H. R. Karadayi * Dept.Physics, Fac. Science, Tech.Univ.Istanbul 80626, Maslak, Istanbul, Turkey

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1 A N MULTIPLICITY RULES AND SCHUR FUNCTIONS H R Karadayi * DeptPhysics, Fac Science, TechUnivIstanbul 80626, Maslak, Istanbul, Turkey Abstract arxiv:math-ph/ v2 22 Jul 1998 It is well-known that explicit calculation of the right-hand side of Weyl character formula could often be problematic in practice We show that a specialization in this formula can be carried out in such a way that its right-hand side becomes simply a Schur Function For this, we need the use of some properly chosen set of weights which we call fundamental weights by the aid of which elements of Weyl orbits are expressed conveniently In the generic definition, an Elementary Schur Function S Q (x 1, x 2,, x Q ) of degree Q is known to be defined by some polynomial of Q indeterminates x 1, x 2,, x Q It is also known that definition of Elementary Schur Functions can be generalized in such a way that for any partition (Q k ) of weight Q and length k one has a Generalized Schur Function S (Qk )(x 1, x 2,, x Q ) When they are considered for A N 1 Lie algebras, on the other hand, a kind of degeneration occurs for these generic definitions This is mainly due to the fact that, for an A N 1 Lie algebra, only a finite number of indeterminates, namely (N-1), can be independent This leads us to define Degenerated Schur Functions by taking, for Q > N 1, all the indeterminates x Q to be non-linearly dependent on first (N-1) indeterminates x 1, x 2,, x N 1 With this in mind, we show that for each and every dominant weight of A N 1 we always have a (Degenerated) Schur Function which provides the right-hand side of Weyl character formula for irreducible representation corresponding to this dominant weight For instance, an Elementary Schur Function S Q (x 1, x 2,, x Q ) is, for Q < N, nothing but the character of tensorial representation with Q symmetrical indices for A N 1 The same is also true for Q N but there corresponds a Degenerated Elementary Schur Function Generalized Schur Functions are known to be expressed by determinants of some matrices of Elementary Schur Functions We would like to call these expressions multiplicity rules This is mainly due to the fact that, to calculate weight multiplicities, these rules give us an efficient method which works equally well no matter how big is the rank of algebras or the dimensions of representations * karadayi@ituedutr

2 2 I INTRODUCTION The calculation of weight multiplicities is essential in many problems encountered in group theory applications and also in theoretical high energy physics Due to seminal works of Freudenthal [1], Racah [2] and Kostant [3] the problem seems to be solved to great extent for finite Lie algebras One must however note that great difficulties arise in practical applications (*) of these multiplicity formulas when the rank of algebras and also the dimensions of representations grow high There are therefore quite many works[4] trying to reconsider the problem Beside these formulas, one can say that Weyl character formula [5] gives a unified framework to calculate the weight multiplicities for finite and also infinite dimensional Lie algebras [6] As well as for finite Lie algebras [7] there are numerous applications of Weyl-Kac character formula [8] for affine Lie algebras [9] though quite a little is known beyond affine Lie algebras Both for finite or affine Lie algebras, Weyl-Kac character formula involve sums over Weyl groups of finite Lie algebras [10] As we will show in a subsequent paper, the sum over the Weyl group of a finite Lie algebra of rank N can always be cast into a sum over the Weyl group of an A N 1 algebra One can thus conclude that the weight multiplicities for any finite Lie algebra are known by calculating only A N weight multiplicities Therefore, it will be suitable to consider the problem first for A N Lie algebras We want to show here that there is an intriguing interplay between A N weight multiplicities and Schur functions which arise in the litterature in a different context Within this framework, the definition of Schur functions plays in fact the role of a kind of multiplicity formula by giving the right hand side of the Weyl character formula directly For any degree Q=1,2,, Elementary Schur Functions S Q (x 1, x 2,, x Q ) are defined [11] by the formula k Z S k (x 1, x 2,, x k ) z k Exp i=1 x i z i (I1) with the understanding that S 0 = 1 and also S Q 0 Note here that this definition includes only some restricted class of Schur functions of degree Q and which are polynomials of Q indeterminates x 1, x 2, x Q Let us define, on the other hand, a partition (Q k ) with weight Q and length k (=1,2,,Q) as in the following: q 1 + q q k Q, q 1 q 2 q k > 0 (I2) For any such partition (Q k ), another generalization S (Qk )(x 1, x 2,, x Q ) of Schur functions are known to be defined by determinant of the following k k matrix: S q1 0 S q1+1 S q1+2 S q1+k 1 S q2 1 S q2+0 S q2+1 S q2+k 2 S q3 2 S q3 1 S q3+0 S q3+k 3 (I3) and hence it is natural to call them Generalized Schur Functions By decomposing the determinant of matrix (I3) in terms of its minors, one obtains an equivalent expression for S (Qk )(x 1, x 2,, x N ) in terms of products of Elementary Schur functions We will call these decompositions the multiplicity rules because, with a suitable generalization, it will be seen in the following that they are nothing but the right hand side of the Weyl character formula and hence provide a powerful method in calculating weight multiplicities By experience, anyone can easily see in fact that this will give us a quite fast method to calculate weight multiplicities in comparison with all the known methods This will be exemplified in the last section (*) AUKlymik has pointed out by that the use of Gelfand-Zeitlin basis proves also useful in calculations of weight multiplicities

3 II CHARACTERS OF A N 1 WEYL ORBITS 3 As it is known, characters are conventionally defined for irreducible representations of finite Lie algebras and also a huge class of representations of affine Lie algebras Now, rather than representations, we give a convenient extension of the concept for Weyl orbits It will be seen that, with an appropriate specialization in Weyl character formula, this allows us to recognize generalized Schur functions S Qk (x 1, x 2,, x N 1 ) directly as A N 1 characters Here, it is essential to use fundamental weights µ I (I=1,2, N) which are defined, for A N 1, by µ 1 λ 1 µ i µ i 1 α i 1, i = 2, 3, N (II1) or conversely by together with the condition that λ i = µ 1 + µ µ i, i = 1, 2, N 1 (II2) µ 1 + µ µ N 0 (II3) λ i s and α i s (i=1,2,n-1) are fundamental dominant weights and simple roots of A N 1 Lie algebras For an excellent study of Lie algebra technology we refer to the book of Humphreys [13] We know that there is an irreducible A N 1 representation for each and every dominant weight Λ + which can be expressed by K Λ + = q i µ i, q 1 q 2 q k > 0 (II4) providing in general that k N 1 Its height will also be defined by i=1 h(λ + ) k i=1 q i (II5) One thus concludes that there is a dominant weight Λ + for each and every partition (Q k ) with weight Q = h(λ + ) Note here that there could be in general more than one dominant weight with the same length For A N 1 Lie algebras, we also define Sub(Q λ 1 ) to be the set of all dominant weights corresponding, for k=1,2,q, to all the partitions (Q k ) with, modulo N, the same weigth Q = h(λ + ) The weight structure of the corresponding irreducible representation R(Λ + ) can then be expressed by the aid of the following orbital decomposition: R(Λ + ) = λ + Sub(Q λ 1) m Λ +(λ + ) W(λ + ) (II6) where m Λ +(λ + ) is the multiplicity of λ + within R(Λ + ) Note here that one always has m Λ +(Λ + ) = 1 and also m Λ +(λ + ) = 0, K k if Λ + and λ + correspond, respectively, to partitions (Q K ) and (Q k ) For instance, the tensor representations with Q completely symmetric and antisymmetric indices correspond, respectively, to two extreme cases which come from the partitions (Q 1 ) and (Q Q ) provided that and for any λ + Sub(Q λ 1 ) m Q λ1 (λ + ) = 1 m λq (λ + ) = 0, λ + λ Q (II7) (II8)

4 4 The left-hand side of Weyl character formula can now be expressed by ChR(Λ + ) = m Λ +(λ + ) ChW(λ + ) λ + Sub(Q λ 1) (II9) In view of orbital decomposition (II6), it is clear that (II9) allows us to define the characters ChW(Λ + ) µ W(Λ + ) e(µ) (II10) for Weyl orbits W(Λ + ) Formal exponentials e(µ) are defined, for any weight µ, as in the book of Kac [8] For any partition (Q k ) with weight Q and length k, it will be useful to introduce here the class functions K (Qk )(u 1, u 2,, u M ) which are defined to be polynomials, say, of N indeterminates u 1, u 2,, u N as in the following: N K (Qk )(u 1, u 2,, u N ) (u j1 ) q1 (u j2 ) q2 (u jk ) q k (II11) j 1,j 2,j k =1 In (II11), no two of indices j 1, j 2, j k shall take the same value for each particular monomial Of particular importance, one must note also that, on the contrary to generic definitions of Elementary or Generalized Schur Functions, the values of Q and N need not be correlated here Now, to be the main result of this section, if Λ + corresponds to a partition (Q k ), we can state, for A N 1, that ChW(Λ + ) = K (Qk )(u 1, u 2,, u N ) (II12) in the specialization e(µ i ) u i, i = 1, 2, N (II13) of (II10) One knows here that indeterminates u 1, u 2,, u N are restricted, due to (II3), by the condition that u 1 u 2 u N 1 (II14) In order to investigate (II12), we refer to a permutational lemma which we introduced quite previously [12] In view of this lemma, for any dominant weight Λ + which corresponds to a partition (Q k ), all other weights of the Weyl orbit W(Λ + ) are to be obtained by all permutations of parameters q 1, q 2,, q k III REDUCTION RULES AND SCHUR FUNCTIONS In this section, we want to expose the explicit correspondence which governs on one side A N 1 characters defined as in (II9) and (Generalized) Schur Functions S (Qk )(x 1, x 2,, x N 1 ) on the other side For this, we first emphasize that the set of indeterminates x 1, x 2, are conceptually different from indeterminates u 1, u 2, which give us the specialization (II13) in Weyl character formula Within the framework of our work, we however suggest a correspondence determined by the replacements K(Q) Q x Q, Q = 1, 2, (III1) In the notation of (II11), the generators K(Q) are defined, for partition (Q 1 ), as K(Q) K (Q1)(u 1, u 2,, u N ) (III2) The reason why we call K(Q) s generators is the fact that they generate all other class functions K (Qk )(u 1, u 2,, u N ) by the aid of some reduction rules By suppressing explicit u i dependences, these

5 5 reduction rules are given by K (q1,q 2) = K(q 1 ) K(q 2 ) K(q 1 + q 2 ), q 1 > q 2 K (q1,q 1) = 1 2 K(q 1) K(q 1 ) 1 2 K(q 1 + q 1 ) K (q1,q 2,q 3) = K(q 1 ) K (q2,q 3) K (q1+q 2,q 3) K (q1+q 3,q 2), q 1 > q 2 > q 3 K (q1,q 2,q 2) = K(q 1 ) K (q2,q 2) K (q1+q 2,q 2), q 1 > q 2 (III3) K (q1,q 1,q 2) = 1 2 K(q 1) K (q1,q 2) 1 2 K (q 1+q 1,q 2) 1 2 K (q 1+q 2,q 1), q 1 > q 2 K (q1,q 1,q 1) = 1 3 K(q 1) K (q1,q 1) 1 3 K (q 1+q 1,q 1) for the first three orders For higher orders, the reduction rules can be obtained similarly as in above This allows us to express everything in terms of generators K(Q) and hence the name generators In view of (II9) and (II7), one obtains for any given A N 1 ChR(Q λ 1 ) = K (Qk )(u 1, u 2,, u N ) (III4) where the sum is over all partitions (Q k ) with weight Q and length k=1,2,q For the right-hand side of (III4), an equivalent expression can be provided here by polynomials h k (u 1, u 2,, u k ) defined in k=0 h k (u 1, u 2,, u k ) z k i=1 1 (1 z u i ) (III5) By applying reduction rules mentioned above, one clearly sees, in view of (III1), the equivalence and hence one obtains h Q (u 1, u 2,, u Q ) S Q (x 1, x 2,, x Q ) ChR(Q λ 1 ) S Q (x 1, x 2,, x Q ) (III6) Now, it will be instructive to give here some clarifications on how we use Schur functions in applications of Weyl character formula for A N 1 Lie algebras For this, we first note that in their generic definitions (I1) there is no reference to any Lie algebra and hence S Q (x 1, x 2,, x N 1 ) can only be defined providing Q = N-1, ie a Schur function of degree Q can only be expressed in terms of Q indeterminates For the reader s convenience, we called these Elementary Schur Functions In our applications for A N 1, due to influence of condition (II3), we are in the situation to introduce the concept of Degenerated Schur Functions As the first example, one can say that S Q (x 1, x 2,, x N 1 ) is degenerated for Q = N in the sense that if it is represented by a polinomial for A N 1 it will be represented by a different one for all other Lie algebras A N, A N+1, To this end, let us consider K (NN)(u 1, u 2,, u N ) 1 (III7) which is nothing but the condition (II14) In view of (III1), one then sees that x N is non-linearly depended on the first (N-1) of x i s In the light of generalization K (QN)(u 1, u 2,, u N ) K(Q N) (III8) of (III7), one can also see that all other Schur functions of degree Q > N are degenerated for A N 1 Lie algebras This will also be exemplified in the last section IV WEYL CHARACTER FORMULA AND GENERALIZED SCHUR FUNCTIONS Throughout the sections IV and V, Λ + is an A N 1 dominant weight which corresponds, via (II4), to a partition (Q k ) of weight Q and length k We have two sets of indeterminates one of which is {u i, i = 1, 2, N}

6 6 which come from the specialization (II13) and the other one is {x i, i = 1, 2, N 1} which are created by the replacements (III1) The former set is constrained by the condition (II14) It will be instructive to emphasize here that Schur functions S (Qk )(x 1, x 2,, x N 1 ) depend on x i s while the right hand-side of Weyl formula will be obtained in terms of indeterminates u i s With this understanding, we will now show, as similar as in (III6), that for any other representation R(Λ + ) the Schur function S (Qk )(x 1, x 2,, x N 1 ) comes to be equivalent directly to the right-hand side of Weyl character formula in the specialization (II13) The main object [8] is A(Λ + ) ǫ(ω) e ω(λ+ ) (IV1) ω where the sum is over A N 1 Weyl group and hence ω(λ + ) represents an action of Weyl reflection ω while ǫ(ω) is its sign Our objection here is on the explicit calculation of the sign ǫ(ω) which is known to be defined by ǫ(ω) = ( 1) l(ω) (IV2) where l(ω) is the minimum number of simple reflections to obtain Weyl reflection ω Instead, we can replace (IV2) by ǫ(ω) = ǫ qi1,q i2,,q ik (IV3) for its action ω(λ + ) As is mentioned above, recall here that a reflection ω permutates the parameters q 1, q 2,, q k The tensor ǫ qi1,q i2,,q ik is completely antisymmetric in its indices while its numerical value is given on condition that ǫ q1,q 2,,q k +1, q 1 q 2 q k Weyl character formula now simply says that ChR(Λ + ) = A(ρ + Λ+ ) A(ρ) (IV3) where ρ λ 1 +λ 2 ++λ N 1 is A N 1 Weyl vector and the left-hand side of (IV3) is determined by (II9) For finite Lie algebras, (IV3) means that a miraculous factorization comes out on the right hand side By experience, no one could say that this is to be seen so fast or so easy in practical applications for, especially, higher dimensional representations of Lie algebras of higher rank It would however be greatly helpful to note, in the specialization (II13), that A(ρ) = N j>i=1 (u i u j ) (IV4) and A(ρ + Λ + ) can also be shown to be equivalent to determinant of the following N N matrix: u q1 1+N 1 u q2 2+N u q1 1+N 2 u q2 2+N u q1 1+N N 1 u 2 u u q2 2+N N u qn N+N 1 qn N+N 2 qn N+N N (IV5) where, for notational convenience, q i 0, i > k (IV4) is in fact nothing but the Vandermonde determinant and what is miraculous here is the fact that A(ρ + Λ + ) every time factorizes into (IV4) together with a polynomial which, for dominant Λ +, can be specified as a result of A(ρ + Λ + ) = A(ρ) S (Qk )(x 1, x 2,, x N 1 ) (IV6) In the case of Q λ 1, we already know from above that A(ρ + Q λ 1 ) = A(ρ) S Q (x 1, x 2,, x N 1 ) (IV7)

7 7 This hence brings us back to the fact that Generalized Schur Functions are in fact nothing but the right-hand side of Weyl character formula in the specialization (II13) V A N MULTIPLICITY RULES In the light of above developments, the equivalence between (II9) and (IV3) provides, for A N 1, Weyl character formula in the form of ChR(Λ + ) = S (Qk )(x 1, x 2,, x N 1 ) (V1) and this hence allows us to calculate the multiplicities m + Λ (λ+ ) without an explicit calculation of A(ρ+Λ + ), if one knows a way to calculate the Generalized Schur Functions S (Qk )(x 1, x 2,, x N 1 ) It is so fortunate that this is already provided by (I3) because it serves us to express any Generalized Schur Function in terms of the elementary ones in the form of some kind of reduction formulas which we would like to call multiplicity formulas due to our way of using them in calculating weight multiplicities In view of (I3), second order A N 1 multiplicity rules, for instance, are obtained, for k=2, in the form of S q1,q 2 = S q1 S q2 S q1+1 S q2 1 (V2) while third order multiplicity rules S q1,q 2,q 3 = + S q1+0 (S q2+0 S q3+0 S q2+1 S q3 1) S q1+1 (S q2 1 S q3+0 S q2+1 S q3 2) + S q1+2 (S q2 1 S q3 1 S q2+0 S q3 2) (V3) are also obtained for k=3 The dependences on (N-1) indeterminates (x 1, x 2,, x N 1 ) are suppressed above It is now clear that we need an explicit way to express Degenerated Schur Functions in order to use the multiplicity rules as a powerful tool In applications of these multiplicity rules for A N 1 Lie algebras, the Elementary Schur Functions of degrees M=1,2,N-1 are generic ones which are already obtained by (I1) As is explained above, all other ones are Degenerated Schur Functions and one can easily investigate, in view of (III7) and (III8), that they fulfill the following reduction rules: S M = ( 1) N S M N 1 N k=1 S k S M k, M N (V4) where S k is obtained from S k under replacements x i x i (i=1,2,n-1) In result, (V1) gives us an overall equation in terms of several monomials of (N-1) independent indeterminates x 1, x 2,, x N 1 These monomials are homogeneous of degree Q h(λ + ) and hence their numbers are just as the numbers of elements in Sub(Q λ 1 ) Since coefficients of these monomials are linearly dependent on weight multiplicities, this means that we have, for each and every element of Sub(Q λ 1 ), a system of linear equations which gives us unique solutions for weight multiplicities m Λ +(λ + ) where λ + Sub(Q λ 1 ) Let us visualize the whole procedure in a simple example by studying, for instance, all partitions (7 k ) (k=1,2,6) in the notation of (IV5) :

8 8 (7 1 ) 1 = , 7 λ 1 (7 2 ) 1 = , 5 λ 1 + λ 2 (7 2 ) 2 = , 3 λ λ 2 (7 2 ) 3 = , λ λ 2 (7 3 ) 1 = , 4 λ 1 + λ 3 (7 3 ) 2 = , 2 λ 1 + λ 2 + λ 3 (7 3 ) 3 = , 2 λ 2 + λ 3 (7 3 ) 4 = , λ λ 3 (V5) (7 4 ) 1 = , 3 λ 2 + λ 4 (7 4 ) 2 = , λ 1 + λ 2 + λ 4 (7 4 ) 3 = , λ 3 + λ 4 (7 5 ) 1 = , 2 λ 1 + λ 5 (7 5 ) 2 = , λ 2 + λ 5 (7 6 ) 1 = , λ 1 where an extra sub-index is added here to denote the partitions of the same length and, for A 5, corresponding elements of Sub(7 λ 1 ) are also shown on the right-hand side in view of (II1-3) Weyl orbit characters for these dominant weights can be obtained in terms of generators K(Q) by the aid of the following reduction rules beyond third order: K (q1,q 2,q 2,q 2) = K(q 1 ) K (q2,q 2,q 2) K (q1+q 2,q 2,q 2), q 1 > q 2 K (q1,q 2,q 3,q 3) = K(q 1 ) K (q2,q 3,q 3) K (q1+q 2,q 3,q 3) K (q1+q 3,q 2,q 3), q 1 > q 2 > q 3 K (q1,q 1,q 1,q 2) = 1/3 ( K(q 1 ) K (q1,q 1,q 2) K (q1+q 1,q 1,q 2) K (q1+q 2,q 1,q 1) ), q 1 > q 2 (V6) K (q1,q 2,q 2,q 2,q 2) = K(q 1 ) K (q2,q 2,q 2,q 2) K (q1+q 2,q 2,q 2,q 2), q 1 > q 2 K (q1,q 1,q 2,q 2,q 2) = 1/2 ( K(q 1 ) K (q1,q 2,q 2,q 2) K (q1+q 1,q 2,q 2,q 2) K (q1+q 2,q 2,q 2,q 2) ), q 1 > q 2 All these class functions can therefore be assumed to be polynomials of indeterminates x 1, x 2,, x 5 because we have x 6 = x x4 1 x x2 1 x x x3 1 x 3 x 1 x 2 x x x2 1 x 4 + x 2 x 4 + x 1 x 5 x 7 = x x x5 1 x x3 1 x (V7) 8 x4 1 x x2 1 x 2 x x2 2 x x3 1 x 4 + x 3 x x2 1 x 5 + x 2 x 5 as a result of K (1,1,1,1,1,1) 1, K (2,1,1,1,1,1) K(1) (V7) is sufficient to obtain our first two Degenerated Schur Polynomials: (V8) S 6 = x x2 1 x x3 1 x 3 + x x 2 x x 1 x 5 S 7 =2 x x x5 1 x x3 1 x x4 1 x 3 x 2 1 x 2 x 3 + (V9) x 1 x x2 x3 1 x x 1 x 2 x x 3 x x 2 1 x x 2 x 5 in addition to five generic ones S 1, S 2,, S 5 which are known to be obtained by the aid of (I1) One can easily investigate that all these are compatible with (V4)

9 9 Let us now study Weyl character formula (V1) in terms of above ingredients Its left-hand side is to be expressed in the form of an overall equation which are expressed in terms of monomials x 1, x 7 1, x 5 1 x 2, x 4 1 x 3, x 3 1 x 2 2, x 3 1 x 4, x 2 1 x 5, x 2 1 x 2 x 3, x 1 x 3 2, x 1 x 2 3, x 1 x 2 x 4, x 3 x 4, x 2 1 x 5, x 2 x 5 (V10) with coefficients depending on weight multiplicities linearly As is emphasized above, note here especially that we have 14 monomials in (V10) while we also have 14 dominant weights in Sub(7 λ 1 ) This guarantees that the system of linear equations for multiplicities m + Λ (λ+ ) (λ + Sub(7 λ 1 )) are always consistent for each one of R(Λ + ) with Λ + Sub(7 λ 1 ) For the right-hand side of (V1), on the other hand, the multiplicity rules are sufficient in each one of these cases For instance, as being in line with (V2), S 6,1 = S 6 S 1 S 7 gives equivalently 1 15 ( 15 x 1 + x 5 1 x x 2 1 x 2 x x 3 1 x 4 30 x 3 x 4 30 x 2 x 5 ) (V11) This is, in fact, nothing but a Degenerated Generalized Schur Function and it provides the right-hand side of (V1) by giving rise to solutions (7 λ 1 ) = 0, (5 λ 1 + λ 2 ) = 1, (3 λ λ 2 ) = 1 (λ λ 2 ) = 1, (4 λ 1 + λ 3 ) = 2, (2 λ 1 + λ 2 + λ 3 ) = 2 (2 λ 2 + λ 3 ) = 2, (λ λ 3 ) = 2, (3 λ 2 + λ 4 ) = 3 (λ 1 + λ 2 + λ 4 ) = 3, (λ 3 + λ 4 ) = 3, (2 λ 1 + λ 5 ) = 4 (λ 2 + λ 5 ) = 4, (λ 1 ) = 3 for 1980-dimensional irreducible representation R(5 λ 1 + λ 2 ) of A 5 For further use, we give in the following all other Degenerated Generalized Schur Functions participating in Sub(7 λ 1 ): S (5,2) = ( 720 x 1 + x x 5 1 x 2 60 x 3 1 x x 1 x x 4 1 x x 2 1 x 2 x x 2 2 x x 1 x x 3 1 x x 1 x 2 x x 2 1 x x 2 x 5 ) S (4,3) = (x x 5 1 x x 3 1 x x 4 1 x x 2 1 x 2 x x 2 2 x x 1 x x 3 1 x x 3 x x 2 1 x x 2 x 5 ) S (5,1,1) = (240 x 1 + x x 5 1 x x 3 1 x x 1 x x 4 1 x x 2 1 x 2 x x 2 2 x 3 40 x 3 1 x x 1 x 2 x x 3 x x 2 1 x x 2 x 5 ) S (4,2,1) = (120 x 1 + x x3 1 x x4 1 x x 2 1 x 2 x 3 60 x 2 2 x 3 80 x 3 1 x x 1 x 2 x x 3 x x 2 1 x 5) S (3,3,1) = (x x5 1 x x 3 1 x x 1 x x4 1 x x 2 1 x 2 x x 2 2 x 3 40 x 3 1 x x 1 x 2 x x 3 x x 2 1 x x 2 x 5 )

10 10 S (3,2,2) = (x7 1 2 x 5 1 x x 3 1 x x 1 x x 4 1 x x 2 1 x 2 x x 2 2 x x 3 1 x x 1 x 2 x x 3 x x 2 1 x x 2 x 5 ) S (4,1,1,1) = ( 360 x 1 + x x5 1 x x 3 1 x x4 1 x x 2 1 x 2 x x 2 2 x x 1 x x3 1 x x 3 x x 2 1 x x 2 x 5 ) S (3,2,1,1) = ( 360 x x x3 1 x x4 1 x x 2 1 x 2 x x 2 2 x x 1 x x3 1 x x 3 x x 2 1 x 5) S (2,2,2,1) = (x x5 1 x x 3 1 x x4 1 x x 2 1 x 2 x x 2 2 x x 1 x x3 1 x x 3 x x 2 1 x x 2 x 5 )/360 S (3,1,1,1,1) = (240 x 1 + x x 5 1 x x 3 1 x x 1 x x 4 1 x x 2 2 x x 3 1 x x 1 x 2 x x 2 1 x x 2 x 5 ) S (2,2,1,1,1) = (240 x 1 + x x 5 1 x x 3 1 x x 1 x x 4 1 x x 2 1 x 2 x x 2 2 x x 3 1 x x 1 x 2 x x 2 1 x x 2 x 5 ) We finally add that all these calculations can be performed by the aid of some simple computer programs written, for instance, in the language of Mathematica [14] They can be provided by from the address noted in the first page of this paper REFERENCES [1] HFreudenthal, Indag Math 16 (1964) [2] GRacah, Group Theoretical Concepts and Methods in Elementary Particle Physics, ed FGursey, NY, Gordon and Breach (1964) [3] BKostant, TransAmMathSoc, 93 (1959) [4] RCKing and SPOPlunkett, JPhysA:MathGen, 9 (1976) RCKing and AHA Al-Qubanchi, JPhysA:MathGen, 14 (1981) KKoike, JourAlgebra, 107 (1987) RVMoody and JPatera, BullAmMathSoc, 7 (1982) [5] HWeyl, The Classical Groups, NJ Princeton Univ Press (1946) [6] IBFrenkel, Representations of Kac-Moody Algebras and Dual Resonance Models, Lectures in Applied Mathematics 21 (1985) [7] MBremner, RVMoody and JPatera, Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras, NY, MDekker (1985) [8] VGKac, Infinite Dimensional Lie Algebras, NY, Cambridge Univ Press (1990) [9] SKass, R Moody, JPatera and RSlansky, Affine Kac-Moody Algebras, Weight Multiplicities and Branching Rules, Berkeley, Univ California Press (1990) [10] VGKac and SPeterson, Adv Math, 53 (1984) [11] VGKac and AKRaina, Bombay Lectures on Highest Weight Representations, World Sci, Singapore (1987) [12] HRKaradayi, Anatomy of Grand Unifying Groups I and II, ICTP preprints(unpublished) IC/81/213 and 224 HRKaradayi and MGungormez, JourMathPhys, 38 (1997) [13] JEHumphreys, Introduction to Lie Algebras and Representation Theory, NY, Springer-Verlag (1972) [14] S Wolfram, Mathematica TM, Addison-Wesley (1990)