aa r X i v : . [ m a t h - ph ] O c t ON CHARACTERS OF F LIE ALGEBRAM. Gungormez
Dept. Physics, Fac. Science, Istanbul Tech. Univ.34469, Maslak, Istanbul, Turkeye-mail: [email protected]
H. R. Karadayi
Dept. Physics, Fac. Science, Istanbul Tech. Univ.34469, Maslak, Istanbul, Turkeye-mail: [email protected]. Physics, Fac. Science, Istanbul Kultur University34156, Atakoy, Istanbul, Turkey
Abstract
In a previous work, we have given an explicit method to obtain irreducible charactersof finite Lie algebras without referring to Weyl character formula. Irreducible charactersof G Lie algebra has been given as an example. The work is now extended to somewhatmore complicated case of F Lie algebra, in the same manner.
I. INTRODUCTION
To be complete here, sections I. and II. will be a reminder of our previous work [1] .It is known that the character formula of Weyl [2] gives us a direct way to calculate thecharacter of irreducible representations of finite Lie algebras. For this, let G r be a Liealgebra of rank r, W ( G r ) its Weyl group, α i ’s and λ i ’s be, respectively, its simple rootsand fundamental dominant weights. The notation here and in the following sections willbe as in our previous work [3] . For further reading, we refer to the beautiful book ofHumphreys [4] . A dominant weight Λ + is expressed in the formΛ + = r X i =1 s i λ i ( I. s i ’s are some positive integers (including zero). An irreducible representation V (Λ + )can then be attributed to Λ + . The character Ch (Λ + ) of V (Λ + ) is defined by Ch (Λ + ) ≡ X λ + X µ ∈ W ( λ + ) m Λ + ( µ ) e µ ( I. m Λ + ( µ ) ’s are multiplicities which count the number of times a weight µ is repeatedfor V (Λ + ). The first sum here is over Λ + and all of its sub-dominant weights λ + ’s whilethe second sum is over the elements of their Weyl orbits W ( λ + )’s. Formal exponentialsare taken just as in the book of Kac [5] and in (I.2) we extend the concept for any weight µ , in the form e µ . Note here that multiplicities are invariant under Weyl group actionsand hence it is sufficient to determine only m Λ + ( λ + ) for the whole Weyl orbit W ( λ + ).An equivalent form of (I.2) can be given by Ch (Λ + ) = A ( ρ + Λ + ) A ( ρ ) ( I. ρ is the Weyl vector of G r . (I.3) is the celebrated Weyl Character Formula whichgives us the possibility to calculate characters in the most direct and efficient way. Thecentral objects here are A ( ρ + Λ + )’s which include a sum over the whole Weyl group: A ( ρ + Λ + ) ≡ X σ ∈ W ( G r ) ǫ ( σ ) e σ ( ρ +Λ + ) . ( I. σ denotes an element of Weyl group,i.e. a Weyl reflection, and ǫ ( σ ) is the corre-sponding signature with values either +1 or -1.The structure of Weyl groups is completely known for finite Lie algebras in principle.In practice, however, the problem is not so trivial, especially for Lie algebras of somehigher rank. The order of E Weyl group is, for instance, 696729600 and any applicationof Weyl character formula needs for E an explicit calculation of a sum over 696729600Weyl reflections. Our main point here is to overcome this difficulty in an essential manner.For an actual application of (I.3), an important notice is the specialization of formalexponentials e µ as is called in the book of Kac [5] . In its most general form, we considerhere the specialization e α i ≡ u i , i = 1 , , . . . , r. ( I. A ( ρ ) in the form of A ( ρ ) = P ( u , u , . . . , u r ) ( I. P ( u , u , . . . , u r ) is a polynomial in indeterminates u i ’s. We also have A ( ρ + Λ + ) = P ( u , u , . . . , u r ; s , s , . . . , s r ) ( I. P ( u , u , . . . , u r ; s , s , . . . , s r ) is another polynomial of indeterminates u i ’s andalso parameters s i ’s defined in (I.1). The Weyl formula (I.3) then says that polyno-mial (I.7) always factorizes on polynomial (I.6) leaving us with another polynomial R ( u , u , . . . , u r ; s , s , . . . , s r ) which is nothing but the character polynomial of V (Λ + ).The specialization (I.5) will always be normalized in such a way that (I.3) gives us the Weyl dimension formula [6] , in the limit u i = 1 for all i = 1 , , . . . , r . One also expectsthat P ( u , u , . . . , u r ; 0 , , . . . , ≡ P ( u , u , . . . , u r ) . ( I. II. RECREATING A ( ρ + Λ + ) FROM A ( ρ )In this section, without any reference to Weyl groups, we give a way to calculatepolynomial (I.7) directly from polynomial (I.6). For this, we first give the following explicitexpression for polynomial (I.6): A ( ρ ) = Q α ∈ Φ + ( e α − Q ri =1 ( e α i ) k i ( II. k i ≡
12 ( α i , α i ) ( λ i , ρ ) ( II. + is the positive root system of G r . Exponents k i ’s are due to the fact that themonomial of maximal order is r Y i =1 ( e α i ) k i in the product Q α ∈ Φ + ( e α − λ i , α j ) or ( α i , α i ) are thesymmetrical ones and they are known to be defined via Cartan matrix of a Lie algebra.The crucial point, however, is to see that (II.1) is equivalent to A ( ρ ) = | W ( G r ) | Y A =1 ǫ A ( e α i ) ξ i ( A ) ( II. | W ( G r ) | is the order of Weyl group W ( G r ) and as is emphasized insection I, ǫ A ’s are signatures with values ǫ A = ∓
1. Note here that, by expanding theproduct Q α ∈ Φ + ( e α − ξ i ( A ) in (II.3), let us define R + is composedout of elements of the form β + ≡ r X i =1 n i α i ( II. n i ’s are some positive integers including zero. R + is a subset of the Positive RootLattice of G r . The main emphasis here is on some special roots γ i ( I i ) ∈ R + which are defined by following conditions( λ i − γ i ( I i ) , λ j − γ j ( I j )) = ( λ i , λ j ) , i, j = 1 , , . . . , r. ( II. λ i , λ j )’s are defined by inverse Cartan matrix. For the range of indices I j ’s, we assume that they take values from the set { , , . . . , | I j |} , that is I j ∈ { , , . . . , | I j |} , j = 1 , , . . . , r. We also define the setsΓ( A ) ≡ { γ ( I ( A )) , γ ( I ( A )) , . . . , γ r ( I r ( A )) } , A = 1 , , . . . , D ( II. I j ( A ) ∈ { , , . . . , | I j |} and D is the maximal numberof these sets.Following two statements are then valid:(1) D = | W ( G r ) | (2) | I j | = | W ( λ j ) | where | W ( λ j ) | is the order, i.e. the number of elements, of the Weyl orbit W ( λ j ). As isknown, a Weyl orbit is stable under Weyl reflections and hence all its elements have thesame length. It is interesting to note however that the lengths of any two elements γ i ( j )and γ i ( j ) could, in general, be different while the statement (2) is, still, valid.The exponents in (II.3) can now be defined by ξ i ( A ) ≡
12 ( α i , α i ) ( λ i − γ i ( A ) , ρ ) . ( II. A ( ρ + Λ + ) = | W ( G r ) | Y A =1 ǫ A ( e α i ) ξ i ( A ) ( II. ξ i ( A ) ≡
12 ( α i , α i ) ( λ i − γ i ( A ) , ρ + Λ + ) . ( II. Ch (Λ + ) to the problem of finding solutions to conditions (II.5). It is clear that this ismore manageable than that of using Weyl character formula directly. III. EXPLICIT CONSTRUCTION OF F CHARACTERS
It is known that F is characterized by two different root lengths( α , α ) = ( α , α ) = 4 , ( α , α ) = ( α , α ) = 2 ( III. α , α ) = ( α , α ) = − α , α ) = − III. + = { α , α , α , α , α + α , α + α , α + α , α + 2 α , α + α + α , α + α + α ,α + α + 2 α , α + 2 α + α , α + α + α + α , α + 2 α + 2 α ,α + 2 α + 2 α , α + α + 2 α + α , α + α + 2 α + 2 α ,α + 2 α + 2 α + α , α + 2 α + 2 α + 2 α , α + 2 α + 3 α + α ,α + 2 α + 3 α + 2 α , α + 2 α + 4 α + 2 α , α + 3 α + 4 α + 2 α , α + 3 α + 4 α + 2 α } ( III. γ (1) = 0 γ (2) = α γ (3) = α + α γ (4) = α + α + 2 α γ (5) = α + α + 2 α + 2 α γ (6) = α + 2 α + 2 α γ (7) = α + 2 α + 2 α + 2 α γ (8) = α + 2 α + 4 α + 2 α γ (9) = α + 3 α + 4 α + 2 α γ (10) = 2 α + 2 α + 2 α γ (11) = 2 α + 2 α + 2 α + 2 α γ (12) = 2 α + 2 α + 4 α + 2 α γ (13) = 2 α + 4 α + 4 α + 2 α γ (14) = 2 α + 4 α + 6 α + 2 α γ (15) = 2 α + 4 α + 6 α + 4 α γ (1) = 0 γ (2) = α γ (3) = α + α γ (4) = α + α + α γ (5) = α + 2 α + α γ (6) = α + 2 α + 2 α γ (7) = α + α + α + α γ (8) = α + α + 2 α + α γ (9) = α + α + 2 α + 2 α γ (10) = α + 2 α + 2 α + α γ (11) = α + 2 α + 2 α + 2 α γ (12) = α + 2 α + 3 α + α γ (13) = α + 2 α + 3 α + 3 α γ (14) = α + 2 α + 4 α + 2 α γ (15) = α + 2 α + 4 α + 3 α γ (16) = 3 α + 3 α + 4 α + 2 α γ (17) = 3 α + 4 α + 4 α + 2 α γ (18) = 3 α + 4 α + 6 α + 2 α γ (19) = 3 α + 4 α + 6 α + 4 α γ (20) = 3 α + 5 α + 6 α + 2 α γ (21) = 3 α + 5 α + 6 α + 4 α γ (22) = 3 α + 5 α + 8 α + 4 α γ (23) = 3 α + 6 α + 8 α + 4 α γ (24) = 4 α + 6 α + 8 α + 4 α γ (16) = α + 3 α + 4 α + 2 α γ (17) = α + 3 α + 4 α + 3 α γ (18) = α + 3 α + 5 α + 3 α γ (19) = 2 α + 3 α + 4 α + 2 α γ (20) = 2 α + 3 α + 4 α + 3 α γ (21) = 2 α + 3 α + 5 α + 3 α γ (22) = 2 α + 4 α + 5 α + 3 α γ (23) = 2 α + 4 α + 6 α + 3 α γ (24) = 2 α + 4 α + 6 α + 4 α γ (1) = 0 γ (2) = α γ (3) = α + 2 α γ (4) = α + 2 α + 2 α γ (5) = 2 α + 2 α γ (6) = 2 α + 2 α + 2 α γ (7) = 2 α + 4 α + 2 α γ (8) = 3 α + 4 α + 2 α γ (9) = α + α γ (10) = α + α + 2 α γ (11) = α + α + 2 α + 2 α γ (12) = α + 3 α + 2 α γ (13) = α + 3 α + 2 α + 2 α γ (14) = α + 3 α + 4 α γ (15) = α + 3 α + 4 α + 4 α γ (16) = α + 3 α + 6 α + 2 α γ (17) = α + 3 α + 6 α + 4 α γ (18) = α + 5 α + 6 α + 2 α γ (19) = α + 5 α + 6 α + 4 α γ (20) = α + 5 α + 8 α + 4 α γ (1) = 0 γ (2) = α γ (3) = α + α γ (4) = α + α γ (5) = α + α + α γ (6) = α + 2 α γ (7) = α + 2 α + 2 α γ (8) = α + 3 α + α γ (9) = α + 3 α + 2 α γ (10) = 2 α + 3 α + α γ (11) = 2 α + 3 α + 2 α γ (12) = 2 α + 4 α + 2 α γ (13) = α + α + α γ (14) = α + α + α + α γ (15) = α + α + 2 α γ (16) = α + α + 2 α + 2 α γ (17) = α + α + 3 α + α γ (18) = α + α + 3 α + 2 α γ (19) = α + 2 α + 2 α γ (20) = α + 2 α + 2 α + 2 α γ (21) = 2 α + 2 α + 2 α γ (22) = 2 α + 2 α + 2 α + 2 α γ (23) = 2 α + 2 α + 4 α + 2 α γ (24) = 2 α + 3 α + 2 α γ (25) = 2 α + 3 α + 2 α + 2 α γ (26) = 2 α + 3 α + 4 α γ (27) = 2 α + 3 α + 4 α + 4 α γ (28) = 2 α + 3 α + 6 α + 2 α γ (29) = 2 α + 3 α + 6 α + 4 α γ (30) = 2 α + 4 α + 4 α γ (31) = 2 α + 4 α + 4 α + 4 α γ (32) = 2 α + 4 α + 8 α + 4 α γ (33) = 2 α + 5 α + 4 α + 2 α γ (34) = 2 α + 5 α + 8 α + 2 α γ (35) = 2 α + 5 α + 8 α + 6 α γ (36) = 2 α + 6 α + 6 α + 2 α γ (37) = 2 α + 6 α + 6 α + 4 α γ (38) = 2 α + 6 α + 8 α + 2 α γ (39) = 2 α + 6 α + 8 α + 6 α γ (40) = 2 α + 6 α + 10 α + 4 α γ (41) = 2 α + 6 α + 10 α + 6 α γ (42) = 2 α + 7 α + 8 α + 4 α γ (43) = 2 α + 7 α + 10 α + 4 α γ (44) = 2 α + 7 α + 10 α + 6 α γ (45) = 3 α + 3 α + 4 α + 2 α γ (46) = 3 α + 5 α + 4 α + 2 α γ (47) = 3 α + 5 α + 8 α + 2 α γ (48) = 3 α + 5 α + 8 α + 6 α γ (49) = 3 α + 7 α + 8 α + 2 α γ (50) = 3 α + 7 α + 8 α + 6 α γ (51) = 3 α + 7 α + 12 α + 6 α γ (52) = 3 α + 9 α + 12 α + 6 α γ (53) = 4 α + 5 α + 6 α + 2 α γ (54) = 4 α + 5 α + 6 α + 4 α γ (55) = 4 α + 5 α + 8 α + 4 α γ (56) = 4 α + 6 α + 6 α + 2 α γ (21) = α + 2 α + 3 α γ (22) = α + 2 α + 3 α + 3 α γ (23) = α + 2 α + 5 α + 2 α γ (24) = α + 2 α + 5 α + 3 α γ (25) = α + 3 α + 3 α + α γ (26) = α + 3 α + 3 α + 2 α γ (27) = α + 3 α + 5 α + α γ (28) = α + 3 α + 5 α + 4 α γ (29) = α + 3 α + 6 α + 2 α γ (30) = α + 3 α + 6 α + 4 α γ (31) = α + 4 α + 5 α + 2 α γ (32) = α + 4 α + 5 α + 3 α γ (33) = α + 4 α + 6 α + 2 α γ (34) = α + 4 α + 6 α + 4 α γ (35) = α + 4 α + 7 α + 3 α γ (36) = α + 4 α + 7 α + 4 α γ (37) = 2 α + 2 α + 3 α + α γ (38) = 2 α + 2 α + 3 α + 2 α γ (39) = 2 α + 2 α + 4 α + 2 α γ (40) = 2 α + 3 α + 3 α + α γ (41) = 2 α + 3 α + 3 α + 2 α γ (42) = 2 α + 3 α + 5 α + α γ (43) = 2 α + 3 α + 5 α + 4 α γ (44) = 2 α + 3 α + 6 α + 2 α γ (45) = 2 α + 3 α + 6 α + 4 α γ (46) = 2 α + 4 α + 4 α + 2 α γ (47) = 2 α + 4 α + 5 α + α γ (48) = 2 α + 4 α + 5 α + 4 α γ (49) = 2 α + 4 α + 7 α + 2 α γ (50) = 2 α + 4 α + 7 α + 5 α γ (51) = 2 α + 4 α + 8 α + 4 α γ (52) = 2 α + 5 α + 6 α + 2 α γ (53) = 2 α + 5 α + 6 α + 4 α γ (54) = 2 α + 5 α + 7 α + 2 α γ (55) = 2 α + 5 α + 7 α + 5 α γ (56) = 2 α + 5 α + 9 α + 4 α γ (57) = 4 α + 6 α + 6 α + 4 α γ (58) = 4 α + 6 α + 8 α + 2 α γ (59) = 4 α + 6 α + 8 α + 6 α γ (60) = 4 α + 6 α + 10 α + 4 α γ (61) = 4 α + 6 α + 10 α + 6 α γ (62) = 4 α + 7 α + 8 α + 2 α γ (63) = 4 α + 7 α + 8 α + 6 α γ (64) = 4 α + 7 α + 12 α + 6 α γ (65) = 4 α + 8 α + 8 α + 4 α γ (66) = 4 α + 8 α + 12 α + 4 α γ (67) = 4 α + 8 α + 12 α + 8 α γ (68) = 4 α + 9 α + 10 α + 4 α γ (69) = 4 α + 9 α + 10 α + 6 α γ (70) = 4 α + 9 α + 12 α + 4 α γ (71) = 4 α + 9 α + 12 α + 8 α γ (72) = 4 α + 9 α + 14 α + 6 α γ (73) = 4 α + 9 α + 14 α + 8 α γ (74) = 4 α + 10 α + 12 α + 6 α γ (75) = 4 α + 10 α + 14 α + 6 α γ (76) = 4 α + 10 α + 14 α + 8 α γ (77) = 5 α + 7 α + 8 α + 4 α γ (78) = 5 α + 7 α + 10 α + 4 α γ (79) = 5 α + 7 α + 10 α + 6 α γ (80) = 5 α + 9 α + 10 α + 4 α γ (81) = 5 α + 9 α + 10 α + 6 α γ (82) = 5 α + 9 α + 12 α + 4 α γ (83) = 5 α + 9 α + 12 α + 8 α γ (84) = 5 α + 9 α + 14 α + 6 α γ (85) = 5 α + 9 α + 14 α + 8 α γ (86) = 5 α + 11 α + 14 α + 6 α γ (87) = 5 α + 11 α + 14 α + 8 α γ (88) = 5 α + 11 α + 16 α + 8 α γ (89) = 6 α + 9 α + 12 α + 6 α γ (90) = 6 α + 10 α + 12 α + 6 α γ (57) = 2 α + 5 α + 9 α + 5 α γ (58) = 2 α + 6 α + 8 α + 4 α γ (59) = 2 α + 6 α + 9 α + 4 α γ (60) = 2 α + 6 α + 9 α + 5 α γ (61) = 3 α + 4 α + 5 α + 2 α γ (62) = 3 α + 4 α + 5 α + 3 α γ (63) = 3 α + 4 α + 6 α + 2 α γ (64) = 3 α + 4 α + 6 α + 4 α γ (65) = 3 α + 4 α + 7 α + 3 α γ (66) = 3 α + 4 α + 7 α + 4 α γ (67) = 3 α + 5 α + 6 α + 2 α γ (68) = 3 α + 5 α + 6 α + 4 α γ (69) = 3 α + 5 α + 7 α + 2 α γ (70) = 3 α + 5 α + 7 α + 5 α γ (71) = 3 α + 5 α + 9 α + 4 α γ (72) = 3 α + 5 α + 9 α + 5 α γ (73) = 3 α + 6 α + 7 α + 3 α γ (74) = 3 α + 6 α + 7 α + 4 α γ (75) = 3 α + 6 α + 9 α + 3 α γ (76) = 3 α + 6 α + 9 α + 6 α γ (77) = 3 α + 6 α + 10 α + 4 α γ (78) = 3 α + 6 α + 10 α + 6 α γ (79) = 3 α + 7 α + 9 α + 4 α γ (80) = 3 α + 7 α + 9 α + 5 α γ (81) = 3 α + 7 α + 10 α + 4 α γ (82) = 3 α + 7 α + 10 α + 6 α γ (83) = 3 α + 7 α + 11 α + 5 α γ (84) = 3 α + 7 α + 11 α + 6 α γ (85) = 4 α + 6 α + 8 α + 4 α γ (86) = 4 α + 6 α + 9 α + 4 α γ (87) = 4 α + 6 α + 9 α + 5 α γ (88) = 4 α + 7 α + 9 α + 4 α γ (89) = 4 α + 7 α + 9 α + 5 α γ (90) = 4 α + 7 α + 10 α + 4 α γ (91) = 6 α + 10 α + 14 α + 6 α γ (92) = 6 α + 10 α + 14 α + 8 α γ (93) = 6 α + 11 α + 14 α + 6 α γ (94) = 6 α + 11 α + 14 α + 8 α γ (95) = 6 α + 11 α + 16 α + 8 α γ (96) = 6 α + 12 α + 16 α + 8 α γ (91) = 4 α + 7 α + 10 α + 6 α γ (92) = 4 α + 7 α + 11 α + 5 α γ (93) = 4 α + 7 α + 11 α + 6 α γ (94) = 4 α + 8 α + 11 α + 5 α γ (95) = 4 α + 8 α + 11 α + 6 α γ (96) = 4 α + 8 α + 12 α + 6 α Table-IBy the aid of Table-I, the sets Γ A of (II.6) are given, for A = 1 , , . . . , = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } 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, , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , 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, , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Γ = { , , , } Table-IIIn view of (II.6), the notation in Table-II is briefly given byΓ A = { I , I , I , I } ≡ Γ( A ) = { γ ( I ( A )) , γ ( I ( A )) , γ ( I ( A )) , γ ( I ( A )) } Note here that | W ( F ) | = 1152, | W ( λ ) | = | W ( λ ) | = 24 and | W ( λ ) | = | W ( λ ) | = 96show, for F , the validity of our two statements mentioned above.Corresponding signatures are also given by the following Table-III : ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +19 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +11 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1 ǫ = +1 ǫ = − ǫ = − ǫ = +1Table-IIIThe signatures, given in this Table-III, can be easily found to be true by comparing twoexpressions (II.1) and (II.3) for the case in hand.Let us now choose the most general specialization as in (I.5) : e α = u , e α = u , e α = u , e α = u . As explained in sec.I, corresponding character polynomial for an irreducible F represen-tation originated from a dominant weight Λ + = s λ + s λ + s λ + s λ will be3defined in the following form: R ( u , u , u , u , s , s , s , s ) ≡ P ( u , u , u , u , s , s , s , s ) P ( u , u , u , u , , , ,
0) (
III. P ( u , u , u , u , s , s , s , s )here. It is however given [7] in our website.To exemplify our result, the following moderated specialization will however be themore suitable one : e α = x , e α = x , e α = y , e α = y ( III. P ( x, y, s , s , s , s ) = P + ( x, y, s , s , s , s ) + P − ( x, y, s , s , s , s ) ( III. P − ( x, y, s , s , s , s ) = P + (1 /x, /y, s , s , s , s ) ( III. P + ( x, y, s , s , s , s ) = x − s − s − s − s y − − s − s − s − s ( − x s + s y s + s ( − y s + s )( − y s +4 s +2 s + s )( − y s +4 s +3 s +2 s )+ x s +2 s y s +2 s +3 s +4 s ( − y s +2 s + s )( − y s + s + s )( − y s +4 s +2 s + s ) − x s +2 s y s +2 s ( − y s + s )( − y s +2 s +2 s + s )( − y s +4 s +3 s + s ) − x s +4 s +2 s y s +6 s +3 s ( − y s )( − y s +2 s + s + s )( − y s +2 s +2 s + s )+ x s +4 s +2 s y s +6 s +3 s ( − y s )( − y s + s + s )( − y s +2 s + s )+ x s +2 s +2 s y s +2 s +3 s ( − y s + s )( − y s +2 s + s + s )( − y s +4 s +2 s + s )+ x s +2 s + s y s +2 s + s ( − y s + s + s )( − y s +2 s +2 s + s ) × ( − y s +4 s +3 s +2 s ) − x s +2 s +2 s y s +2 s +4 s +4 s ( − y s +2 s + s ) × ( − y s +2 s + s )( − y s +4 s +3 s + s ) + x s + s + s y s +2 s + s ( − y s ) × ( − y s +4 s +3 s + s )( − y s +4 s +3 s +2 s ) − x s + s +2 s y s +2 s +3 s × ( − y s + s )( − y s + s )( − y s +2 s + s ) − x s + s +2 s y s +6 s +3 s +4 s × ( − y s +2 s + s )( − y s + s )( − y s +2 s +2 s + s ) + x s +2 s + s y s +2 s +4 s + s × ( − y s +2 s + s + s )( − y s +2 s + s )( − y s +4 s +3 s +2 s )4 − x s +2 s +2 s y s +4 s +4 s ( − y s )( − y s +4 s +2 s + s )( − y s +4 s +3 s + s )+ x s +10 s +7 s +4 s y s +18 s +12 s +6 s ( − y s )( − y s )( − y s + s ) − x s +9 s +7 s +4 s y s +14 s +12 s +6 s ( − y s + s )( − y s )( − y s + s + s ) − x s +10 s +7 s +4 s y s +18 s +12 s +6 s ( − y s )( − y s )( − y s + s )+ x s +9 s +6 s +4 s y s +14 s +9 s +6 s ( − y s + s )( − y s + s )( − y s +2 s + s ) − x s +9 s +6 s +3 s y s +14 s +9 s +4 s ( − y s )( − y s + s + s )( − y s +2 s + s ) − x s +8 s +6 s +4 s y s +12 s +9 s +6 s ( − y s )( − y s + s + s )( − y s +2 s + s )+ x s +8 s +7 s +4 s y s +14 s +12 s +6 s ( − y s + s )( − y s )( − y s + s + s )+ x s +9 s +7 s +4 s y s +14 s +12 s +6 s ( − y s +2 s + s )( − y s ) × ( − y s +2 s + s + s ) + x s +8 s +6 s +3 s y s +12 s +9 s +4 s ( − y s + s ) × ( − y s + s )( − y s +2 s + s ) − x s +9 s +6 s +4 s y s +14 s +9 s +6 s × ( − y s +2 s + s )( − y s + s )( − y s +2 s +2 s + s ) − x s +8 s +7 s +4 s × y s +14 s +12 s +6 s ( − y s +2 s + s )( − y s )( − y s +2 s + s + s ) − x s +8 s +5 s +3 s y s +12 s +8 s +4 s ( − y s + s )( − y s )( − y s + s + s ) − x s +8 s +5 s +4 s y s +14 s +9 s +6 s ( − y s + s )( − y s + s )( − y s +2 s + s )+ x s +9 s +6 s +3 s y s +14 s +9 s +4 s ( − y s )( − y s +2 s + s + s ) × ( − y s +2 s +2 s + s ) + x s +7 s +5 s +3 s y s +12 s +8 s +4 s ( − y s )( − y s ) × ( − y s + s ) + x s +8 s +5 s +4 s y s +14 s +9 s +6 s ( − y s +2 s + s )( − y s + s ) × ( − y s +2 s +2 s + s ) + x s +8 s +6 s +4 s y s +12 s +9 s +6 s ( − y s ) × ( − y s +2 s + s + s )( − y s +2 s +2 s + s ) + x s +8 s +5 s +2 s y s +14 s +9 s +4 s × ( − y s )( − y s + s + s )( − y s +2 s + s ) + x s +6 s +5 s +4 s × y s +12 s +9 s +6 s ( − y s )( − y s + s + s )( − y s +2 s + s )+ x s +6 s +6 s +4 s y s +8 s +9 s +6 s ( − y s +2 s + s )( − y s + s + s ) × ( − y s +4 s +2 s + s ) − x s +8 s +6 s +3 s y s +12 s +9 s +4 s ( − y s +2 s + s ) × ( − y s + s )( − y s +2 s +2 s + s ) − x s +8 s +5 s +2 s y s +14 s +9 s +4 s ( − y s ) × ( − y s +2 s + s + s )( − y s +2 s +2 s + s ) − x s +6 s +6 s +3 s y s +8 s +9 s +4 s × ( − y s + s )( − y s +2 s + s + s )( − y s +4 s +2 s + s ) + x s +8 s +5 s +3 s × y s +12 s +8 s +4 s ( − y s +2 s + s )( − y s )( − y s +2 s + s + s ) − x s +6 s +6 s +4 s y s +8 s +9 s +6 s ( − y s + s )( − y s +2 s + s + s ) × ( − y s +4 s +2 s + s ) − x s +6 s +5 s +2 s y s +12 s +9 s +4 s ( − y s + s ) × ( − y s + s )( − y s +2 s + s )5 − x s +6 s +4 s +4 s y s +8 s +6 s +6 s ( − y s +2 s + s )( − y s +2 s + s ) × ( − y s +4 s +3 s + s ) − x s +5 s +5 s +4 s y s +8 s +9 s +6 s ( − y s +2 s + s ) × ( − y s + s + s )( − y s +4 s +2 s + s ) + x s +6 s +6 s +3 s y s +8 s +9 s +4 s × ( − y s +2 s + s )( − y s + s + s )( − y s +4 s +2 s + s ) + x s +5 s +4 s +4 s × y s +8 s +6 s +6 s ( − y s +2 s + s )( − y s +2 s + s )( − y s +4 s +3 s + s )+ x s +6 s +4 s +4 s y s +8 s +6 s +6 s ( − y s + s )( − y s +2 s +2 s + s ) × ( − y s +4 s +3 s + s ) − x s +7 s +5 s +3 s y s +12 s +8 s +4 s ( − y s )( − y s ) × ( − y s + s ) − x s +6 s +5 s +4 s y s +12 s +9 s +6 s ( − y s )( − y s +2 s + s + s ) × ( − y s +2 s +2 s + s ) + x s +6 s +3 s +2 s y s +12 s +8 s +4 s ( − y s + s ) × ( − y s )( − y s + s + s ) + x s +6 s +3 s +3 s y s +8 s +4 s +4 s ( − y s + s ) × ( − y s +2 s +2 s + s )( − y s +4 s +3 s + s ) + x s +5 s +5 s +2 s y s +8 s +9 s +4 s × ( − y s + s )( − y s +2 s + s + s )( − y s +4 s +2 s + s ) + x s +5 s +5 s +4 s × y s +8 s +9 s +6 s ( − y s + s )( − y s +2 s + s + s )( − y s +4 s +2 s + s )+ x s +6 s +4 s + s y s +8 s +6 s + s ( − y s +2 s + s + s )( − y s +2 s + s ) × ( − y s +4 s +3 s +2 s ) − x s +6 s +3 s +3 s y s +8 s +4 s +4 s ( − y s +2 s + s ) × ( − y s +2 s + s )( − y s +4 s +3 s + s ) − x s +4 s +5 s +3 s y s +6 s +8 s +4 s × ( − y s +2 s + s )( − y s )( − y s +2 s + s + s ) − x s +4 s +4 s +4 s × y s +6 s +6 s +6 s ( − y s )( − y s +4 s +2 s + s )( − y s +4 s +3 s + s )+ x s +6 s +5 s +2 s y s +12 s +9 s +4 s ( − y s +2 s + s )( − y s + s ) × ( − y s +2 s +2 s + s ) − x s +5 s +4 s +4 s y s +8 s +6 s +6 s ( − y s + s ) × ( − y s +2 s +2 s + s )( − y s +4 s +3 s + s ) − x s +4 s +3 s +2 s y s +12 s +8 s +4 s × ( − y s )( − y s )( − y s + s ) − x s +6 s +3 s + s y s +8 s +4 s + s × ( − y s + s + s )( − y s +2 s +2 s + s )( − y s +4 s +3 s +2 s ) − x s +5 s +4 s + s × y s +8 s +6 s + s ( − y s +2 s + s + s )( − y s +2 s + s )( − y s +4 s +3 s +2 s ) − x s +6 s +4 s + s y s +8 s +6 s + s ( − y s + s + s )( − y s +2 s +2 s + s ) × ( − y s +4 s +3 s +2 s ) − x s +5 s +5 s +2 s y s +8 s +9 s +4 s ( − y s +2 s + s ) × ( − y s + s + s )( − y s +4 s +2 s + s ) + x s +4 s +5 s +3 s y s +6 s +8 s +4 s × ( − y s + s )( − y s )( − y s + s + s ) + x s +4 s +4 s +4 s y s +6 s +6 s +6 s × ( − y s )( − y s +4 s +2 s + s )( − y s +4 s +3 s + s ) − x s +5 s +2 s +2 s × y s +8 s +4 s +4 s ( − y s + s )( − y s +2 s +2 s + s )( − y s +4 s +3 s + s ) − x s +3 s +3 s +3 s y s +4 s +4 s +4 s ( − y s )( − y s +4 s +2 s + s ) × ( − y s +4 s +3 s + s )6+ x s +6 s +3 s + s y s +8 s +4 s + s ( − y s +2 s + s + s ) × ( − y s +2 s + s )( − y s +4 s +3 s +2 s ) − x s +6 s +3 s +2 s y s +12 s +8 s +4 s × ( − y s +2 s + s )( − y s )( − y s +2 s + s + s ) + x s +5 s +2 s + s y s +8 s +4 s + s × ( − y s + s + s )( − y s +2 s +2 s + s )( − y s +4 s +3 s +2 s ) + x s +4 s +4 s + s × y s +6 s +6 s + s ( − y s + s )( − y s +4 s +2 s + s )( − y s +4 s +3 s +2 s )+ x s +4 s +2 s +3 s y s +6 s +3 s +4 s ( − y s +2 s + s )( − y s + s ) × ( − y s +2 s +2 s + s ) + x s +5 s +4 s + s y s +8 s +6 s + s ( − y s + s + s ) × ( − y s +2 s +2 s + s )( − y s +4 s +3 s +2 s ) + x s +3 s +3 s + s y s +4 s +4 s + s × ( − y s + s )( − y s +4 s +2 s + s )( − y s +4 s +3 s +2 s ) + x s +2 s +3 s +3 s × y s +2 s +4 s +4 s ( − y s +2 s + s )( − y s +2 s + s )( − y s +4 s +3 s + s )+ x s +5 s +2 s +2 s y s +8 s +4 s +4 s ( − y s +2 s + s )( − y s +2 s + s ) × ( − y s +4 s +3 s + s ) − x s +4 s +4 s + s y s +6 s +6 s + s ( − y s + s ) × ( − y s +4 s +2 s + s )( − y s +4 s +3 s +2 s ) − x s +4 s +2 s +3 s y s +6 s +3 s +4 s × ( − y s + s )( − y s + s )( − y s +2 s + s ) + x s +2 s +2 s +2 s y s +4 s +4 s +4 s × ( − y s )( − y s +4 s +2 s + s )( − y s +4 s +3 s + s ) − x s +2 s +2 s +3 s × y s +2 s +3 s +4 s ( − y s +2 s + s )( − y s + s + s )( − y s +4 s +2 s + s ) − x s +5 s +2 s + s y s +8 s +4 s + s ( − y s +2 s + s + s )( − y s +2 s + s ) × ( − y s +4 s +3 s +2 s ) − x s +3 s + s + s y s +4 s +2 s + s ( − y s ) × ( − y s +4 s +3 s + s )( − y s +4 s +3 s +2 s ) − x s +2 s +2 s + s y s +4 s +4 s + s × ( − y s + s )( − y s +4 s +2 s + s )( − y s +4 s +3 s +2 s ) − x s +4 s + s + s × y s +6 s +2 s + s ( − y s )( − y s +4 s +3 s + s )( − y s +4 s +3 s +2 s ) − x s +2 s +3 s + s y s +2 s +4 s + s ( − y s +2 s + s + s )( − y s +2 s + s ) × ( − y s +4 s +3 s +2 s ) + x s +4 s +3 s +2 s y s +12 s +8 s +4 s ( − y s )( − y s ) × ( − y s + s ) + x s +3 s +3 s +3 s y s +4 s +4 s ( − y s )( − y s +4 s +2 s + s ) × ( − y s +4 s +3 s + s ) + x s +2 s + s + s y s +4 s +2 s + s ( − y s ) × ( − y s +4 s +3 s + s )( − y s +4 s +3 s +2 s ) + x s +4 s + s + s y s +6 s +2 s + s × ( − y s )( − y s +4 s +3 s + s )( − y s +4 s +3 s +2 s ) + x s + s +3 s +2 s × y s +6 s +8 s +4 s ( − y s +2 s + s )( − y s )( − y s +2 s + s + s )+ x s + s +2 s +3 s y s +2 s +3 s +4 s ( − y s )( − y s + s + s )( − y s +2 s + s ) − x s +2 s +3 s +3 s y s +4 s +4 s ( − y s + s )( − y s +2 s +2 s + s ) × ( − y s +4 s +3 s + s ) − x s +3 s +3 s + s y s +4 s + s ( − y s + s ) × ( − y s +4 s +2 s + s )( − y s +4 s +3 s +2 s )7+ x s +2 s +2 s +3 s y s +3 s +4 s × ( − y s + s )( − y s +2 s + s + s )( − y s +4 s +2 s + s ) + x s +2 s +3 s + s × y s +4 s + s ( − y s + s + s )( − y s +2 s +2 s + s )( − y s +4 s +3 s +2 s ) − x s + s +3 s +2 s y s +6 s +8 s +4 s ( − y s + s )( − y s )( − y s + s + s ) − x s + s +2 s +3 s y s +3 s +4 s ( − y s )( − y s +2 s + s + s )( − y s +2 s +2 s + s )+ x s +3 s + s + s y s +2 s + s ( − y s )( − y s +4 s +3 s + s ) × ( − y s +4 s +3 s +2 s )) ( III. P ( x, y, , , ,
0) = 1 x y (1 + x )(1 + y )( − x ) ( − y ) ( − xy ) (1 + xy ) ( − x y ) × ( − xy ) (1 + xy )(1 + xy + x y )( − x y )( − xy )( − x y ) (1 + x y ) × ( − xy )(1 + xy + x y )( − x y ) ( − x y )( − x y ) ( III. Ch ( λ ) = R ( x, y, , , ,
0) = 1 x y (1 + x y )(1 + x + x + x y + x y + x y + x y + x y + 2 x y + x y + x y + x y + x y + x y + x y + 2 x y + x y + x y + x y + x y + x y + x y + x y + x y ) Ch ( λ + λ ) = R ( x, y, , , ,
1) = 1 x y (1 + y ) (1 + xy ) (1 + x y )(1 + y ) × (1 + xy )(1 + x y )(1 − xy + x y )(1 + xy )(1 + x y )(1 + x y )(1 + x y )A useful application of (III.4) is in the calculation of tensor coupling coefficients [8] .We note, for instance, that the coefficients in decomposition V ( λ ) ⊗ V ( λ + λ ) = V ( λ + λ + λ ) ⊕ V ( λ + 2 λ ) ⊕ V (2 λ ) ⊕ V ( λ + λ ) ⊕ V ( λ + 2 λ ) ⊕ V ( λ + λ ) ⊕ V ( λ + λ ) ⊕ V (3 λ ) ⊕ V ( λ ) ⊕ V ( λ + λ ) ⊕ V (2 λ ) ⊕ V ( λ )can be calculated easily by R ( x, y, , , , ⊗ R ( x, y, , , ,
1) = R ( x, y, , , , ⊕ R ( x, y, , , , ⊕ R ( x, y, , , , ⊕ R ( x, y, , , , ⊕ R ( x, y, , , , ⊕ R ( x, y, , , , ⊕ R ( x, y, , , , ⊕ R ( x, y, , , , ⊕ R ( x, y, , , , ⊕ R ( x, y, , , , ⊕ R ( x, y, , , , ⊕ R ( x, y, , , , CONCLUSION
As is promised, the case for F Lie algebra is studied here explicitly. In a subsequentwork, we will study the case for E Lie algebra. This first three cases complete theexceptional chain of finite Lie algebras. Beyond these, although applicable, the methodseem to be unpractical. We have shown however that there is another method [9] whichworks equally well for E and E . REFERENCES [1] M.Gungormez and H.R.Karadayi, J. Geometry and Physics 57 (2007) 2533-2538[2] H.Weyl, The Classical Groups, N.J. Princeton Univ. Press (1946)[3] H.R.Karadayi and M.Gungormez, J.Phys.A:Math.Gen. 32 (1999) 1701-1707[4] J.E.Humphreys, Introduction to Lie Algebras and Representation Theory,N.Y., Springer-Verlag (1972)[5] V.G.Kac, Infinite Dimensional Lie Algebras, N.Y., Cambridge Univ. Press (1990)[6] Weyl dimension formula cited in p.139 of ref [4][7] http://atlas.cc.itu.edu.tr/ ∼ gungorm[8] Steinberg formula cited in p.140 of ref [4][9] H.R.Karadayi and M.Gungormez, Summing over the Weyl Groups of E and E8