Tensor powers for non-simply laced Lie Algebras B 2 case
TTensor powers for non-simply laced Lie algebras B -case P P Kulish , V D Lyakhovsky , and O V Postnova Sankt-Petersburg Branch ofV A Steklov Mathematical Institute RAS , Sankt-Petersburg State University,High Energy Physics and Elementary Particles Department e-mail: [email protected], [email protected] [email protected] 10, 2018 Abstract
We study the decomposition problem for tensor powers of B -fundamentalmodules. To solve this problem singular weight technique and injection fanalgorithms are applied. Properties of multiplicity coefficients are formu-lated in terms of multiplicity functions. These functions are constructedshowing explicitly the dependence of multiplicity coefficients on the high-est weight coordinates and the tensor power parameter. It is thus possibleto study general properties of multiplicity coefficients for powers of thefundamental B - modules. Consider an analog of the Brauer centralizer algebras for the spinor groups anddefine the subspaces of the tensor space ( ⊗ p V n ) on which the symmetric group S k and Spin( n ) act as the dual pair (in a direct product form). Here V n isthe fundamental representation of Spin( n ). Namely the centralizer algebra ofthe orthogonal group in ( ⊗ p V n ) is generated by the symmetric group S k andthe contractions and the immersions of the invariant form and is called theBrauer centralizer algebras. To proceed further one needs the list of Spin( n )-irreducible subspaces in the decomposition ( ⊗ p V n ) = (cid:80) µ ∈ P m pµ V ( µ ) ( P is theSpin( n ) weight space) and their multiplicities m pµ . As far as we are interestedin an arbitrary power p of the fundamental module V n Spin( n ) our main prob-lem is to find multiplicities of submodules in a form of multiplicity functions a r X i v : . [ m a t h . R T ] J un ( µ, p ) explicitly depending on the corresponding highest weight µ and thepower parameter p .There are numerous combinatorial studies of the problem [1, 2, 3, 4] and alsoseries of works dealing with fermonic formulas, some of them based on crystalbasis approach [5, 6, 7, 8]. On this way important general results were obtained[9, 10, 11]. On the other hand practical computations with the correspondingformulas are scarcely possible for all but the simplest examples. In most of thesestudies the simply laced algebras are considered and as a rule the multiplicitiesformulas are connected with complicated path countings.We must mention also an algorithm for tensor product decompositions pro-posed in [12] and improved in [13]. It is used in our investigations.Summing up, we are to find multiplicities m pµ in the decomposition ( ⊗ p V n ) = (cid:80) µ ∈ P m pµ V ( µ ) as a function of µ and p . To solve this problem we propose analgorithm based on singular weights properties [14] and the injection fan tech-nique [15, 16]. We study the multiplicities m pµ formulated in terms of multiplicityfunctions M g ( µ, p ). The latter have the weight space L P for the domain of def-inition. On the sublattice P ++ of dominant weights the multiplicity functiongives us the desired multiplicities, M ( µ, p ) | µ ∈ P ++ = m pµ . In this paper we shallshow how to adopt these tools to non-simply laced algebras and shall demon-strate how they work by studying the tensor powers (cid:0) L ω i B n (cid:1) ⊗ p of the fundamentalmodule L ω i B of B . g – simple Lie algebra of the series B n , L µ – the integrable module of g with the highest weight µ ; r – the rank of the algebra g ;∆ – the root system; ∆ + – the positive root system for g ; N µ – the weight diagram of L µ ; W – the Weyl group; C (0) – the fundamental Weyl chamber, C (0) – its closure; ρ – the Weyl vector; (cid:15) ( w ) := det ( w ) , w ∈ W ; α i – the i -th simple root for g ; i = 0 , . . . , r ; ω i – the i -th fundamental weight for g ; i = 0 , . . . , r ; L ω i g – the i -th fundamental module; (cid:8) e i (cid:9) | i =1 ,...,r – the natural Euclidean basis of the weight space (the e -basis), { v i } – the coordinates of a weight in the e -basis; P – the weight lattice L P – the weight space; Q – the root lattice; E – the group algebra of the group P ;Ψ ( µ ) := (cid:80) w ∈ W (cid:15) ( w ) e w ◦ ( µ + ρ ) − ρ – the singular element for the g -module L µ ;2 Ψ ( µ ) – the set of singular weights ψ ∈ P for the module L µ with the coordi-nates ( ψ, (cid:15) ( w ( ψ ))) | ψ = w ( ψ ) ◦ ( µ + ρ ) − ρ ;ch ( L µ ) – the formal character of L µ ;ch ( L µ ) = (cid:80) w ∈ W (cid:15) ( w ) e w ◦ ( µ + ρ ) − ρ (cid:81) α ∈ ∆+ (1 − e − α ) = Ψ ( µ ) Ψ (0) – the Weyl formula; R := (cid:81) α ∈ ∆ + (1 − e − α ) = Ψ (0) – the denominator; M ω i g ( µ, p ) – the multiplicity function corresponding to the decomposition (cid:0) L ω i g (cid:1) ⊗ p = (cid:80) m ( i ) pµ L µ g , M ω i g ( µ, p ) | µ ∈ P ++ = m ( i ) pµ . Lemma 1
The projection Ψ ( ν,ξ ) ↓ g of the singular element Ψ ( ν,ξ ) g ⊕ g for the irre-ducible representation L ( ν ,...,ν r ) ⊗ L ( ξ ,...,ξ r ) of the direct sum g ⊕ g on the weightspace of the diagonal subalgebra g → g ⊕ g . is equal to the product Ψ ( ν,ξ ) ↓ g = Ψ ν g Ψ ξ g , Let (cid:110) ψ ( ν ) k | ψ ( ν ) k ∈ Ψ ( ν ) , k = 1 , . . . , W (cid:111) | and (cid:110) ψ ( ξ ) p | ψ ( ξ ) p ∈ Ψ ( ξ ) , p = 1 , . . . , W (cid:111) be the sets of singular weights for the modules L ( ν ,...,ν r ) and L ( ξ ,...,ξ r ) corre-spondingly then the set (cid:92) Ψ ( ν,ξ ) ↓ g consists of the weights (cid:110) ψ ( ν ) k + ψ ( ξ ) p , (cid:15) (cid:16) w (cid:16) ψ ( ν ) k (cid:17)(cid:17) (cid:15) (cid:16) w (cid:16) ψ ( ξ ) p (cid:17)(cid:17)(cid:111) . Proof.
Let (cid:8) e , e , . . . , e r , e r +1 , . . . e r (cid:9) be the weight space basis for L P ( g ⊕ g ),( ν , . . . , ν r , ξ , . . . , ξ r ) – the coordinates for the highest weight ( ν, ξ ) naturallybelonging to the space L P ( g ⊕ g ). The weights v ( ν ) ∈ N ν , u ( ξ ) ∈ N ξ and thesingular vectors ψ ( ν ) k and ψ ( ξ ) p also are lifted to the space L P ( g ⊕ g ) lv ( ν ) a ⇒ (cid:16) ν ( ν ) a , . . . , ν ( ν ) ar , , . . . , (cid:17) ; lu ( ξ ) b ⇒ (cid:16) , . . . , , u ( η ) b , . . . , u ( η ) br (cid:17) lψ ( ν ) k ⇒ (cid:16) ψ ( ν ) k , . . . , ψ ( ν ) kr , , . . . , (cid:17) ; lψ ( ξ ) p ⇒ (cid:16) , . . . , , ψ ( ξ ) p , . . . , ψ ( ξ ) pr (cid:17) The set (cid:8) lv a + lu b | a = 1 , . . . , dim ( L ν ) , b = 1 , . . . , dim (cid:0) L ξ (cid:1)(cid:9) forms the weightdiagram N ( ν,ξ ) g ⊕ g of L ( ν,ξ ) g ⊕ g . As far as for the Weyl group W g ⊕ g we have W g ⊕ g = W × W and the Weyl vector is ρ g ⊕ g = ( ρ, ρ ) , the set of singular weights (cid:92) Ψ ( ν,ξ ) g ⊕ g isformed by the vectors whose first 2 r coordinates are (cid:110) lψ ( ν ) k + lψ ( ξ ) p | k, p = 1 , . . . , W (cid:111) and the last one is equal to the product (cid:15) (cid:16) w (cid:16) ψ ( ν ) k (cid:17)(cid:17) (cid:15) (cid:16) w (cid:16) ψ ( ξ ) p (cid:17)(cid:17) : (cid:92) Ψ ( ν,ξ ) g ⊕ g = (cid:110) lψ ( ν ) k + lψ ( ξ ) p , (cid:15) (cid:16) w (cid:16) ψ ( ν ) k (cid:17)(cid:17) (cid:15) (cid:16) w (cid:16) ψ ( ξ ) p (cid:17)(cid:17) | k, p = 1 , . . . , W (cid:111) . The vector ( c , . . . , c r , c r +1 , . . . , c r ) ∈ L P ( g ⊕ g ) being projected to the diago-nal subalgebra weight space L P in the basis (cid:8)(cid:0) e + e r +1 (cid:1) / , . . . , (cid:0) e r + e r (cid:1) / (cid:9) c + c r +1 ) , . . . , ( c r + c r )) . The latter means that (cid:92) Ψ ( ν,ξ ) ↓ g = (cid:110) ψ ( ν ) k + ψ ( ξ ) p , (cid:15) (cid:16) w (cid:16) ψ ( ν ) k (cid:17)(cid:17) (cid:15) (cid:16) w (cid:16) ψ ( ξ ) p (cid:17)(cid:17)(cid:111) ,k, p = 1 , . . . , W. Q.E.D.One of the main tools to study the decomposition properties is the injectionfan Γ g diag → g ⊕ g . [15, 16] . To use this instrument we consider the decompositionof tensor products as a special case of branching. The latter corresponds to theinjection of the diagonal subalgebra into the direct sum: g diag → g ⊕ g . Lemma 2
The vectors of the injection fan Γ for g diag → g ⊕ g consists of theopposites to the singular weights of the trivial module L (0) g Γ g diag → g ⊕ g = − S ◦ (cid:100) Ψ (0) \ (0 , . . . , , (here S is the full reflection). Proof.
According to the definition [15] the vectors γ of the fan Γ g diag → g ⊕ g are fixed by the relation 1 − (cid:81) α ∈ (cid:16) ∆ + g ⊕ g ↓ g diag (cid:17) (1 − e − α ) mult g ⊕ g ( α ) − mult g diag ( α ) = (cid:80) γ ∈ Γ g diag → g ⊕ g s ( γ ) e − γ . The projections of the g ⊕ g -roots to the diagonalsubalgebra obviously reproduce the set { α i } i =0 ,...,r in L P ( g ) :( α i , ↓ g (0 , α i ) ↓ g (cid:27) = α i . Thus mult g ( α ) = 2 while mult g ⊕ g ( α ) = 1 and we have (cid:88) γ ∈ Γ g diag → g ⊕ g s ( γ ) e − γ = 1 − (cid:89) α ∈ ( ∆ + g ⊕ g ↓ g ) (cid:0) − e − α (cid:1) mult g ⊕ g ( α ) − mult g ( α ) == 1 − (cid:89) α ∈ (∆ + ) (cid:0) − e − α (cid:1) . Q.E.D.
Lemma 3
The singular element Ψ ( ξ ) g for the module L µ ⊗ L ν can be presentedin two equivalent forms: ch (cid:0) L µ g (cid:1) Ψ ν g = Ψ µ g ch (cid:0) L ν g (cid:1) . (1) Proof.
In the Weyl formula for L µ ⊗ L ν ,ch ( L µ ⊗ L ν ) ↓ P + g = (cid:88) ξ ∈ P ++ g m µνξ ch (cid:0) L ξ (cid:1) , (cid:32) Ψ ( ν,ξ ) g ⊕ g Ψ g ⊕ g (cid:33) ↓ P g = Ψ µ g Ψ ν g Ψ g Ψ g = (cid:88) ξ ∈ P ++ g m µνξ Ψ ξ g Ψ g . (cid:16)(cid:0) Ψ g (cid:1) − Ψ µ g (cid:17) Ψ ν g = Ψ µ g (cid:16)(cid:0) Ψ g (cid:1) − Ψ ν g (cid:17) = (cid:88) ξ ∈ P ++ g m µνξ Ψ ξ g . Thus we have (cid:88) ξ ∈ P ++ g m µνξ Ψ ξ g = ch (cid:0) L µ g (cid:1) Ψ ν g = Ψ µ g ch (cid:0) L ν g (cid:1) . (2)Q.E.D.Now put µ = ω , ν = ( p − ω ch (cid:16) L ( ω ) (cid:17) Ψ( ⊗ ( p − ω ) = (cid:88) ξ ∈ P M ω g ( ξ, p ) Ψ ( ξ ) , (3) M ω g ( ξ, p ) defines the singular element Ψ ( ⊗ p ω ) . On the other hand these equa-tions can be considered as a system of recurrent relations for the multiplicityfunction M ω g ( ξ, p ) . Conjecture 4
Let g = B , L µ and L ω = L vect be the highest weight moduleswith the highest weights µ and ω (the first fundamental weight). Then thetensor product decomposition ( L µ ⊗ L vect ) ↓ g diag = ⊕ γ L γ is multiplicity free. Proof.
According to Lemma 1 the projected singular element for a g ⊕ g -module L µ ⊗ L vect is Ψ ( µ,vect ) g ⊕ g ↓ g diag = Ψ µ g Ψ vect g and the set of singular weights is (cid:92) Ψ ( µ,vect ) ↓ g diag = (cid:110) ψ ( µ ) k + ψ ( vect ) p , (cid:15) (cid:16) w (cid:16) ψ ( µ ) k (cid:17)(cid:17) (cid:15) (cid:16) w (cid:16) ψ ( vect ) p (cid:17)(cid:17)(cid:111) ,k, p = 1 , . . . , W. Suppose µ = ( µ , µ ) is greater than ω = (1 ,
0) and µ > µ ≥
1. The singularweights of L µ in the fundamental chamber and its nearest neighbours are (cid:110) ψ ( µ ) s (cid:111) = ( µ , µ , (+1)) , ( µ , − µ − , ( − , ( µ − , µ + 1 , ( − s = 1 , , (cid:110) ψ ( µ ) s + ψ ( vect ) p (cid:111) = ( µ , µ , (+1)) + ψ ( vect ) p , ( µ , − µ − , ( − ψ ( vect ) p , ( µ − , µ + 1 , ( − ψ ( vect ) p s = 1 , , p = 1 , . . . , W. g diag → g ⊕ g to the set (cid:92) Ψ ( µ,vect ) ↓ g diag in the three se-lected chambers (starting with the highest weight ( µ + 1 , µ , (+1))) we findthe weights:( µ , µ , (+1)) , ( µ ± , µ , (+1)) , ( µ , µ ± , (+1)) , ( µ , − µ − , ( − , ( µ ± , − µ − , ( − , ( µ , − µ , ( − , ( µ , − µ − , ( − , ( µ − , µ + 1 , ( − , ( µ , µ + 1 , ( − , ( µ − , µ + 1 , ( − , ( µ − , µ + 2 , ( − , ( µ − , µ , ( − . Only the first 5 weights are in C (0) . They are the highest singular weights for( L µ ⊗ L vect ) ↓ g diag and we have the decomposition: (cid:0) L µ ⊗ L vect (cid:1) ↓ g diag = L µ ⊕ L ( µ +1 ,µ ) ⊕ L ( µ − ,µ ) ⊕ L ( µ ,µ +1) ⊕ L ( µ ,µ − . There are two special cases where the highest weights are on the borders of C (0) .For µ = nω = ( n,
0) and for µ = nω = ( n/ , n/
2) the same algorithm gives: (cid:0) L µ ⊗ L vect (cid:1) ↓ g diag = L ( µ +1 , ⊕ L ( µ − , ⊕ L ( µ ,µ +1) , (cid:0) L µ ⊗ L vect (cid:1) ↓ g diag = L µ ⊕ L ( µ +1 ,µ ) ⊕ L ( µ ,µ − . correspondingly. Q.E.D. B -case For g = B , r = 2 the simple roots in e -basis are α = e − e , α = e ,the fundamental weights are ω = ( e + e ) , ω = e and the fundamentalmodules – L ω (spinor) and dim L ω = 4 , L ω (vector), dim L ω = 5.Consider the modules ( L ω i ) ⊗ p | p ∈ Z + ,i =1 , and the decompositions ( L ω i ) ⊗ p = (cid:80) ν m ( i ) pν L ν . Our aim is to find multiplicities m ( i ) pν as functions of ν and p . To solve theproblem we propose to use the singular elements formalism [14], the polynomialdependence property that is a consequence of the relation (3) and the injectionfan technique [15] [16]. Γ B diag2 →⊕ p B Consider the injection B diag2 → ⊕ p B . The fan Γ B diag2 →⊕ p B ≡ Γ p is the ( p − (0) B . Conjecture 5
Place the origin of the space L P at the end of the lowest weightvector of the fan. The structure of the fan Γ p is as follows: . Along the line pα in the k -th root lattice point the multiplicity is ( − k C k − p ; k = 1 , . . . , p + 1 .
2. Each weight with the coordinates ( k − , − k ) is an origin of the set S ( k − , − k ) of singular weights described below.3. The set S ( k − , − k ) is composed of the tensor product of the singular ele-ments Ψ [0] Let Γ p be the fan with the properties described above. Rememberthat the set S ( k − , − k ) has itself the multiplicity ( − k C k − p . Multiply the fanΓ p by the element Ψ (0) , i.e. pass to the power ( p + 1). This means that the set S ( k − , − k ) will be transformed to( − k C k − p (cid:16) Ψ [ v ] 00 otherwise . Finally we getΓ p = (cid:88) a,b γ p ( a, b ) e ( a,b ) ; (cid:26) a = k − , . . . , p − k − ,b = 1 − k, . . . , p + k − ,γ p ( a, b ) = p (cid:88) k =1 k (cid:88) l k =1 p − k +1 (cid:88) m k =1 e ( a,b ) ( − k + a + b − l k + m k ) ×× (cid:98) C k − p − (cid:98) C l k − k − (cid:98) C m k − p − k (cid:98) C b + k − l k +2 p − k (cid:98) C a − k − m k +4 k − , (4)7ere the fan is a function of the parameter p and the coordinates ( a, b ) of thehighest weight. The zero point has the multiplicity ( − µ ∈ P the singular weights diagram Ψ (cid:16)(cid:80) ν m ( i ) pν L ν (cid:17) = (cid:80) e ( µ ) (cid:15) ( w ( ψ ( µ )))has the following fundamental property: (cid:88) a,b γ p ( a, b ) (cid:15) ( w ( ψ ( µ + ( a, b )))) = 0 , described by the fan Γ p . ( L ω ) ⊗ p – the spinor case Let us construct the singular element for the p -th power of the second (spinor)fundamental module L ω (coordinates of singular weights here are half-integerand the W -invariant vector is (0 , S k consisting of blocks enumerated by a pair of indices( l k , m k ) where l k = 1 , . . . , k + 1 and m k = 1 , . . . , p − k + 2 and attachedto the points (cid:0) p − k + 1 − m k − , p + k − − l k − (cid:1) . The multi-plicities of these blocks are ( − l k + m k − (cid:98) C l k − k (cid:98) C m k − p − k +1 2. Localize the systems S k along the line ( p , p ) − pα : the first system S hasthe origin at the point ( p , p ), the k -th – at the point (cid:0) p − k + 1 , p + k − (cid:1) ,the last one, S p +1 , – at (cid:0) − p, p (cid:1) . These systems have the multiplicities( − k − C k − p ; k = 1 , . . . , p + 1 . 3. The numbers (cid:0) p − k + 1 − m k − , p + k − − l k − (cid:1) are the co-ordinates of the upper right corner of the ( l k , m k )-block. The blocks havethe structure dual to the structure of the system S k but the intervals inthe blocks are doubled. The weights in the block are enumerated by theindices ( i k , j k ) where j k = 1 , . . . , k + 1 and i k = 1 , . . . , p − k + 2.4. Thus the ( l k , m k )-block in S k has the form: p − k +2 (cid:88) i k =1 k +1 (cid:88) j k =1 ( − i k + j k − (cid:98) C j k − k (cid:98) C i k − p − k +1 e ( p − k +1 − m k − j k +7 , p + k − − l k − i k +5 ) . 5. Now the system S k can be composed: p − k +2 (cid:88) i k ,m k =1 k +1 (cid:88) j k ,l k =1 ( − l k + m k + i k + j k − ×× (cid:98) C l k − k (cid:98) C m k − p − k +1 (cid:98) C j k − k (cid:98) C i k − p − k +1 e ( p − k +1 − m k − j k +7 , p + k − − l k − i k +5 ) . 8. Finally, the singular element Ψ( ( ω ) ⊗ p ) ≡ Ψ( ( s ) ⊗ p ) is fixed asΨ( ( s ) ⊗ p ) = p +1 (cid:88) k =1 p − k +2 (cid:88) i k ,m k =1 k +1 (cid:88) j k ,l k =1 ( − l k + m k + i k + j k − ×× (cid:98) C l k − k (cid:98) C m k − p − k +1 (cid:98) C j k − k (cid:98) C i k − p − k +1 ×× e ( p − k +1 − m k − j k +7 , p + k − − l k − i k +5 )The singular multiplicities are functions of p and c, d : ψ ( ( s ) ⊗ p ) ( c, d ) = e ( c,d ) p +1 (cid:88) k =1 k (cid:88) l k =1 p − k +2 (cid:88) m k =1 ( − k + ( c − d ) − ( l k + m k )+1 ×× (cid:98) C k − p (cid:98) C l k − k (cid:98) C ( − m k ) − k + c − p +1 ) k ×× (cid:98) C m k − p − k +1 (cid:98) C ( − m k + k − d + p +1 ) k . ( L ω ) ⊗ p – the vector case The construction procedure in the vector case is analogous to that of the spinor.It results in obtaining the expression:Ψ( ( ω ) ⊗ p ) ≡ Ψ( ( v ) ⊗ p ) = p +1 (cid:88) k =1 k − (cid:88) j,n =0 p − k +1 (cid:88) i,m =0 e (2 k + j +5 m − p − , − p + i − k +5 n +1) ×× ( − i + j + k + m + n − (cid:98) C mp − k +1 (cid:98) C nk − (cid:98) C jp − k (cid:98) C k − p (cid:98) C ip − k +1 . The corresponding singular multiplicities function depending on p and c, d are ψ ( ( v ) ⊗ p ) ( c, d ) = p +1 (cid:88) k =1 k (cid:88) l k =1 p − k +2 (cid:88) m k =1 e ( c,d ) ×× ( − k − d − c + p − l k + m k )+7 ×× (cid:98) C k − p (cid:98) C l k − k − (cid:98) C m k − p − k +1 (cid:98) C − d +2 k − l k − − p − k +1 (cid:98) C p − c − k − m k − k − . (5) To illustrate the recursive algorithm we perform calculations that must give usthe multiplicities m (1) pν . The starting value is always known – this is the mul-tiplicity of the highest weight ν = ( p, 0) of ( L ω ) ⊗ p that equals 1. Suppose wehave found the values of the multiplicity function m (1) pν for the 14 first weights: c \ d − − p − p − − p p − p − p ( p − 3) 09pplying the fan Γ p (4) we find the multiplicity for the next weight in thethird line, it has the coordinates ( p − , p − × p ( p − , the second line – − ( p − 1) ( p − × ( p − (cid:18) + 12 ( p − 1) ( p − 2) ( p − (cid:19) × (+1) ++ (cid:18) ( p − − 12 ( p − 1) ( p − (cid:19) × ( − ψ ( ( v ) ⊗ p ) ( p − , 1) for the weight ( p − , ψ ( ( v ) ⊗ p ) ( p − , 1) = p ( p − . The multiplicity m (1) p ( p − , is the sum: m (1) p ( p − , = ( p − (cid:18) p ( p − − ( p − 1) ( p − 2) + p + ( p − 2) + ( p − 2) ( p − − (cid:19) = 12 ( p − 1) ( p − . Notice that here we consider the line ν = ( p − , 1) in the space P × R . As aresult we obtain polynomials characterizing the p -dependence of the multiplicityfor a fixed distance between the highest weight and the weight ν .This example shows that the tools elaborated above (the injection fan andsingular elements) are effective in solving the reduction problem for tensor powermodules ( L ω i ) ⊗ p . According to Lemma 3 the multiplicity coefficients have additional recurrenceproperties generated by the fundamental module weights system N (cid:0) L ( ω i ) (cid:1) : (cid:88) µ ∈ P ++ m ( i ) pµ Ψ ( µ ) = ch (cid:16) L ( ω i ) (cid:17) Ψ( ⊗ ( p − ω i ) . (6)This relation can be decomposed using the multiplicity functions (defined on P ), (cid:88) µ ∈ P M ω i ( µ, p ) e µ = ch (cid:16) L ( ω i ) (cid:17) Ψ( ⊗ ( p − ω i ) , (7)10emember that M ω i ( µ, p ) | µ ∈ C (0) = m ( i ) pµ . Thus instead of the highest weightsearch for each singular element Ψ ( µ ) we use their anti-symmetry properties.This leads to the recurrent relation: M ω i ( µ, p ) = (cid:88) ζ ∈ N (cid:16) L ( ωi ) (cid:17) n ζ (cid:16) L ( ω i ) (cid:17) M ω i ( µ − ζ, p − , (8)where n ζ (cid:0) L ( ω i ) (cid:1) = mult L ( ωi ) ( ζ ). Obviously such relations are especially usefulwhen dim (cid:0) L ( ω i ) (cid:1) is small, thus for our needs (when the module L ( ω i ) is fun-damental with the trivial weights multiplicities n ζ (cid:0) L ( ω i ) (cid:1) = 1) the obtainedrecurrence must be effective.Formula (8) tells us what happens when we pass from the ( p − p -th. In particular for the spinor module L ω to find the value of M ω ( µ, p )the coordinates must be shifted by the vectors of N (cid:0) L ( ω ) (cid:1) diagram: M ω ( µ, p ) = (cid:88) ζ = N ( L ( ω ) M ω ( µ − ζ, p − . (9)In the recurrence starting point the value of the multiplicity function is known M ω ( pω , p ) = 1 and for all ν > pω it has zero values. In the natural coordi-nates this means: M ω (( a, b ) , p ) = (cid:88) λ =( a,b ) − { ( , )( − , )( , − )( − , − ) } M ω ( λ, p − . For the vector module L ω and its tensor powers we can construct similarrelations: M ω (( a, b ) , p ) = (cid:88) ζ = N ( L ( ω ) M ω (( a, b ) − ζ, p − 1) = (cid:88) λ =( a,b ) −{ (1 , , − , , − , (0 , } M ω ( λ, p − , (10)with the similar boundary condition M ω (( p, , p ) = 1 . (11)The obtained recurrence relations indicate an important property of M ω i ( µ, p ): Conjecture 6 The multiplicity function M ω i ( µ, p ) is a polynomial on p over Q (rational numbers). Notice that when µ belongs to the correlated (different!) boundaries of thearea where the function is nontrivial the values M ω i ( µ, p ) for i = 1 and i = 2coinside. 11 onjecture 7 The multiplicities M ω ( µ, p ) of the ”upper diagonal” highestweights ( µ = ( p, − nα ) for ( L ω ) ⊗ p coinside with the multiplicities M ω ( µ, p ) of the ”upper line” highest weights ( µ = (cid:0) p , p (cid:1) − nα ) for ( L ω ) ⊗ p . On the left boundary of the Weyl chamber C (0) the multiplicities M ω ( µ, p )are subject to the presence of the ”reflected” singular weights in the left adjacentWeyl chamber. This observation is important because for an analogous bound-ary in the spinor case the situation is different: the adjacent Weyl chamberhad no influence on the values of M ω ( µ, p ) with µ ∈ C (0) (on the correspond-ing subdiagonal the function has zero values). In particular this results in thefollowing property. Conjecture 8 The ”second diagonal” of the highest weights for ( L ω ) ⊗ p startswith zero: M ω (( p − ω , p ) = 0 . We have found out that the multiplicity functions M ω i ( µ, p ) are subject to aninfinite system of coupled algebraic equations with simple and obvious boundaryconditions. They can be solved step by step.For example consider the Bratteli-like diagram for B vector module L ω and let p = 1 , . . . , 6. The maximal number of paths that connect a point in the p − p -th one is five.Notice that the path counting procedure here is very complicated because ofthe boundary effects. Such complexities grow up considerably if we try to applythat counting procedures to algebras with higher rank. This fact stimulates spe-cial interest to direct studies of the recurrence relations systems. Moreover if thecorresponding equations could be solved this will give an explicit p -dependenceof the multiplicity function – the result that scaresly could be achieved by com-binatorial methods.We can construct the solution for the recurrence equations successively andthe answer is limited only by the number of equations in the system solved .This gives the explicit multiplicity dependence on p but for a finite number ofsuccessive weights. Thus having solved the first five equations for the spinorcase we get the following table of functions M ω (( a, b ) , p ) (here the coordinates( a, b ) are fixed by the relation µ = pω − aα − ( b − α ): b \ a p ( p − p − ( p − ·· ( p + 1) ( p − p ( p − 1) 00 0 03 ( p − 1) ( p − ·· ( p − 3) ( p + 2) 012orrespondingly having solved the first fifteen equations for the vector casewe get the following table of functions M ω (( a, b ) , p ), b \ a p p − p − p − p ( p − ( p − 1) ( p − p ( p − 3) 0 p − ( p − p − 2) ( p − p ( p − p − p ( p − p − p ( p − p − When the algebra is simply laced, for example g = A n , the Weyl symmetrywas proven to be a highly effective tool to solve the set of recurrences equationsfor the powers of the first fundamental module L ω A n [17]. In the simplest case g = A the complete set of multiplicity functions for powers of an arbitraryirreducible module were thus constructed.In the case of B the difficulties start when the vector fundamental moduleis tensored. The recurrence equation can be solved successively, as was shown inthe previous section, but the complete solution for the function M ω (( a, b ) , p )was not found.Nevertheless the recurrence property (9) permits to describe the generaldependence of the multiplicities on one of the coordinates. To see this considerthe coordinates ( s, t ) defined by the relation µ = pω − tα − ( s − e . The t -dependence will be explicitly described, but only for limited values of s =1 , , . . . . This description is based on the fact that the Conjecture 7 gives usan explicit answer to the ”first diagonal” of multiplicities (cid:110) m (1) p (1 ,t ) | t = 0 , , . . . (cid:111) .Starting with this expression and using the relation (9) reformulated for the”diagonal lines” of functions we can find explicit expressions for any such lineprovided the previous lines are known: M ω ((1 , t ) , p ) = M ω (cid:16)(cid:16) p , p − t (cid:17) , p (cid:17) = Γ ( p + 1) ( p + 1 − t )Γ ( p + 2 − t ) Γ ( t + 1) ,M ω ((2 , t ) , p ) = Γ ( p + 1) ( p − t ) ( p − t )Γ ( p + 2 − t ) Γ ( t ) ( t + 1) ,M ω ((3 , t ) , p ) = Γ ( p + 1) ( p − t − p − t ) Γ ( t + 1) ,M ω ((4 , t ) , p ) = Γ ( p + 1) ( p − t − p + 1 − t ) Γ ( t + 3) ·· (cid:18) (cid:0) t + 6 t + 2 (cid:1) p − t + 2) ( t + 1) p ++ t + 4 t + 8 t + 8 t + 6 (cid:19) , ω ((5 , t ) , p ) = Γ ( p + 1) ( p − t − p − t ) Γ ( t + 3) ·· (cid:18) (cid:0) t + 11 t + 6 (cid:1) p − (2 t + 4) ( t + 1) ( t + 2) p ++ (cid:0) t + 4 t + 8 t + 8 t + 6 (cid:1) (cid:19) ,M ω ((6 , t ) , p ) = Γ ( p + 1) ( p − t − p − t ) Γ ( t + 4) ·· (cid:0) t + 21 t + 86 t + 36 (cid:1) p −− (cid:0) t + 54 t + 309 t + 654 t + 276 (cid:1) p ++ (cid:0) t + 45 t + 326 t + 1086 t + 1408 t + 516 (cid:1) p −− ( t + 1) ( t + 3) (cid:0) t + 8 t + 68 t + 208 t + 12 (cid:1) , and so on. We see that beginning from M ω B ((4 , t ) , p ) only some factor ofthe multiplicity function can be presented as a product of simple binomialslike ( p − x ). In the forthcoming publications we shall discuss this property indetails. M ω ((4 , b ) , p ) The tensor powers decomposition algorithm based on singular weights and in-jection fan technique was proven to be an effective tool in multiplicity propertystudies. Its abilities were demonstrated on tensor powers decompositions of B -fundamental modules. This algorithm is universal and can be applied toinvestigate decomposition properties in case of an arbitrary simple Lie algebraand its arbitrary module.As it was predicted in [17] in non-simply laced case the Weyl symmetryproperties are insufficient to provide the final solution for the corresponding setof recurrence relations for multiplicity functions (at least this appeared to betrue for the vector fundamental modules). Nevertheless (this was shown abovein our studies of fundamental B -modules) important properties of multiplicitycoefficients for any highest weight ν can be found by constructing the functions M ω i (( ν ) , p ) successively i.e. by constructing the solution for a final part of thefull set of recurrence relations. Supported by the Russian Foundation for Fundamental Research grant N 09-01-00504 and the ”Dynasty” Foundation. References [1] A. N. Kirillov and N. Yu. Reshetikhin. Formulas for Multiplicities of Oc-curence of Irreducible Components in the Tensor Product of Representa- ions of Simple Lie Algebras. Journal of Mathematical Sciences, Vol. 80,No. 3, 1996.[2] M. Kleber, Combinatorial Structure of Finite Dimensional Representationsof Yangians: the Simply-Laced Case . arXiv:q-alg/9611032v2.[3] M. Kleber, Finite Dimensional Representations of Quantum Affine Alge-bras . arXiv:math/9809087v1 [math.QA].[4] V. 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