ..., , − m, , ..., . e − α ←→ q e − α ←→ q ... ... e − α m − ←→ q e − α m ←→ q − m e − α m +1 ←→ q ... ... e − α m − ←→ q e − δ ←→ q m . After calculation | γ | = ( γ | γ ) = (7.5) = 2 κ ( k , ..., k m − ) + 2 k m − + 4 m − m and the fact that mult nδ always equals dim h = 4 m − F s ( P γ ∈ Q +Λ e Λ + γ − | γ | δ Q n ≥ (1 − e − nδ ) multnδ )= q const P k ,...,k m − q (3 m ) κ ( k ,...,k m − )+ f lin ( k ,...,k m − ) [ ϕ ( q m )] m − (7.6)where f lin ( k , ..., k m − ) = − k −· · ·− k m − +(3 m − k m − k m +1 −· · ·− k m − +(3 m − k m − . The right hand side of the formula (5.2) for the mentioned settings (7.3) lookslike q const Y j ≥ (1 − q (3 m ) j ) P k + k =4 m − q m ( k m + k m ) Q j ≥ (1 − q (3 m ) jm ) Q j ≥ (1 − q (3 m ) j m ) == q const ϕ ( q m ) P k + k =4 m − q (3 k + k ) ϕ ( q ) ϕ ( q ) . k = (4 m − − k the calculations X k + k =4 m − q (3 k + k ) = X k ∈ Z q [3 k +((4 m − − k ) ] = q (4 m − X k ∈ Z q (4 k − m − k ) = q (4 m − X k ∈ Z q k − m ) +( k − m ) − m + m = q (4 m − − m + m X k ∈ Z q k − m ) +( k − m ) ( Gauss q m − m +12 · ϕ ( q ) ϕ ( q )implicate that the right hand side of the formula (5.2) has the form q const ϕ ( q m ) P k + k =4 m − q (3 k + k ) ϕ ( q ) ϕ ( q ) = q const ϕ ( q m ) ϕ ( q ) ϕ ( q ) ϕ ( q ) . (7.7)Now, from (7.6) and (7.7) it is obvious that the series-product identity (7.1)holds for (7.2). Remark 7.2
Notice that for m = 1 series-product identities (1.2) and (1.3)(i.e. (6.1) and (7.1)) are the same. References [1] Kac, V.G.: