On a property of branching coefficients for affine Lie algebras
aa r X i v : . [ m a t h . R T ] D ec On a property of branching coefficientsfor affine Lie algebras
Mikhail IlyinTheoretical Department, SPb State University,198904, Sankt-Petersburg, RussiaPetr Kulish ∗ Sankt-Petersburg Department ofSteklov Institute of MathematicsFontanka 27, 191023, Sankt-Petersburg, RussiaVladimir Lyakhovsky † Theoretical Department, SPb State University,198904, Sankt-Petersburg, Russiae-mail:[email protected] 8, 2018
Abstract
It is demonstrated that decompositions of integrable highest weightmodules of a simple Lie algebra with respect to its reductive subalgebraobey the set of algebraic relations leading to the recursive properties forthe corresponding branching coefficients. These properties are encodedin the special element Γ g ⊃ a of the formal algebra E a that describes theinjection and is called the fan. In the simplest case, when a = h ( g ), therecursion procedure generates the weight diagram of a module L g . Whenapplied to a reduction of highest weight modules the recursion describedby the fan provides a highly effective tool to obtain the explicit values ofbranching coefficients. We consider integrable modules L µ of affine Lie algebra g with the highestweight µ and the reduced modules L µ g ↓ a with respect to a reductive subalgebra a ⊂ g . In particular when the Cartan subalgebra a = h is studied, the branching ∗ Supported by RFFI grant N 06-01-00451 † Supported by RFFI grant N 06-01-00451 and the National Project RNP.2.1.1.1112 A (1) n ⊂ A (1) n − p − ⊕ A (1) p ) arise in the computation of the localstate probabilities for solvable models on square lattice [1]. Irreducible highestweight modules with dominant integral weights appear also in application of thequantum inverse scattering method [2] where solvable spin chains are studied inthe framework of the AdS/CFT correspondence conjecture of the super-stringtheory (see [3, 4] and references therein).There are different ways to find branching coefficients. One can use the BGGresolution [7] (for Kac-Moody algebras the algorithm is described in [5, 6]), theSchure function series [8], the BRST cohomology [9], Kac-Peterson formulas [5]or the combinatorial methods applied in [10]. We want to obtain the recursiveformulas for weight multiplicities and branching coefficients using the purelyalgebraic approach. From the Weyl-Kac character formula [5]ch L µ ( g ) = P w ∈ W ǫ ( w ) e w ◦ ( µ + ρ ) − ρ Q α ∈ ∆ + (1 − e − α ) mult( α ) , (1)we derive the special set of relations for branching coefficients. These relationscan be used both to construct a representation and to reduce it with respectto a subalgebra a ⊂ g that is to find the corresponding branching rules. Eachrelation of the set deals with a finite collection of weights. Among them itis always possible to fix the lowest one (with respect to the natural orderingof weights induced by basic roots). Thus it is possible to use the relationsof the set as recurrent relations for branching coefficients. It is demonstratedthat branching is governed by a certain system of weights (called ”the fan ofinjection”) that depends only on the algebra and the injection morphism andcan be used to decompose the highest weight modules.For finite dimensional classical Lie algebras the case of regular injectionswas considered in [11] where the recurrent relations were constructed using theproperties of the Kostant-Heckman partition function. The same method wasused for regular injections of affine Lie algebras [12]. In this study we presenta different approach and find that for any reductive subalgebra a of an affineLie algebra g such that h ∗ a ⊂ h ∗ g and h ∗ ◦ a ⊂ h ∗ ◦ g the branching coefficients obeythe set of properties that give rise to recurrent relations. The latter provide acompact and effective method to construct the corresponding branching rules.The results are illustrated by examples. Consider the affine Lie algebras g and a with the underlying finite-dimensionalsubalgebras ◦ g and ◦ a and an injection a −→ g such that a is a reductive subalgebra a ⊂ g with correlated root spaces: h ∗ a ⊂ h ∗ g and h ∗ ◦ a ⊂ h ∗ ◦ g .The following notation will also be used:2 µ ( L ν a ) – the integrable module of g with the highest weight µ ; (resp.integrable a -module with the highest weight ν ); r , ( r a ) – the rank of the algebra g (resp. a ) ;∆ (∆ a )– the root system; ∆ + (resp. ∆ + a )– the positive root system (of g and a respectively);mult ( α ) (mult a ( α )) – the multiplicity of the root α in ∆ (resp. in (∆ a )); ◦ ∆ , (cid:18) ◦ ∆ a (cid:19) – the finite root system of the subalgebra ◦ g (resp. ◦ a ); N µ , ( N ν a ) – the weight diagram of L µ (resp. L ν a ) ; W , ( W a )– the corresponding Weyl group; C , ( C a )– the fundamental Weyl chamber; ρ , ( ρ a ) – the Weyl vector; ǫ ( w ) := det ( w ) ; α i , (cid:0) α ( a ) j (cid:1) – the i -th (resp. j -th) basic root for g (resp. a ); i = 0 , . . . , r ,( j = 0 , . . . , r a ); δ – the imaginary root of g (and of a if any); α ∨ i , (cid:16) α ∨ ( a ) j (cid:17) – the basic coroot for g (resp. a ) , i = 0 , . . . , r ; ( j = 0 , . . . , r a ); ◦ ξ , ◦ ξ ( a ) – the finite (classical) part of the weight ξ ∈ P , (cid:16) resp. ξ ( a ) ∈ P a (cid:17) ; λ = (cid:18) ◦ λ ; k ; n (cid:19) – the decomposition of an affine weight indicating the finitepart ◦ λ , level k and grade n . P (resp. P a ) – the weight lattice; M (resp. M a ) :== ( P ri =1 Z α ∨ i (cid:16) resp. P ri =1 Z α ∨ ( a ) i (cid:17) for untwisted algebras or A (2)2 r , P ri =1 Z α i (cid:0) resp. P ri =1 Z α ( a ) i (cid:1) for A ( u ≥ r and A = A (2)2 r , ) ; E , ( E a )– the group algebra of the group P (resp. P a );Θ λ := e − | λ | k δ P α ∈ M e t α ◦ λ – the classical theta-function;Θ ( a ) ν := e − | ν | k a δ P β ∈ M a e t β ◦ ν ;notice that when the injection is considered the level k a must be correlated withthe corresponding rescaling of roots; A λ := P s ∈ ◦ W ǫ ( s )Θ s ◦ λ resp. A ( a ) ν := P s ∈ ◦ W a ǫ ( s )Θ ( a ) s ◦ ν ;Ψ ( µ ) := e | µ + ρ | k δ − ρ A µ + ρ = e | µ + ρ | k δ − ρ P s ∈ ◦ W ǫ ( s )Θ s ◦ ( µ + ρ ) == P w ∈ W ǫ ( w ) e w ◦ ( µ + ρ ) − ρ – the singular weight element for the g -module L µ ;Ψ ( ν )( a ) := e | ν + ρ a | k a δ − ρ a A ( a ) ν + ρ a = e | ν + ρ a | k a δ − ρ a P s ∈ ◦ W a ǫ ( s )Θ ( a ) s ◦ ( ν + ρ a ) =3 P w ∈ W a ǫ ( w ) e w ◦ ( ν + ρ a ) − ρ a – the corresponding singular weight element for the a -module L ν a ; d Ψ ( µ ) (cid:18) d Ψ ( ν )( a ) (cid:19) – the set of singular weights ξ ∈ P (resp. ∈ P a ) for the module L µ (resp. L ν a ) with the coordinates (cid:18) ◦ ξ, k, n, ǫ ( w ( ξ )) (cid:19) | ξ = w ( ξ ) ◦ ( µ + ρ ) − ρ , (resp. (cid:18) ◦ ξ, k, n, ǫ ( w a ( ξ )) (cid:19) | ξ = w a ( ξ ) ◦ ( ν + ρ a ) − ρ a ), (this set is similar to P ′ nice ( µ ) in [6]) m ( µ ) ξ , (cid:16) m ( ν ) ξ (cid:17) – the multiplicity of the weight ξ ∈ P (resp. ∈ P a ) in themodule L µ , (resp. ξ ∈ L ν a ); ch ( L µ ) (resp. ch ( L ν a ))– the formal character of L µ (resp. L ν a ); ch ( L µ ) = P w ∈ W ǫ ( w ) e w ◦ ( µ + ρ ) − ρ Q α ∈ ∆+ (1 − e − α ) mult( α ) = Ψ ( µ ) Ψ (0) – the Weyl-Kac formula. R := Q α ∈ ∆ + (1 − e − α ) mult( α ) = Ψ (0) (cid:16) resp. R a := Q α ∈ ∆ + a (1 − e − α ) mult a ( α ) = Ψ (0) a (cid:17) – the denominator. For the injection a −→ g consider the reduced module L µ g ↓ a = M ν ∈ P + a b ( µ ) ν L ν a (2)with the branching coefficients b ( µ ) ν . The character reduction π a ◦ (cid:0) ch L µ g (cid:1) = X ν ∈ P + a b ( µ ) ν ch L ν a (3)involves the projection operator π a : P −→ P a .The denominator identity can be applied to redress the relation (3), π a ◦ (cid:0)P w ∈ W ǫ ( w ) e w ◦ ( µ + ρ ) − ρ (cid:1) π a ◦ (cid:16)Q α ∈ ∆ + (1 − e − α ) mult( α ) (cid:17) = X ν ∈ P + a b ( µ ) ν P w ∈ W a ǫ ( w ) e w ◦ ( ν + ρ a ) − ρ a Q β ∈ ∆ + a (1 − e − β ) mult a ( β ) , (4)and to rewrite it as π a ◦ (cid:0)P w ∈ W ǫ ( w ) e w ◦ ( µ + ρ ) − ρ (cid:1) == π a ◦ “Q α ∈ ∆+ ( − e − α ) mult( α ) ”Q β ∈ ∆+ a (1 − e − β ) mult a ( β ) P ν ∈ P + a b ( µ ) ν P w ∈ W a ǫ ( w ) e w ◦ ( ν + ρ a ) − ρ a , (5)4or the trivial g -module L with µ = 0 we have π a ◦ (cid:0)P w ∈ W ǫ ( w ) e w ◦ ρ − ρ (cid:1) π a ◦ (cid:16)Q α ∈ ∆ + (1 − e − α ) mult( α ) (cid:17) = P w ∈ W a ǫ ( w ) e w ◦ ρ a − ρ a Q β ∈ ∆ + a (1 − e − β ) mult a ( β ) ,π a ◦ (cid:16)Q α ∈ ∆ + (1 − e − α ) mult( α ) (cid:17)Q β ∈ ∆ + a (1 − e − β ) mult a ( β ) = π a ◦ (cid:0)P w ∈ W ǫ ( w ) e w ◦ ρ − ρ (cid:1)P w ∈ W a ǫ ( w ) e w ◦ ρ a − ρ a . The relation (5) takes the form π a ◦ (cid:0)P w ∈ W ǫ ( w ) e w ◦ ( µ + ρ ) − ρ (cid:1) P w ∈ W a ǫ ( w ) e w ◦ ρ a − ρ a == π a ◦ (cid:0)P w ∈ W ǫ ( w ) e w ◦ ρ − ρ (cid:1) P ν ∈ P + a b ( µ ) ν P w ∈ W a ǫ ( w ) e w ◦ ( ν + ρ a ) − ρ a . (6)Consider the expression P ν ∈ P + a b ( µ ) ν Ψ ( ν )( a ) and introduce the numbers k ( µ ) λ , called the anomalous branching coefficients , – the multiplicities of submodules L ν a timesthe determinants ǫ ( w ) contained in Ψ ( ν )( a ) . X ν ∈ P + a b ( µ ) ν Ψ ( ν )( a ) = X λ ∈ P a k ( µ ) λ e λ (7)In these terms the expression (6) reads π a ◦ (cid:0)P w ∈ W ǫ ( w ) e w ◦ ( µ + ρ ) − ρ (cid:1) P w ∈ W a ǫ ( w ) e w ◦ ρ a − ρ a == π a ◦ (cid:0)P w ∈ W ǫ ( w ) e w ◦ ρ − ρ (cid:1) P ξ ∈ P a k ( µ ) ξ e ξ , (8)or P w ∈ W,v ∈ W a ǫ ( w ) ǫ ( v ) e π a ◦ ( w ◦ ( µ + ρ ) − ρ )+ v ◦ ρ a − ρ a == P ξ ∈ P a P w ∈ W ǫ ( w ) e π a ◦ ( w ◦ ρ − ρ )+ ξ k ( µ ) ξ . (9)Thus we have proved the statement: Proposition 1
Let L µ be the integrable highest weight module of g , a ⊂ g , h a ⊂ h g , h ∗ a ⊂ h ∗ g and π a – a projection P −→ P a – then for any point ξ ∈ P a the following relation holds: X w ∈ W ǫ ( w ) k ( µ ) ξ − π a ◦ ( w ◦ ρ − ρ ) = X w ∈ W,v ∈ W a ǫ ( w ) ǫ ( v ) δ π a ◦ ( w ◦ ( µ + ρ ) − ρ ) ,ξ + ρ a − v ◦ ρ a . (10)5his can be rearranged to produce a recurrent relation for the anomalousmultiplicities, k ( µ ) ξ = − X w ∈ W \ e ǫ ( w ) k ( µ ) ξ − π a ◦ ( w ◦ ρ − ρ ) + X w ∈ W,v ∈ W a ǫ ( w ) ǫ ( v ) δ π a ◦ ( w ◦ ( µ + ρ ) − ρ ) ,ξ + ρ a − v ◦ ρ a . (11)This formula can be applied to find the branching coefficients b ( µ ) ν due to thefact that being restricted to the fundamental Weyl chamber ( C a ) the anomalousbranching coefficients coincide with the branching coefficients k ( µ ) ξ = b ( µ ) ξ for ξ ∈ C a . The relation (11) contains the standard system of shifts, ξ −→ ξ − π a ◦ ( w ◦ ρ − ρ ), defined by the singular weights of the trivial module, the corre-sponding element of the algebra E being Ψ (0) = e | ρ | k δ − ρ × P s ∈ ◦ W ǫ ( s )Θ s ◦ ( ρ ) . Atthe same time the second term in the r.h.s. contains the summation in both W and W a . Below we demonstrate that this relation can be simplified byintroducing the different system of shifts.Let us return to the relation (5). The conditions a −→ g and h a ⊂ h g guarantee the inclusion ∆ + a ⊂ ∆ + . Thus the first factor in the r. h. s. being anelement of E can be written as π a ◦ (cid:16)Q α ∈ ∆ + (1 − e − α ) mult( α ) (cid:17)Q β ∈ ∆ + a (1 − e − β ) mult a ( β ) = Y α ∈ ( π a ◦ ∆ + ) (cid:0) − e − α (cid:1) mult( α ) − mult a ( α ) == − X γ ∈ P a s ( γ ) e − γ . For the coefficient function s ( γ ) define Φ a ⊂ g ⊂ P a as its carrier:Φ a ⊂ g = { γ ∈ P a | s ( γ ) = 0 } ; (12) Y α ∈ ( π a ◦ ∆ + ) (cid:0) − e − α (cid:1) mult( α ) − mult a ( α ) = − X γ ∈ Φ a ⊂ g s ( γ ) e − γ . (13)When the second factor in the r.h.s. of (5) is also decomposed we obtain therelation X w ∈ W ǫ ( w ) e π a ◦ ( w ◦ ( µ + ρ ) − ρ ) = − X γ ∈ Φ a ⊂ g s ( γ ) e − γ X λ ∈ P a k ( µ ) λ e λ = − X γ ∈ Φ a ⊂ g X λ ∈ P a s ( γ ) k ( µ ) λ e λ − γ and the new property X w ∈ W ǫ ( w ) δ ξ,π a ◦ ( w ◦ ( µ + ρ ) − ρ ) + X γ ∈ Φ a ⊂ g s ( γ ) k ( µ ) ξ + γ = 0; ξ ∈ P a . (14)Thus the following statement is true:6 roposition 2 Let L µ be the integrable highest weight module of g , a ⊂ g , h a ⊂ h g , h ∗ a ⊂ h ∗ g and π a – a projection P −→ P a then for any vector ξ ∈ P a thesum P γ ∈ Φ a ⊂ g − s ( γ ) k ( µ ) ξ + γ is equal to the anomalous multiplicity of the weight ξ in the module π a ◦ L µ g . This property (14) also produces recurrent relations for the anomalous mul-tiplicities. Returning to the relation (13) we see that Φ a ⊂ g contains vectorswith nonnegative grade and is a subset in the carrier of the singular weightselement Ψ ( µ ) = e | µ + ρ | k δ − ρ P s ∈ ◦ W ǫ ( s )Θ s ◦ ( µ + ρ ) . In each grade the set Φ a ⊂ g hasfinite number of vectors [5]. In particular a ⊂ g ) n =0 = ◦ W .
In (Φ a ⊂ g ) n =0 let γ be the lowest vector with respect to the natural orderingin ◦ ∆ a . Decomposing the defining relation (13), Y α ∈ ( π a ◦ ∆ + ) (cid:0) − e − α (cid:1) mult( α ) − mult a ( α ) = − s ( γ ) e − γ − X γ ∈ Φ a ⊂ g \ γ s ( γ ) e − γ , (15)in (14) we obtain k ( µ ) ξ = − s ( γ ) X w ∈ W ǫ ( w ) δ ξ,π a ◦ ( w ◦ ( µ + ρ ) − ρ )+ γ + X γ ∈ Γ a ⊂ g s ( γ + γ ) k ( µ ) ξ + γ (16)where the set Γ a ⊂ g = { ξ − γ | ξ ∈ Φ a ⊂ g } \ { } . (17)was introduced called the fan of the injection a ⊂ g . The equality (16) canbe considered as a recurrent relation for anomalous branching coefficients k ( µ ) ξ .Contrary to the relation (11) here only the summation over W is applied.When r = r a the positive roots ∆ + a can always be chosen so that γ = 0,the relation (15) indicates that s ( γ ) = −
1. Thus in this special caseΓ a ⊂ g = Φ a ⊂ g \ { } , and the recurrent relation acquires the form k ( µ ) ξ = X γ ∈ Γ a ⊂ g s ( γ ) k ( µ ) ξ + γ + X w ∈ W ǫ ( w ) δ ξ,π a ◦ ( w ◦ ( µ + ρ ) − ρ ) . (18)Comments:1. The sets Φ a ⊂ g and Γ a ⊂ g do not depend on the representation L µ ( g ) anddescribe the injection of the subalgebra a into the algebra g .7. Let the set of singular weights of the projected module π a ◦ L µ be con-structed. Then the sets Ψ (0) and Ψ (0)( a ) define the anomalous branchingcoefficients for the reduced module L µ g ↓ a by means of relation (11). Thesame information can be obtained using the fan Γ a ⊂ g via the relations(16) or (18).3. The set of branching coefficients n b ( µ ) ν o is the subset of the anomalousbranching coefficients n k ( µ ) ξ o : n b ( µ ) ν | ν ∈ P + a o = n k ( µ ) ξ | ξ ∈ C a o . Thus the recurrent relations (11),(16) and (18) supply us with the branch-ing coefficients as well.Let us apply the obtained results to the case where a is a Cartan subalgebraof g , a = h g . Then the Weyl group W a and the projector π a are trivial andin the formulas (11) and (18) the anomalous coefficient k ( µ ) ξ is the multiplicityof the only singular weight that contributes to the element Ψ ( ξ ) h g – the highestweight of the h g -submodule. This means that here the number k ( µ ) ξ is alwaysnonnegative and coincides with the multiplicity m ( µ ) ξ of the weight ξ in themodule L µ ( g ). The relations (11) and (18) directly lead to Corollary 3
In the integrable highest weight module L µ ( g ) of an affine Lie al-gebra g the multiplicity m ( µ ) ξ of the weight ξ (considered as a numerical functionon P g ) obeys the relation m ( µ ) ξ = − X w ∈ W \ e ǫ ( w ) m ( µ ) ξ − ( w ◦ ρ − ρ ) + X w ∈ W ǫ ( w ) δ ( w ◦ ( µ + ρ ) − ρ ) ,ξ . (19)In implicit form this relation can be found for affine Lie algebras in [5] (Ch.11,the second formula in Ex. 11.14) and for finite dimensional algebras in [13].Usually the truncated formula (without the second term in the r. h. s.) is pre-sented as the recurrent relation for the multiplicities of weights (see for example[14], Ch. VIII, Sect. 9.3). From our point of view it is highly important to dealwith the recurrent relation in its full form. The reason is that the relation (19)can be applied for any reducible module and is valid in any domain of P , notonly inside the diagram N µ \ µ with the single highest weight µ .This relation gives the possibility to construct recursively the module L µ ( g )provided the elements Ψ ( µ ) and Ψ (0) are known.8 Examples
Example 1
Consider the finite dimensional Lie algebras A ⊂ g . The rootsystem ∆ is generated by the simple roots α (the long) and α (the short) withthe angle π between them. ∆ + = { α , α , α + α , α + 2 α , α + 3 α , α + 3 α } (20)∆ + sl (3) = { α , α + 3 α , α + 3 α } (21) From (12), (13) and (17) we obtain Φ A ⊂ g = { , α , α + α , α + 3 α , α + 3 α , α + 4 α } (22)Γ A ⊂ g = { α , α + α , α + 3 α , α + 3 α , α + 4 α } (23) Consider the adjoint module L α +3 α . Its singular weights are { α + 3 α , α , − α + 2 α , − α , − α − α , − α − α , − α − α , − α − α , − α − α , − α − α , − α − α , α + 2 α } (24) to each of them corresponds the Weyl transformation w ( ψ ) and the value ǫ ( w ) : { ǫ ( w ( ψ )) } = { +1 , − , +1 , +1 , − , − , +1 , − , +1 , − , +1 , − } (25) In the closure of the fundamental chamber C a the relation (18) defines threenonzero branching coefficients b ( µ )2 α +3 α = +1 , b ( µ ) α +2 α = +1 , b ( µ ) α + α = +1 . (26) corresponding to the adjoint and two fundamental submodules of sl (3) in thedecomposition L α +3 α ↓ sl (3) ( Notice that we need the singular weight α + 2 α with s (2 α + 2 α ) = − to be used in the above calculations.) Example 2
Consider the special injection of the algebra B into A . Let α and α be the simple roots of A , ∆ + = { α , α , α + α } (27) The only positive root of B is ∆ + B = (cid:26) β := 12 α (cid:27) (28) From (12) it follows that Φ B ⊂ A = (cid:26) , α , − α , α (cid:27) = { , β, − β, β } (29) For these vectors the function s B ⊂ A has the values s B ⊂ A = {− , +1 , +1 , − } (30)9 he minimal vector γ γ B ⊂ A = − βs B ⊂ A ( − β ) = +1 . (31) The fan is formed by eliminating γ from Φ B ⊂ A and shifting the remainingvectors by − γ : Γ B ⊂ A = { β, β, β } s B ⊂ A ( γ + γ ) = {− , − , + } ; γ ∈ Γ B ⊂ A (32) Consider the module L α + α . Its singular weights are { α + α , − α + α , − α − α , − α , − α − α , α − α } (33) with the values { ǫ ( ξ ) } = { +1 , − , +1 , − , +1 , − } (34) We rewrite their projections to P B in terms of β : {− β, − β, − β, , , +2 β }{ ǫ ( ξ ) } = { +1 , − , +1 , − , +1 , − } (35) In the closure of the fundamental chamber (cid:0) C a (cid:1) the relation (16) defines twononzero branching coefficients b ( µ )2 β = +1 , b ( µ ) β = +1 . (36) corresponding to the submodule of the adjoint subrepresentation and the 5-dimensional spin 2 submodule of B in the reduced module L α + α ↓ B (Notice thatthe singular vector ” ” with s (0) = − has the multiplicity 2.) Example 3
For the affine algebra A (1)2 consider the twisted subalgebra A (2)2 .For the level k sublattice P k introduce the normalized basic vectors { e , e , e } with | e j | j =1 , , = 1 and δ with | δ | = 0 . For A (1)2 we fix the simple roots α = e − e ; α = e − e ; α = δ − e + e ; The positive roots are as follows: ∆ + = α j + lδ ; j = 1 , , l ∈ Z ≥ − α j + pδ ; j = 1 , , p ∈ Z > pδ ; mult ( pδ ) = 2; p ∈ Z > , the classical positive roots being ◦ ∆ + = { α = e − e ; α = e − e ; α = e − e ; } . he fundamental weights ω = 13 (2 e − e − e ) + k ; ω = 13 ( e + e − e ) + k ; ω = k ; and the Weyl vector ρ = ( α + α , , . The Weyl group is generated by the classical reflections s α , s α and (in accord with M = P ri =1 Z α ∨ i for untwisted algebras) the translations t α , t α . Consider the module L ω . Notice that to obtain the branching rules weneed only the projected singular element π a ◦ Ψ ( ω ) of this module and the set Γ A (2)2 ⊂ A (1)2 and do not need any other properties of the module itself. Let usdescribe the element Ψ ( ω ) by the set \ Ψ ( ω ) of singular weights of the module L ω : { ( λ , λ , λ , n, ǫ ( w )) | λ i ∈ Z , n ∈ Z ≤ , ǫ ( w ) = ± } , (the level is always k = 1 and is not indicated). Then for n > − the set \ Ψ ( ω ) contains the following 54 vectors: \ Ψ ( ω ) = { (0 , , , , − , ( − , , , , , (0 , − , , , , ( − , − , , , − , ( − , , , , − , ( − , , , , , (2 , − , , − , − , ( − , , , − , − , ( − , , , − , , ( − , − , , − , , ( − , , − , − , − , (2 , , − , − , , ( − , , , − , , ( − , − , , − , , (0 , − , , − , − , ( − , − , , − , − , ( − , , , − , − , ( − , , , − , − , (3 , − , , − , , (2 , − , , − , , (0 , , − , − , , ( − , , − , − , , (3 , − , − , − , − , (2 , , − , − , − , ( − , − , , − , − , ( − , , , − , − , (0 , − , , − , , ( − , − , , − , , ( − , , , − , , ( − , , , − , , (4 , − , , − , − , (3 , − , , − , − , (0 , , − , − , − , ( − , , − , − , − , (3 , , − , − , , (4 , − , − , − , , ( − , , , − , , ( − , − , , − , − , ( − , , , − , − , (4 , − , , − , , ( − , , − , − , , (4 , , − , − , − , ( − , − , , − , − , ( − , − , , − , , ( − , , , − , − , ( − , , , − , , (3 , − , , − , , (2 , − , , − , − , ( − , , − , − , , ( − , , − , − , − , (6 , − , − , − , − , (6 , − , − , − , , (3 , , − , − , − , (2 , , − , − , , . . . } . he π a projection leads to the following set of vectors: ✻❄✛ − n ◦ λ − ✈ − ✈ +1 ✈ +2 ✈ − ✈ +2 ✈ − ✈ − ✈ +1 ✈ +2 ✈ − ✈ − ✈ +2 ✈ +2 ✈ − ✈ − ✈ +2 ✈ +2 ✈ − ✈ − ✈ +2 ✈ +1 ✈ − ✈ +2 ✈ − ✈ +2 ✈ − ✈ − ✈ +2 ✈ +2 ✈ − n the figure the grade values increase to the right. The vertical line unit is thebasic root vector β of A (2)2 .For the subalgebra A (2)2 the basic roots are β = (1 , ,
0) ; β = δ − β = ( − , ,
1) ; with θ = 2 β and the normalization | β | = 1 , | β | = 4 , ( β , β ) = − . (37) The fundamental weights ω = 1 / β + k = (1 / , , , ω = 2 k = (0 , , and the Weyl vector ρ = 1 / β + 3 k = (1 / , , . The positive roots are ∆ + A (2)2 = (cid:26) β + nδ, ± β + (2 n + 1) δ ; n ∈ Z ≥ − β + mδ ; mδ m ∈ Z > (cid:27) and have the multiplicity one. The Weyl group W A (2)2 is generated by the classicalreflection s β and the translations t ∈ T A (2)2 ⊂ W A (2)2 along the coroot α ∨ =1 / δ − β = ( − , , / : T A (2)2 = (cid:8) t lα ∨ , l ∈ Z (cid:9) .The injection A (2)2 −→ A (1)2 is governed by its classical part – the specialinjection B −→ A . The latter means that when we construct the subset ∆ A (2)2 in the root space of A (1)2 the roots in ∆ A (2)2 are scaled: β = α / , K A (2)2 = 2 K A (1)2 . (38) (So in the modules L µ of the level k the A (2)2 -submodules have the level k .)According to (13) the set Φ A (2)2 ⊂ A (1)2 is defined by the opposite vectors in thenonzero components of the element Q α ∈ „ π A (2)2 ◦ ∆ + « (1 − e − α ) mult( α ) − mult a ( α ) .Taking into account the scaling (38) we obtain the element (cid:0) − e − β (cid:1) (cid:0) − e β (cid:1) × Y κ = ± ∞ Y n =1 (cid:0) − e κ β +2 nδ (cid:1) (cid:0) − e κ β + nδ (cid:1) ∞ Y m =1 (cid:0) − e + mδ (cid:1) generated by the set Φ A (2)2 ⊂ A (1)2 . Here the lowest vector γ is − β and the fan is Γ A (2)2 ⊂ A (1)2 = n ξ + β | ξ ∈ Φ A (2)2 ⊂ A (1)2 o \ { } . he structure of the fan can be illustrated by the following figure presenting thevectors γ ∈ Γ A (2)2 ⊂ A (1)2 with the grade n ≤ and their multiplicities s ( γ ) . ✻❄ ✲ n − β ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ -1 +2 -2 +2 -3 +4 -5 +6 -7 ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ -1 -1 +2 -2 +2 -3 +4 -5 +6 -7 ✈ ✈ ✈ ✈ ✈ ✈ ✈ -1 -1 +2 -2 +2 -3 +4 ✈ -1 ✈ ✈ ✈ ✈ ✈ ✈ ✈ -1 -1 +2 -2 +2 -3 +4 ✈ -1 ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ +1 -1 +1 -1 +2 -3 +3 -4 +6 -7 +8 ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ +1 -1 +1 -1 +2 -3 +3 -4 +6 -7 +8 ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ +1 -1 +1 -1 +2 -3 +3 -4 +6 -7 ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ +1 -1 +1 -1 +2 -3 +3 -4 +6 ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ +1 -1 +1 -1 +2 -3 +3 -4 +6 -7 ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ +1 -1 +1 -1 +2 -3 +3 -4 +6 ✈ ✈ ✈ ✈ ✈ ✈ +1 -1 +1 -1 +2 -3 ✈ ✈ ✈ ✈ ✈ ✈ +1 -1 +1 -1 +2 -3 ✈ ✈ ✈ ✈ +1 -1 +1 -1 ✈ ✈ ✈ ✈ +1 -1 +1 -1 Now we are able to construct the branching rules and explicitly reduce themodule L ω with respect to the subalgebra A (2)2 . Remember that in terms of A (2)2 the diagram N ω is located in the subspace of level k = 2 . Applying theformula (16) in the sublattice with n = 0 we find the first nontrivial value forthe weight with the highest (for n = 0 ) anomalous multiplicity , that is for the ector (1 , , where the branching coefficient is evident, k ( µ )(1 , , = 1 . Imple-menting the recurrence procedure we obtain the set \ Ψ A (2)2 of singular weights: ✻❄✛ C A (2)2 − n β − t ∗ ∗ ∗ ∗ ∗ -4 -3 -2 -1 -1 ∗ ∗ ∗ ∗ ∗ +4 +3 +2 +1 +1 ∗ ∗ ∗ ∗ -3 -2 -1 -1 ∗ ∗ ∗ ∗ +3 +2 +1 +1 ∗ ∗ ∗ ∗ +3 +2 +1 +1 ∗ ∗ ∗ ∗ -3 -2 -1 -1 ∗ ∗ ∗ ∗ ∗ +5 +4 +2 +2 +1 ∗ ∗ ∗ ∗ ∗ -5 -4 -2 -2 -1 ∗ ∗ ∗ ∗ ∗ +5 +4 +2 +2 +1 ∗ ∗ ∗ ∗ ∗ -5 -4 -2 -2 -1 ∗ ∗ ∗ +2 +2 +1 ∗ ∗ ∗ -2 -2 -1 ∗ ∗ -2 -1 ∗ ∗ +2 +1 − t +1 t +2 t − t +2 t − t − t +1 t +2 t − t − t +2 t +2 t − t − t +2 t +2 t − t − t +2 t +1 t − t +2 t − t +2 t − t − t +2 t +2 ∗ -1 ∗ +1 e have performed the branching in terms of the singular weights \ Ψ ( ξ ) A (2)2 ofthe submodules L ξA (2)2 in the decomposition L µA (1)2 ↓ A (2)2 = L ξ ∈ P + A (2)2 b ( µ ) ξ L ξA (2)2 . Nowit is quite easy to extract the branching coefficients b ( µ ) ξ . The intersection \ Ψ A (2)2 \ C A (2)2 gives the set of highest weights and their multiplicities b ( µ ) ξ and the branching is L ω A (1)2 ↓ A (2)2 = L ω A (2)2 (0) ⊕ L ω A (2)2 ( − ⊕ L ω A (2)2 ( − ⊕ L ω A (2)2 ( − ⊕ L ω A (2)2 ( − ⊕ L ω A (2)2 ( − ⊕ L ω A (2)2 ( − ⊕ L ω A (2)2 ( − ⊕ L ω A (2)2 ( − ⊕ L ω A (2)2 ( − ⊕ . . . (Notice that as far as we have shifted the set Φ A (2)2 ⊂ A (1)2 the Weyl chamber C A (2)2 is also shifted correspondingly.) The result can be presented in terms of twobranching functions b ( µ ) I ( q ) = 1 + q + 2 q + 3 q + 4 q + . . .b ( µ ) II ( q ) = q + 2 q + 2 q + 4 q + 5 q + . . . We have demonstrated that the decompositions of integrable highest weightmodules of a simple Lie algebra (classical or affine) with respect to its reductivesubalgebra obey the (infinite) set of algebraic relations. These relations originatefrom the properties of the singular vectors of the module L g considered as thehighest weights of the Verma modules M a . This gives rise to the recursionrelations for the branching coefficients.The properties stated above are encoded in the subset Γ g ⊃ a of the weightlattice P a called the fan of the injection. The fan depends only on the map a −→ g . It describes the injection (whenever it is regular or special) just as theroot system describes the injection h ( g ) −→ g of a Cartan subalgebra. Thus inthe simplest case, when a = h ( g ), the recursion procedure produces the weightdiagram of a module L g .When applied to a reduction of highest weight modules the recursion de-scribed by the fan provides a highly effective tool to obtain the explicit valuesof branching coefficients. 16 eferences [1] E Date, M Jimbo, A Kuniba, T Miwa and M Okado, ”One dimensionalconfiguration sums in vertex models and affine Lie algerba characters”,Preprint RIMS-631, 1988; Lett. Math. Phys. 17, pp. 69-77, 1989[2] L D Faddeev, ”How Bethe ansatz works for integrable model”, in: Quan-tum Symmetries/Symetries Quantique. Proc. 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