The Lawrence-Krammer-Bigelow Representations of the Braid Groups via Quantum SL_2
aa r X i v : . [ m a t h . G T ] M a y The Lawrence-Krammer-BigelowRepresentations of the Braid Groups via U q ( sl ) Craig Jackson and Thomas Kerler
October 10, 2018
Abstract:
We construct representations of the braid groups B n on n strands on free Z [ q ± , s ± ] -modules W n,l using generic Verma modulesfor an integral version of U q ( sl ) . We prove that the W n, are isomor-phic to the faithful Lawrence Krammer Bigelow representations of B n after appropriate identification of parameters of Laurent polynomial ringsby constructing explicit integral bases and isomorphism. We also provethat the B n -representations W n,l are irreducible over the fractional field Q ( q, s ) . † Introduction
In recent years the representation theory of the braid groups B n on n strandshas attracted attention due to two groundbreaking developments. One of them is inthe work of Bigelow and Krammer [1, 13], who managed to resolve the long stand-ing problem of the linearity of the braid groups by showing that a two-parametergeneralization of the classical Burau representation is faithful. The second devel-opment is the emergence of vast families of braid group representations that areconstructed from quantum algebras (see [10] and references therein) and conformalfield theories [15, 12, 20]. Intriguing relationships between these seemingly verydifferent approaches have been discovered and they remain a fascinating area ofstudy.In this article we give an explicit construction and proof of an isomorphismbetween the faithful representation H n, of B n considered by Bigelow and Krammerand the submodule of the R-matrix representations on V ⊗ n for the generic Vermamodule V of the quantum group U q ( sl ) .For the purpose of this article we will consider Krammer’s version H n, , asdefined in [13] and restated in [3], and refer to it as the Lawrence-Krammer-Bigelowrepresentation or LKB representation. It is defined over the ring Z [ q ± , t ± ] of twovariable Laurent polynomials with integral coefficients. The parameters q and t areassociated to Deck transformations of a covering ˜ C n → C n , where C n is the two-point configuration space on a disc with n -punctures. The natural representationof B n on H ( ˜ C n ) as a Z [ q ± , t ± ] is isomorphic to H n, over Q ( q ± , t ± ) , see[1]. While these modules are not isomorphic over Z [ q ± , t ± ] (see [18]), Bigelowconjectures in [3] that the relative homology H ( ˜ C n , ˜ ν ) is isomorphic to H n, over † Z [ q ± , t ± ] , where ˜ ν may be understood as a piece in the boundary of a certaincompactification of ˜ C n .The first obstacle in finding such an isomorphism is that the braid group repre-sentations obtained from quantum groups are originally defined over the complexnumbers rather than integral two-variable Laurent polynomials. To this end wewill define U q ( sl ) as an algebra U over Z [ q, q − ] , and introduce the generic Vermamodule V over L = Z [ q ± , s ± ] , where s may be thought of as the exponential high-est weight s = q λ . The braid group action commutes with the U -action so that thehighest weight spaces W n,l ⊂ V ⊗ n of U , corresponding to weights sq − l = q λ − l ,are again B n -invariant. We prove in Section 3 that the W n,l are free L -modules,and construct explicit bases(1) W n,l = { w ~α | ~α = ( α j , . . . , α n ) with j > and n X i = j α i = l − } such that W n,l is the L -span of W n,l . Specifically, we find Theorem 1.
The highest weight space W n,l ⊂ V ⊗ n is a free module over L = Z [ q ± , s ± ] with explicitly given basis W n,l as in (1). Hence, for each l ≥ we ob-tain a representation of the braid group B n in n -strands given by a homomorphismas follows: (2) ρ n,l : B n −→ GL (cid:0) (cid:0) n + l − l (cid:1) , L (cid:1) ∼ = Aut L ( W n,l ) The identification of the quantum representation on W n, from Theorem 1 withthe LKB representation H n, further requires an identification of parameters whichwe give by the following monomorphism between Laurent polynomials.(3) θ : Z [ q ± , t ± ] −→ Z [ s ± , q ± ] = L : q s t
7→ − q − Consider also the involutive automorphism ι of B n defined on the generators by ι ( σ i ) = σ − i (given by switching all crossing or reflection at the plane of projectionof a braid), and denote by H † n,k the representation given by pre-composing theaction on H n,k with ι . With these conventions the main result of this article,which we will prove in Section 4, can be formulated as follows: Theorem 2.
For every n > there is a isomorphism of B n -representations over L (4) W n, ∼ = −→ H † n, ⊗ θ L which maps the basis W n, to the fork basis from [1] . In [22] Zinno manages to find a different identification of the LKB representa-tion with a quantum algebraic object, namely the quotient of the Birman-Wenzl-Murakami algebra similarly defined over Z [ q ± , t ± ] . This representation can,by [21], be understood as the one arising from the quantum orthogonal groups U ζ ( so ( k + 1)) acting on the n -fold tensor product of the fundamental representa-tion. Since the representation in [21] is irreducible this implies that H n, and hence W n, are irreducible for all n > . ‡ In Section 7 we generalize this result in our case to obtain
Theorem 3.
For all n ≥ and l ≥ the B n -representation W n,l is irreducibleover the fraction field L = Q ( q, s ) . Faithfulness of W n,l for l ≥ is still an open question, as are identifications ofthese representation with geometrically constructed ones analogous to Theorem 2.Obvious candidates for a generalization of Theorem 2 are the B n -representationsconstructed by Lawrence in [16]. The starting point there is again the configurationspace Y n,l of l points in the plane with n holes. The braid group action is thennaturally defined on H n,l = H l ( ˜ Y n,l ) , where ˜ Y n,l is the canonical cover of Y n,l withcovering group Z . The latter makes the representation spaces into Z [ q ± , t ± ] -modules. Conjecture 4.
The spaces H l ( ˜ Y n,l ) are free Z [ q ± , t ± ] -modules which carry an(irreducible) action of B n as defined in [16] . They are isomorphic to the repre-sentations of B n on weight spaces W n,l over L after appropriate identifications ofparameters in the Laurent polynomial rings. The first obvious piece evidence for this conjecture is that it holds for l ≤ .Indeed, for l = 1 both H n, = H ( ˜ Y n, ) and W n, can be readily identified withthe classical Burau representation of B n . For more details see the beginning ofSection 4.It has been observed, both by Lawrence (Section 4 of [16]) and by Bigelow(Section 6 of [3]), that for l = 2 and the parameter specialization t = − q − theLKB-representation has as a factor the Temperley-Lieb representation associatedto the two-row partition [ n − , . In the former case it occurs as a quotient andin the latter setting as a sub-module.In Section 5 we will explain the occurrence of the Temperley-Lieb factor from thepoint of view of quantum- sl representations. Particularly, the respective identifi-cation s = q will correspond to specializing the highest weight of the fundamentalrepresentation of quantum- sl within the Verma module. The exact sequence of B n -modules we establish in (59) reflects the cohomological picture of [16].In Lemma 12 we will also identify the irreducible n -dimensional quotient ofthe specialized LKB-representation by the Temperley-Lieb sub-representation, andprove that the sequence in (59) does not split. Consequently, although the represen-tation becomes reducible in the t = − q − specialization it remains indecomposable.We also discuss in Section 5 the construction of braid elements in the kernel of theTemperley-Lieb representation in order to underscore the loss of information in theparameter specialization. ‡ Note, however, that in the symmetric group specialization with s = 1 and q = 1 these repre-sentation are clearly reducible for all l ≥ Theorem 2 as well as its generalization in Conjecture 4 are inspired by [6] and[23] where quantum- sl actions on the homology of local systems over similar con-figuration spaces are constructed. Acknowledgments:
The second author thanks Giovanni Felder for very usefuldiscussions about [6] which motivated this article. We also thank the anonymousreferees for numerous suggestions that helped to improve the article and led to theaddition of Section 5.2.
From Topological to Integral Braid Group Representation
In this section we review the basic definitions and constructions of quantum sl which lead to the relevant representations of the braid groups. We will start fromthe framework of quasi-triangular topological Hopf algebras due to Drinfeld [5] overrings of power series. An exposition and further development of Drinfeld’s theorycan be found in Kassel’s textbook [11] which we will use as main reference.We start with the definition of the algebra U ~ over a power series ring P [[ ~ ]] where P is some commutative ring containing the rational numbers Q . The inde-terminate is related to h used in [11] by ~ = h . The generators of U ~ are E , F ,and H with relations(5) [ H, E ] = 2 E [ H, F ] = − F [ E, F ] = sinh( ~ H )sinh( ~ ) . The algebra U ~ is given by formal power series P n a n ~ n where each coefficient a n is a finite combination of monomials in the generators E , F , and H over P ⊇ Q . Itis easy to see that the expression for [ E, F ] can indeed be written in this way. Inaddition, the comultiplication on U ~ is defined by(6) ∆( E ) = E ⊗ e ~ H + 1 ⊗ E ∆( F ) = F ⊗ e − ~ H ⊗ F , and ∆( H ) = H ⊗ ⊗ H Formally, the coproduct is a homomorphism ∆ : U ~ → U ~ e ⊗ U ~ , where the tensorcompletion is described in Section XVI.3 of [11]. We introduce the usual set ofnotations for q -numbers, q -factorials, and q -binomial coefficients:(7) q = e ~ [ n ] q = q n − q − n q − q − = sinh ( ~ n )sinh ( ~ ) [ n ] q ! = [ n ] q [ n − q . . . [2] q [1] q (cid:2) nj (cid:3) q = [ n ] q ![ n − j ] q ![ j ] q ! Note that all of these quantities are invertible in P [[ ~ ]] for n = 0 . A universalR-matrix for U ~ is now given as in Theorem XVII.4.2 of [11] by(8) R = e ~ ( H ⊗ H ) · (cid:16) ∞ X n =0 q n ( n − ( q − q − ) n [ n ] q ! E n ⊗ F n (cid:17) ∈ U ~ e ⊗ U ~ Drinfeld’s construction from [5] as described in [11] implies that the R -matrixfrom (8) makes U ~ into a quasi-triangular topological Hopf algebra. Particu-larly, this implies that R obeys the Yang-Baxter relation given as an equationin U ~ e ⊗ U ~ e ⊗ U ~ by(9) R R R = R R R . Moreover, R fulfills the usual commutation relation in U ~ e ⊗ U ~ given by(10) R ∆( x ) = ∆ opp ( x ) R ∀ x ∈ U ~ . In order to construct representations of the braid groups we will need to considerfirst representations of U ~ . Instead of distinguishing many representations by theirhighest weights we consider only one representation and “absorb” the highest weightas a parameter in the underlying coefficient ring as follows.In [11] the coefficient ring was chosen as P = C , yet all calculations and state-ments there clearly also apply for any other choice of P ⊇ Q . For our purposes wewill choose the coefficient ring to be P = Q [ λ ] , that is, the polynomial ring withrational coefficients in one indeterminate λ which may be thought of as a generichighest weight. U ~ is thus an algebra over Q [ λ ][[ ~ ]] – the ring of power series in ~ whose coef-ficients are rational polynomials in λ . In this setting U ~ admits a special highestweight module over the same ring described as follows.Consider the Q [ λ ] -module Q freely generated by an infinite sequence of vectorsdenoted by { v , v , . . . } . The generic Verma module V ~ = Q [[ ~ ]] is then the asso-ciate topologically free module by in the sense of Section XVI.2 of [11]. The actionof U ~ on V ~ is given by H.v j = ( λ − j ) v j E.v j = v j − (11) F.v j = [ j + 1] q · [ λ − j ] q v j +1 Note here that indeed [ λ − j ] q = sinh( ~ ( λ − j ))sinh( ~ ) ∈ Q [ λ ][[ ~ ]] . The module is simi-lar to the standard highest weight module obtained as the induced representationassociated to the one-dimensional representation of the Borel algebra generated by E and H acting on v . It is, however, not equivalent to this module since theelements [ λ − j ] q are not invertible in Q [ λ ][[ ~ ]] . (Ring evaluations λ = m createadditional highest weight vectors for the traditional Verma module, but additionallowest weight vectors for the representation in (11). The modules are equivalent,and irreducible, only for evaluations λ N .)As described in the end of Sections XVI.4 any topological U ~ -module such as V ~ now entails a solution to the Yang-Baxter equation on V ~ e ⊗ V ~ e ⊗ V ~ by (9), whichcommutes with the action of U ~ on the same space by (10). As an endomorphismon V ~ e ⊗ V ~ we define the action of a braid group generator by(12) R : V ~ e ⊗ → V ~ e ⊗ : v e ⊗ w e − ~ λ T ( R . ( v e ⊗ w )) Here R acts as an element of U ~ e ⊗ U ~ on V ~ e ⊗ V ~ , and T denotes the usualtransposition T ( v ⊗ w ) = w ⊗ v . We also multiply the map by the unit e − ~ λ ∈ Q [ λ ][[ ~ ]] which also yields a solution to the Yang Baxter equation since this relationis homogeneous.The braid group B n = h σ . . . σ n − | σ i σ i +1 σ i = σ i +1 σ i σ i +1 , σ i σ j = σ j σ i for | i − j | > i is now represented on V ~ e ⊗ n by the assignment(13) σ i e σ i = 1l e ⊗ i − e ⊗ R e ⊗ e ⊗ n − i − The goal of the following constructions is to identify a sublattice in V ~ e ⊗ n whichis invariant under this action of the braid group B n . This lattice will be a freemodule over a subring L ⊂ Q [ λ ][[ ~ ]] which is characterized as follows. Considerfirst the following ring homomorphism from the two-variable Laurent polynomialsto the power series ring:(14) i ~ : L = Z [ q, q − , s, s − ] −→ Q [ λ ][[ ~ ]] : q e ~ s e ~ λ . It is clear that i ~ is well defined by inspection of the power series expansion in ~ , and that i ~ is a monomorphism since ( ~ , λ ) ( e ~ , e ~ λ ) has dense image in C .We will thus denote the image of i ~ also by L = Z [ q ± , s ± ] ⊂ Q [ λ ][[ ~ ]] with theidentification of parameters as prescribed in (14).In order to find a suitable subalgebra over this ring we define next a set of specialgenerators in U ~ by(15) K = e ~ H , K − = e − ~ H , and F ( n ) = ( q − q − ) n [ n ] q ! F n . The generators F ( n ) are similar to the divided powers introduced by Lusztig in [17]but differ by the additional ( q − q − ) factors. The following relations readily followfrom the ones given in (5): KK − = K − K = 1 , KEK − = q E , KF ( n ) K − = q − n F ( n ) F ( n ) F ( m ) = h n + mn i q F ( n + m ) , and [ E, F ( n +1) ] = F ( n ) ( q − n K − q n K − ) . (16)Let now U ⊂ U ~ be the subalgebra over L generated by the set of elements { K, E, F ( n ) } . As a sublattice U is the free L -module spanned by the PBW basis { K l E m F ( n ) : l ∈ Z , m, n ∈ N ∪ { }} . In fact, U is isomorphic to the algebra over L defined abstractly by generators { K ± , E, F ( n ) } and the relations given in (16). The coproduct and antipode evaluated on the generators of U are readily com-puted: ∆( K ) = K ⊗ K ∆( E ) = E ⊗ K + 1 ⊗ E ∆( F ( n ) ) = n X j =0 q − j ( n − j ) K j − n F ( j ) ⊗ F ( n − j ) S ( K ) = K − S ( E ) = − EK − S ( F ( n ) ) = ( − n q n ( n − K n F ( n ) (17)These formulas immediately imply that the coproduct is in fact a map ∆ : U → U ⊗ U with ⊗ taken over L . Consequently, U is a Hopf subalgebra of U ~ , andthus a Hopf algebra over L by itself.Next, let V ⊂ V ~ be the free L -module generated by the basis vectors { v , v , . . . } .That is, an element in V is given by P j p j v j with p j ∈ L = Z [ q ± , s ± ] and onlyfinitely many p j are non-zero. The actions of the generators of U on the basisvectors v j are easily worked out from the action of U ~ to be the following: K.v j = sq − j v j E.v j = v j − F ( n ) .v j = h n + jj i q n − Y k =0 ( sq − k − j − s − q k + j ) ! v j + n . (18)Observe that all coefficients in these formulae lie in the subring L = Z [ q ± , s ± ] and contain only a finite number (one) of vectors. This immediately implies thefollowing: Lemma 5.
The subspace V ⊂ V ~ is invariant under the action of the subalgebra U ⊂ U ~ . This also implies that the natural actions of U as well as U ⊗ n on V ~ e ⊗ n map therespective subspace V ⊗ n ⊂ V ~ e ⊗ n to itself. The main observation of this section isthat the same is true for the braid group action. Lemma 6.
The map R , as defined in (12), maps the subspace V ⊗ ⊂ V ~ e ⊗ toitself.Proof. We first note that the map R can be written as the composite of three maps(19) R = T ◦ C ◦ P , where T is the usual transposition as in (12). The operator C is given by theaction of the factor e ~ ( H ⊗ H ) from the expression in (8) for the universal R -matrixmultiplied by the extra term e − ~ λ that occurs in (12). Finally, P is given byapplication of the remaining summation in parentheses in (8). We prove that each of these three operators in (19) preserves V ⊗ as a subspace.This is trivially true for T . For the action of C we compute C . ( v j ⊗ v k ) = e − ~ λ e ~ ( H ⊗ H ) v j ⊗ v k = e − ~ λ e ~ ( λ − j )( λ − k ) v j ⊗ v k = e − ~ λ ( j + k )+2 ~ jk v j ⊗ v k = s − ( j + k ) q jk v j ⊗ v k (20)Thus C . ( v j ⊗ v k ) ∈ V ⊗ and the claim follows for C .For P we first rewrite the summation expression for the universal R -matrix in(8) in terms of the generators of U .(21) P = ∞ X n =0 q n ( n − ( q − q − ) n [ n ] q ! E n ⊗ F n = ∞ X n =0 q n ( n − E n ⊗ F ( n ) . The fact that the action of E on V is locally nilpotent together with the observationthat any finite truncation of the summation in (21) yields an element in U ⊗ U implythe claim for P . More specifically, the action of P can be worked out explicitly tobe the following.(22) P . ( v i ⊗ v j ) = i X n =0 q n ( n − h n + jj i q n − Y k =0 ( sq − k − j − s − q k + j ) v i − n ⊗ v j + n . Since the summation is a finite one and all coefficients are in Z [ q ± , s ± ] we cannow infer that P . ( v i ⊗ v j ) ∈ V ⊗ . Consequently, all three operators T , C , and P map the subspace V ⊗ to itself, which proves the lemma. (cid:3) For future use let us also record the explicit formula for the action of R on V ⊗ . R . ( v i ⊗ v j ) = s − ( i + j ) i X n =0 q i − n )( j + n ) q n ( n − h n + jj i q n − Y k =0 ( sq − k − j − s − q k + j ) v j + n ⊗ v i − n . (23)Let us summarize our finding of this section in the following theorem: Theorem 7.
The maps σ i = 1l ⊗ i − ⊗ R ⊗ ⊗ n − i − , with the R as in (23), definea representation of the braid group B n on V ⊗ n , as a free Z [ s ± , q ± ] -module. Themaps σ i also commute with the action of U on V ⊗ n and preserve the natural Z grading.Proof. The fact that the maps σ i preserve V ⊗ n is immediate from Lemma 6. Theyfulfill the braid group relations since they are restrictions of the maps e σ i from (13)which fulfill these relations by construction. Moreover, these maps commute withthe action of U ~ and hence also with the action of U . (cid:3) Integrality of Highest Weight Spaces
The main purpose of this section is to prove the assertion in Theorem 1, namely,that the highest weight spaces are free L -modules.In order to define these highest weight spaces let V n,l = ker( K − s n q − l ) ⊂ V ⊗ n be the weight space corresponding to the weight s n q − l . Recall that x ∈ U actson V ⊗ n by ∆ ( n ) x , where ∆ ( n ) : U → U ⊗ n is defined recursively by ∆ (2) = ∆ and ∆ ( n ) = (∆ ( n − ⊗ . By (17) and (18), V n,l is the L -span of the vectors v α ⊗ · · · ⊗ v α n where α + · · · + α n = l . We now define(24) W n,l = ker( E ) ∩ V n,l . The space W n,l is the so-called highest weight space corresponding to the weight s n q − l . Since the representation of B n on V ⊗ n commutes with the U -action, wesee that both V n,l and W n,l are also B n representations.Let us also define A n,l , B n,l ⊂ V n,l for l ≥ by A n,l = L -span of A n,l with A n,l = { v α ⊗ · · · ⊗ v α n | ∃ k such that α k = 1 and α j = 0 ∀ j < k } (25)and B n,l = L -span of B n,l with B n,l = { v α ⊗ · · · ⊗ v α n | ∃ k such that α k > and α j = 0 ∀ j < k } . (26)We immediately see that V n,l = A n,l ⊕ B n,l . Given a multi-index ~α = ( α j , . . . , α n ) for some j > such that P ni = j α i = l − , we can define an element of A n,l by(27) a ~α = v ⊗ ( j − ⊗ v ⊗ v ~α where v ~α = v α j ⊗ · · · ⊗ v α n ∈ V n − j +1 , l − . Clearly, letting ~α vary among all suchmulti-indices gives the basis A n,l of A n,l . Lemma 8.
For all n ≥ and l ≥ , the map E | B n,l : B n,l → V n,l − is an L -linearisomorphism.Proof. To show that E | B n,l : B n,l → V n,l − is surjective we need to show that forevery v ~α = v α ⊗ · · · ⊗ v α n ∈ V n,l − there is some b ∈ B n,l such that E.b = v ~α .We proceed by induction on j = l − α k , where α k is the first nonzero entry in themulti-index ~α = ( α , . . . , α n ) . The initial case, when j = 1 , occurs when α k = l − and is handled simply by observing that E.v l = v l − . To prove the induction steplet us take v ~α = v ⊗ ( k − ⊗ v α k ⊗ · · · ⊗ v α n ∈ V n,l − such that l − α k = j + 1 . Setting b = v ⊗ ( k − ⊗ v α k +1 ⊗ · · · ⊗ v α n we see that b ∈ B n,l and(28) E.b = ( unit ) v ~α + ( other terms ) where the first nonzero index in each of the other terms is α k + 1 . Hence, the otherterms satisfy the induction hypothesis and so are in the image of E . From this itfollows that v ~α is in the image of E . To show that E | B n,l has no kernel take some = b ∈ B n,l . Then b will havesome minimal term in its expression, namely, some v α ⊗ · · · ⊗ v α k ⊗ · · · ⊗ v α n where α i = 0 for all i < k , α k ≥ , and if v β ⊗ · · · ⊗ v β n is in the expression for b then β i = 0 for all i < k and either β k = 0 or β k ≥ α k . Then comparing theterms in the expression for E.b we see that it is impossible to cancel out the term v α ⊗ · · · ⊗ v α k − ⊗ · · · ⊗ v α n . Hence, E.b = 0 . (cid:3) Since E | B n,l is an isomorphism, we seek a way to parametrize Ker ( E ) by A n,l .This parametrization is accomplished with an L -linear map Φ : V n,l → V n,l ,constructed in such a way that E ◦ Φ vanishes on A n,l (see Lemma 10). Hence, for l ≥ , define Φ on basis elements a ~α = v ⊗ ( j − ⊗ v ⊗ v ~α ∈ A n,l and b ∈ B n,l asfollows: Φ( a ~α ) = l X k =0 b ~α,k v ⊗ ( j − ⊗ v k ⊗ E k − v ~α Φ( b ) = b . (29)The coefficients are given by(30) b ~α,k = ( − k − s ( k − j − n − q ( k − l − k − . Notice that when k = 0 in (29) we have a multiple of the term v ⊗ ( j − ⊗ E − v ~α .By E − v ~α we mean the unique element η ∈ B n − j +1 ,l such that Eη = v ~α . Such anelement η exists and is unique because of Lemma 8. Lemma 9.
We have (Φ − = 0 so that Φ is an automorphism of V n,l .Proof. Clearly, we have (Φ − b ) = 0 for b ∈ B n,l . For k = 1 we have b ~α, = 1 sothat (Φ − a ~α ) = Φ( a ~α ) − a ~α ∈ B n,l and hence (Φ − ( a ~α ) = 0 . The nilpotencyrelation immediately implies that Φ − = 2 − Φ is an inverse. (cid:3) Under the change of basis on V n,l given by Φ the operator E has a simple form. Lemma 10.
For all n ≥ and l ≥ the composite E ◦ Φ vanishes on A n,l and isinjective on B n,l with (31) E ◦ Φ = 0 ⊕ E | B n,l : A n,l ⊕ B n,l −→ V n,l − . This implies that the following is an isomorphism of L -modules: (32) Φ : A n,l ∼ = −→ W n,l . Proof.
The first half of the action in (31) is to show E ◦ Φ is zero on any element a ~α which we verify by explicit computation: E ◦ Φ( a ~α ) = ∆ ( n ) ( E ) . X k ≥ b ~α,k v ⊗ ( j − ⊗ v k ⊗ E k − v ~α = X k ≥ s n − j +1 q − l − k ) b ~α,k v ⊗ ( j − ⊗ v k − ⊗ E k − v ~α + X k ≥ b ~α,k v ⊗ ( j − ⊗ v k ⊗ E k v ~α = X k ≥ (cid:0) s n − j +1 q − l − k − b ~α,k +1 + b ~α,k (cid:1) v ⊗ ( j − ⊗ v k ⊗ E k v ~α = 0 . Here we use that (30) implies the recursion s n − j +1 q − l − k − b ~α,k +1 = − b ~α,k . Now (31) follows from (29), where Φ is defined to be identity on B n,l . By Lemma 8we have that E | B n,l is injective so that ker( E ◦ Φ) ∩ V n,l = A n,l . Since, byLemma 9, Φ is an automorphism of L -modules this implies (32). (cid:3) Let us also describe the case l = 1 more explicitly. A basis of V n, is given by(33) c i = v ⊗ ( i − ⊗ v ⊗ v ⊗ ( n − i )0 for i = 1 , . . . , n The subspaces defined in (25) and (26) are defined slightly different for l = 1 ,namely A n, = L -span of A n, with A n, = { c i | i = 1 , . . . , n − } B n, = L -span of B n, with B n, = { c n } . (34)In this setting we have E − ( v ⊗ m ) = v ⊗ ( m − ⊗ v ∈ B m, . Formula (29) thus yieldsa basis for W n, given by vectors(35) w i = Φ( c i ) = c i − s ( n − i ) c n with i = 1 , . . . , n − . With these conventions it is easy to see that all previous lemmas in this sectionalso apply to the case l = 1 (and trivially so to the case l = 0 ).We are now in a position to prove Theorem 1, namely, that the highest weightspaces are free L -modules. Proof of Theorem 1.
Since (32) is an isomorphism of L -modules and A n,l is clearlya free module, also W n,l has to be a free L -module. The rank is given by thenumber of vectors in the set of spanning vectors given in (25), which is given by (cid:0) n + l − l (cid:1) . (cid:3) Since the generators σ i , as defined in (13), map (by U -equivariance) each W n,l subspace to itself, Lemma 10 implies that the conjugate maps σ Φ j = Φ − ◦ σ i ◦ Φ map A n,l to itself. Thus the representation of B n over L given by σ j | W n,l is equivalentto the representation given by the maps σ Φ j | A n,l . Suppose π A is the projection of V n,l onto A n,l along B n,l . Observe also that Φ − | W n,l = π A | W n,l . This yields the basic but useful formula:(36) σ Φ j | A n,l = π A ◦ σ j ◦ Φ . Implicit to this formula is the method of calculating the action of a braid generator σ Φ j on a particular basis vector: For a basis vector a ~α ∈ A n,l determine Φ( a ~α ) ∈ W n,l by (29). Use (23) and (13) to determine the image σ j (Φ( a ~α )) . Write σ j (Φ( a ~α )) in the standard basis A n,l ∪ B n,l and eliminate the compo-nents of B n,l leaving an L -linear combination of vectors from A n,l .In the following we also consider the action of B n directly on W n,l ⊂ V n,l . Anatural basis is given by W n,l = Φ( A n,l ) = { w ~α = Φ( a ~α ) } . By construction theexplicit action of the braid group generators σ j in this basis is exactly the same asthe action of the σ Φ j in the basis A n,l so that the computations remain the same.4. The Lawrence Krammer Bigelow Representation
Here we prove that the representation of B n on W n, is isomorphic the LKBrepresentation which was recently shown in [1] and [13] to be faithful. As prepa-ration let us show first that the representation W n, is isomorphic to the classical,reduced Burau representation over Z [ t , t − ] .The formula for the R -matrix in (23) implies R . ( v ⊗ v ) = v ⊗ v , R . ( v ⊗ v ) = s − v ⊗ v , and R . ( v ⊗ v ) = s − v ⊗ v + (1 − s − ) v ⊗ v . Applied to the basis { c i } of V n, from (33) this implies the following action of B n on V n, : σ i .c j = c j j = i, i + 1 σ i .c i = s − c i +1 + (1 − s − ) c i σ i .c i +1 = s − c i . (37)Using the rescaled basis { d j = s − j c j | ≤ i < n } and with a substitution ofparameter s − t the action from (37) turns out to yield exactly the unreduced Burau representation f H n, of dimension n as described, for example, in (3-23) of[4]. Thus we have by identification of basis vectors that(38) V n, ∼ = f H n, ⊗ t = s − L . Now, the basis for W n, from (35) may also be rescaled as(39) u j = s j w j = s j d j − s n d n = t − j d j − t − n d n with j = 1 , . . . , n − . Recall that the reduced
Burau representation H n, of dimension ( n − is givenby the kernel of the map f H n, → Z [ t ± ] : d j t j . Clearly, the basis described in(39) is thus a basis also of H n, and we obtain the following relation. Lemma 11. W n, ∼ = H n, ⊗ t = s − L . Let us now turn to the l = 2 case. The basis A n, from (27) is given by elements(40) a i,j = v ⊗ ( i − ⊗ v ⊗ v ⊗ ( j − i − ⊗ v ⊗ v ⊗ ( n − j )0 for ≤ i < j ≤ n . Correspondingly, the B n, consists of the following elements:(41) b k = v ⊗ ( k − ⊗ v ⊗ v ⊗ ( n − k )0 for ≤ k ≤ n . The basis W n, for W n, is given by application of the map in (29) to A n, whichyields the following set of elements:(42) w i,j = Φ( a i,j ) = a i,j − s j − i q − b j − s i − j b i for ≤ i < j ≤ n . The action of the braid group B n on these vectors is now computed using thestep by step procedure following (36). In addition to the formulae in the previousparagraph this also involves calculating expressions for R . ( v i ⊗ v j ) with i + j = 2 .In each of these expressions only the coefficients of the v ⊗ v -term needs to beconsidered since the contributions of the v ⊗ v -terms and v ⊗ v -terms will beprojected out by π A . The relevant relations are thus the following:(43) R . ( v ⊗ v ) = 0 mod h v ⊗ v , v ⊗ v i R . ( v ⊗ v ) = q s − ( v ⊗ v ) mod h v ⊗ v , v ⊗ v i R . ( v ⊗ v ) = q ( s − − s − )( v ⊗ v ) mod h v ⊗ v , v ⊗ v i Applying (43) to the elements in (40) and (41), and combining expressions in (42)we can compute the action of B n on the basis vectors in W n, according to theprocedure given at the end of the previous section. The resulting formulae for thegenerators of B n are listed next where we assume that { i, i + 1 } ∩ { j, k } = ∅ :(44) σ i .w j,k = w j,k σ i .w i +1 ,j = s − w i,j σ i .w j,i +1 = s − w j,i σ i .w i,j = s − w i +1 ,j + (1 − s − ) w i,j − s i − j − (1 − s − ) q w i,i +1 σ i .w i,i +1 = s − q w i,i +1 σ i .w j,i = s − w j,i +1 + (1 − s − ) w j,i − s i − j − (1 − s − ) w i,i +1 . For comparison we consider the explicit Lawrence Krammer Bigelow representa-tion H n, of B n as given in Section 5.2 of [3]. (Note that the representation given in[1] contains a sign error which is corrected in [3]). There the space H n, is describedas the free Z [ t ± , q ± ] -module spanned by basis elements { F i,j : 1 ≤ i < j ≤ n } .From the formulae in [3] the actions of the inverses of the generators of B n on H n, are readily worked out to be as follows:(45) σ − i .F j,k = F j,k σ − i .F i +1 ,j = F i,j σ − i .F j,i +1 = F j,i σ − i .F i,j = q − F i +1 ,j + (1 − q − ) F i,j + t − ( q − − q − ) F i,i +1 σ − i .F i,i +1 = − t − q − F i,i +1 σ − i .F j,i = q − F j,i +1 + (1 − q − ) F j,i − ( q − − q − ) F i,i +1 . Proof of Theorem 2.
Let us define the map F : H n, → W n, by F ( F i,j ) = s i + j w i,j and F ( pv + qw ) = θ ( p ) F ( v ) + θ ( q ) F ( w ) , where p, q ∈ Z [ t ± , q ± ] , v, w ∈ H n, , and θ is the ring homomorphism given in(3). It follows now by direct computation from the equations in (44) and (45) that F σ − i = σ i F ∀ i ∈ { , . . . , n } so that F ι ( β ) = β F ∀ β ∈ B n where ι is the involution described in the introduction. Hence F : H † n, → W n, is B n -equivariant by definition. Since it also maps basis vectors to basis vectors offree modules and θ is a monomorphism, H † n, can be considered a B n -submodule of W n, whose L -span is again W n, . This implies the isomorphism in Theorem 2. (cid:3) The Temperley-Lieb Specialization
In Section 6 of [3] Bigelow considers the parameter specialization qt = − fora version of the LKB-representation, and recovers a sub-module on which the B n -action factors through the respective Temperley-Lieb algebra with representationassociated to a two-row Young tableau. The latter, in turn, are closely related tothe representation theory of quantum- sl via Schur-Weyl duality.In this section we will show how the Temperley-Lieb submodule structure nat-urally follows by extracting finite-dimensional highest or lowest weight modules ofquantum- sl from the generic Verma modules used in Theorem 7 for respective pa-rameter identifications in the ground ring.. The Temperley-Lieb algebra then arisesas the centralizer in the case of the tensor powers of the 2-dimensional fundamentalrepresentation of quantum- sl .The topological and representation theoretic derivations of the same submodulestructure in Theorem 6.1 of [3] and Lemma 12 below, respectively, give thus anotherinsight into the topological content of quantum- sl actions. In addition, we willaddress in Lemma 12 the splitting property and complementary module structure,and conclude with general remarks on the loss of information in the Temperley-Liebreduction.In order to construct finite dimensional quantum- sl representations we fix apositive integer ℓ ∈ N and consider the module with ring quotient into Z [ s ± , q ± ] → Z [ q, q − ] that sends s q ℓ . This yields U -modules over Z [ q, q − ] defined as follows:(46) ˘ V ℓ = V ⊗ s = q ℓ Z [ q, q − ] . Clearly, ˘ V ℓ is still a free Z [ q, q − ] -module with basis { v , v , . . . } . It is immediatefrom (18) that(47) F ( n ) .v j = 0 for j + n > ℓ Suppose I ℓ ⊂ ˘ V ℓ is the free Z [ q, q − ] -submodule spanned by { v , v , . . . , v ℓ } . Itfollows easily from (46) and (18) that I ℓ is also a U -submodule, that is, U . I ℓ = I ℓ .It may be thought of as the irreducible lowest weight module whose lowest weightvector v ℓ has the properties K.v ℓ = q − ℓ v ℓ and F ( n ) .v ℓ = 0 for n ≥ . It alsofollows readily, for example from (23), that(48) R . ( I ℓ ⊗ I ℓ ) ⊆ I ℓ ⊗ I ℓ . Thus we can specialize and restrict the braid group representations from Theorem 7to the following finite rank module over Z [ q, q − ] .(49) I ⊗ nℓ ⊆ ˘ V ⊗ nℓ = V ⊗ n ⊗ s = q ℓ Z [ q, q − ] . These braid group representations are equivalent over Q ( q ) to the ones obtainedfrom the standard R -matrix construction for the ( ℓ +1) -dimensional representationsof quantum- sl (for example, Section VIII.3 in [11]), and also correspond to limitsof solutions to the Yang-Baxter equation given in [14]. As before, the highestweight constructions yield respective sub-representations of the braid groups. Forthe following discussion let us instead consider all relevant modules over the fieldof fractions Q ( q ) : ˘ V n,k,ℓ = V n,k ⊗ s = q ℓ Q ( q ) and ˘ W n,k,ℓ = W n,k ⊗ s = q ℓ Q ( q ) L n,k,ℓ = ˘ W n,k,ℓ ∩ I ⊗ nℓ with I ℓ = I ℓ ⊗ Q ( q ) (50)Of particular interest is the specialization ℓ = 1 , that is, s = q , which correspondsto the fundamental representation of quantum- sl . In this case I = Q ( q ) v ⊕ Q ( q ) v so that the R -matrix acts on a 4-dimensional space spanned by v ⊗ v , v ⊗ v , v ⊗ v , and v ⊗ v . The action is more conveniently described in terms of(51) E := q ( R − ⊗ ) for which we can compute readily from the explicit formula (23) that E ( v ⊗ v ) = 0 = E ( v ⊗ v ) E ( v ⊗ v ) = v ⊗ v − q ( v ⊗ v ) E ( v ⊗ v ) = v ⊗ v − q − ( v ⊗ v ) (52) The formulae in (52) can, in turn, be used to verify the following relations: E = − ( q + q − ) E ( E ⊗ ⊗ E )( E ⊗ E ⊗ ⊗ E )( E ⊗ ⊗ E ) = 1l ⊗ E (53)These relations are easily recognized as those of the Temperley-Lieb algebra A n,q .Over the fraction field Q ( q ) (or over C with q specialized to a value that is nota root of unity) it is well known that the images of A n,q and U in End( I ⊗ n ) via theobvious representations are semisimple and each others commutants, see [8]. Thisimplies the quantum analogue of Schur-Weyl duality, namely that the n -fold tensorproduct is isomorphic over Q ( q ) to(54) I ⊗ n ∼ = ⌊ n ⌋ X k =0 F [ n − k,k ] ⊗ π [ n − k,k ] as a U × A n,q -module. Here F [ n − k,k ] is the representation of highest weight q ( n − k ) ,and π [ n − k,k ] the A n,q representation associated to the partition [ n − k, k ] in analogyto the symmetric group [9]. The dimensions of the factors are the same as in theclassical theory (see for example Section 9 of [7]):(55) dim( π [ n − k,k ] ) = (cid:18) nk (cid:19) − (cid:18) nk − (cid:19) and dim( F [ n − k,k ] ) = n + 1 − k. Suppose v k is the highest weight vector of F [ n − k,k ] . It follows readily from (54) thatthe space of highest weight vectors of weight q ( n − k ) corresponds to h v k i ⊗ π [ n − k,k ] .Thus with definitions from (24) and (50) we obtain the following identification of A n,q -modules:(56) π [ n − k,k ] ∼ = L n,k, ⊆ ˘ W n,k, . In order to apply this to the situation of the LKB representation let us denote by τ the following ring homomorphism(57) τ : Z [ q ± , t ± ] −→ Q ( q ) : q q t
7→ − q − We also introduce an n -dimensional representation C n ( λ ) . To this end, let B n → Z be the Abelian quotient map (with σ i ) and B n → S n : b b the symmetricgroup quotient. Then let C n ( λ ) = h e , . . . , e n i where elements of B n acts as(58) σ j .e j = λe j +1 , σ j .e j +1 = e j , and σ j .e i = e i for i
6∈ { j, j + 1 } . We can now state the following relation of the LKB representation with the Temperley-Lieb representation theory.
Lemma 12.
Reducing the ground ring of the LKB representation by τ to Q ( q ) asin (57) we obtain for n ≥ the following short exact sequence of B n -modules (59) → π [ n − , ֒ → H † n, ⊗ τ Q ( q ) ։ C n ( q − ) → where the Q ( q )[ B n ] -action on the first summand factors through A n,q and the B n -action on the second through the combined quotient Z × S n . For n ≥ the sequencein (59) is not split.Proof. The inclusion given the second map in (59) is the same as the inclusionin (56) via the identifications π [ n − ,
2] ( ) ∼ = L n, , ) = ˘ W n, , ∩ I ⊗ n ֒ → ˘ W n, , ) = W n, ⊗ q = s Q ( q ) ( ) ∼ = ( H † n, ⊗ θ Z [ s ± , q ± ]) ⊗ q = s Q ( q ) = H † n, ⊗ τ Q ( q ) . The cokernel of thisinclusion naturally maps to the following quotient of weight spaces:(60) J : ˘ W n, , ˘ W n, , ∩ I ⊗ n −→ ˘ V n, , ˘ V n, , ∩ I ⊗ n A basis over Q ( q ) of ˘ V n, , is given by the A n, = { a i,j } ≤ i 6∈ { k, k + 1 } . Upon setting λ = q − and after renormalization of the basis(63) e j = − q j b j this is precisely the same action as the one described in (58), and hence proves theexact sequence in (59).In order to show that this sequence is not split for n ≥ it suffices to showthat → ˘ V n, , ∩ I ⊗ n → ˘ V n, , → C n ( q − ) → is not split since any splittinghomomorphism for (59) can be composed with the inclusion ˘ W n, , ֒ → ˘ V n, , .Such a splitting would imply the existence of generators e j ∈ ˘ V n, , for j = 1 , . . . , n with a B n action as prescribed in (58) for λ = q − and with e j ≡ − q j b j mod I ⊗ n . The minimal polynomial of σ i on ˘ V n, , is given by µ ( x ) = ( x − q − )( x − since this is the minimal polynomial of R on h v i ⊗ v j | i + j ≤ i . Thus if we consideractions of ρ i = σ i − q − and ε i = σ i − on ˘ V n, , we have im( ε i ) = ker( ρ i ) and ker( ε i ) = im( ρ i ) over Q ( q ) (only if q − = 1 ). The action of (58) implies that e ∈ ker( ρ ) = im( ε ) . The latter space is spanned by generators r = b − q [2] a , , r = b − a , , as well as r j = a ,j − qa ,j for j = 3 , . . . , n . Since e has to bemapped to b in the quotient it is thus a linear combinations of the form e = − q r + P i ≥ α i r i .Now the relations in (58) for λ = q − also imply that e ∈ ker( ε i ) for i ≥ ,which leads to additional constraints that determine the α i and hence e uniquely:(64) e = − q b + q [2] a , + q [2] n X k ≥ q − k ( a ,k − qa ,k ) . The action of σ on C n ( q − ) now implies that(65) e = q σ .e = − q b + q [2] a , − q [2] n X k ≥ q − k ( a ,k − qa ,k ) . From this it subsequently follows that(66) ρ .e = q − q [2] n X k ≥ q − k ( qa ,k + a ,k ) . However, by (58) we must have ρ .e = 0 which leads to a contradiction for n ≥ and q = 1 . (cid:3) Let us next point out some relations of this lemma to the the topological con-struction of the Temperley-Lieb representation given in Section 6 of [3].The identification q = s was motivated in our case by choosing a fundamentalhighest weight for quantum- sl and translates via (3) directly to the specialization qt = − considered by Bigelow in [3] as well as Lawrence in [16]. In terms of thesevariables and pre-composing representations with the involutive automorphism ι on B n given by ι ( σ i ) = σ − i we find from Lemma 12 the Temperley-Lieb representation π [ n − , as the kernel of the following map of B n -modules.(67) H ,n ⊗ t = − q − Q ( q ) −→ C † n ( q − ) : F i,j e i + q − e j . Here the action of B n on C † n ( q − ) is given explicitly by(68) σ j .e j = e j +1 , σ j .e j +1 = q e j , and σ j .e i = e i for i 6∈ { j, j + 1 } . The Temperley-Lieb representation is found as the kernel of the map (67) also byLawrence (see page 170 in [16]), however, in the dual or cohomological version ofthe Lawrence representation. Consequently, in the homology picture of [16] π [ n − , is described as a quotient by an n -dimensional sub-representation.Bigelow finds in Theorem 6.3 of [3] the module π [ n − , as the image of H ( ˜ Y n, ) ⊗ R in H ( ˜ Y n, , ˜ ν ) ⊗ R by the map induced by the inclusion of pairs, where ˜ ν is a limit of configurations in which one of the points of configuration in ˜ Y n, approachesa puncture or both points approach each other. This suggests that the module C † n ( q − ) is somehow related to the first homology of ˜ ν , although it is not naïvelyobtained from the long exact sequence associated to ( ˜ Y n, , ˜ ν ) .The sequence of B n -representations in Lemma 12 fails to split essentially due tothe failure of I ⊂ ˘ V to split off as a quantum- sl representation. Again it wouldbe interesting to understand this as an obstruction in the context of the topologicalconstructions in [3] and [16] where it contributes to subtle distinctions between var-ious types of homological and cohomological variants of the LKB-representations.More generally, the q = s specialization of the W n,k representations will containthe A n,q -representations π [ n − k,k ] of dimension (cid:0) nk (cid:1) − (cid:0) nk − (cid:1) as summands by the samearguments used for the case k = 2 above. This reproduces the Temperley-Liebrepresentations described at the end of Section 5.2 in [16]. One may expect thatthey are again not direct summands as B n -modules as in the case of k = 2 .The behavior of the representation W n,k is very different if we consider themover Q ( q, s ) (where s − q ℓ is invertible). In particular, we will show in the followingsections that W n,k ⊗ Q ( q, s ) is irreducible for all n and k . This indicates thatthe 2-parameter representation over Z [ q ± , s ± ] contains significantly more infor-mation than the one-parameter specialization discussed above and, especially, theTemperley-Lieb sub-representation.The loss of complexity in the specialization to the Temperley-Lieb representationis exemplified also by the fact that H ,n is faithful, while the representation π [ n − , has a non-trivial kernel. For π [2 , elements in the kernel are specified in Section 3of [2].More complicated elements in the kernel of the Temperley-Lieb representationsare constructed in [19]. In this article Piwocki and Traczyk represent the Temperley-Lieb algebra TL n ≡ A n,q in terms of Kauffman diagrams, introduce ideals I n,i generated by diagrams with more than i caps and cups, and consider the kernels ofthe composite morphism J n,i : B n → TL n → TL n,i = TL n / I n,i .In order to relate this to elements in the kernel of π [ n − , note that the generatorfrom (52) can be written as E = C ◦ C ∨ with maps C : Z [ q ± ] → I ⊗ : 1 v ⊗ v − q ( v ⊗ v ) C ∨ : I ⊗ → Z [ q ± ] : C ∨ ( v , v ) = C ∨ ( v , v ) = 0 , C ∨ ( v , v ) = 1 , and C ∨ ( v , v ) = − q − . (69)The action of TL n on I ⊗ n can now be extended by associating to planar diagramswith a start and b end points a map from I ⊗ a to I ⊗ b by assigning the tensors C and C ∨ to cups and caps in respective tensor positions. Note a diagram in I n, must haveeither at least three cups or three caps. This corresponds to the application, forexample, of three contractions of pairs of tensor factors in I ⊗ n with C ∨ . Restricted to I ⊗ n ∩ ˘ W n, , ∼ = π [ n − , all such contractions are zero for degree reasons. Similarly,insertion of three or more tensors with C cannot have image in π [ n − , .We conclude that the ideal I n, acts trivially on π [ n − , and hence, as a braidgroup representation, the latter factors through J n, : B n → TL n, . In [19] Piwockiand Traczyk find a non-trivial 380-crossing braid in the kernel of J , . Using Theo-rem 1 in [19] this can be used to construct of a 1520-crossing braid β in the kernelof J , and hence also in the kernel of π [15 , .Once it is verified that β = 1 (for example, by evaluating it in the LKB repre-sentation) this proves that π [15 , is not a faithful representation of B . A moreaccessible candidate may be the 11-crossing braid in ker( J , ) which yields a 44-crossing element in ker( J , ) ⊆ ker( π [19 , ) . We will not engage in the remainingcomputations in this article, however, and leave them for future work.6. Structure of the Verma representations V n,l In this section we look more closely at the structure and decomposition of theVerma module representations V n,l . More specifically, we look at eigenspace decom-positions of V n,l under the operators E t F ( t ) . The main purpose of these decompo-sitions is to allow us to prove the irreducibility of the highest weight representationsin the next section.Recall that we have previously defined L = Q ( q, s ) , the fraction field of L . Inwhat follows, we will often speak of V n,l as a vector space over L . Of course whatwe really mean is L ⊗ L V n,l , but we will usually make no distinction. We could, inthe interest of generality, carry out our calculations over a smaller ring, essentiallyinverting only those elements of L that are necessary, but this level of generalityadds little to the discussion at hand. Lemma 13. The weight space V n,l splits as a L [ B n ] -module into a direct sum ofhighest weight spaces: (70) V n,l = l M k =0 F ( k ) W n,l − k ∼ = l M k =0 W n,l − k . Proof. We already know that V n,l = W n,l ⊕ B n,l ∼ = W n,l ⊕ V n,l − as L -modules.So V n,l does decompose into a direct sum of highest weight spaces L lk =0 W n,l − k .This decomposition does not preserve the braid group action, however.To prove the decomposition V n,l = L lk =0 F ( k ) W n,l − k we proceed by induc-tion on l . For l = 0 we have an obvious identity. Suppose now that V n,l = L lk =0 F ( k ) W n,l − k and take v = F ( k ) w ∈ V n,l for some w ∈ W n,l − k . We apply E t F ( t ) to v to obtain E t F ( t ) v = h t + kk i q E t F ( t + k ) w (71) = h t + kk i q F ( k ) (cid:16) t Y j =1 ( q j − k − t K − q k + t − j K − ) (cid:17) w = h t + kk i q µ n,lt,k v where µ n,lt,k ∈ L is the nonzero constant given by(72) µ n,lt,k = t Y j =1 ( s n q − l + k − t + j − s − n q l − k + t − j ) . Thus, in particular EF (1) v = [ k + 1] q µ n,l ,k v . Since the constants [ k + 1] q µ n,l ,k ∈ L aredistinct for distinct k , we see that the decomposition V n,l = L lk =0 F ( k ) W n,l − k is theeigenspace decomposition of the transformation EF (1) . The eigenvalues [ k + 1] q µ n,l ,k are each nonzero, so we see that the map F (1) : V n,l → V n,l +1 is injective. Theimage of this map (over the fraction field) is Im ( F (1) ) = L l +1 k =1 F ( k ) W n,l +1 − k and itis clear from (71) that Im ( F (1) ) ∩ W n,l +1 = 0 . Counting dimensions, we see that V n,l +1 = L l +1 k =0 F ( k ) W n,l +1 − k . (cid:3) Having obtained a decomposition of V n,l , we would now like to obtain a similardecomposition of the highest weight spaces W n,l by restricting the braid action.Consider the B n +1 -action on W n +1 ,l . The map V ⊗ n → V ⊗ ( n +1) defined by v ~α v ⊗ v ~α gives us an inclusion W n,l ֒ → W n +1 ,l . In the standard basis of W n +1 ,l the elements of W n,l correspond to the vectors Φ( a ~α ) where ~α = ( α j , . . . , α n ) for j > (see (27) and (29)). We also have the inclusion B n ֒ → B n +1 that takes σ i ∈ B n to σ i +1 ∈ B n +1 . With this identification the inclusion W n,l ֒ → W n +1 ,l is B n -equivariant. The quotient W n +1 ,l / W n,l is isomorphic to V n,l − as an L [ B n ] -module. The isomorphism is given by(73) Φ( a ~α ) v ~α . Let ψ : W n +1 ,l → V n,l − be the composition of the quotient map W n +1 ,l → W n +1 ,l / W n,l with the isomorphism given in (73). We seek a splitting of ψ . Definition 14. Let c k,j ∈ L be recursively defined by setting c k, = 1 and (74) c k,j +1 = s − n − q l − k + j − − s n +1 q − l + k − j +1 s n q − l − k ) c k,j For each k = 1 , , . . . , l we define a map α k : W n,l − k → V n +1 ,l by (75) α k : w k X j =0 c k,j F ( k − j ) ( v j ⊗ w ) . Let us take w ∈ W n,l − k and compute the action of E on α k ( w ) : Eα k ( w ) = k X j =0 c k,j EF ( k − j ) ( v j ⊗ w )= k X j =0 c k,j (cid:0) F ( k − j ) E + F ( k − j − ( q − k + j K − q k − j − K − ) (cid:1) ( v j ⊗ w )= k X j =1 c k,j s n q − l − k ) F ( k − j ) ( v j − ⊗ w )+ k − X j =0 c k,j (cid:0) s n +1 q − l + k − j +1 − s − n − q l − k + j − (cid:1) F ( k − j − ( v j ⊗ w )= k − X j =0 (cid:0) c k,j +1 s n q − l − k ) + c k,j ( s n +1 q − l + k − j +1 − s − n − q l − k + j − ) (cid:1) F ( k − j − ( v j ⊗ w )= 0 . Thus, α k actually maps W n,l − k into W n +1 ,l ⊂ V n +1 ,l . Notice, in the last equalitywe see the reason behind the definition of the coefficients c k,j in Definition 14.Namely, they have been defined to allow E ◦ α k to vanish on W n,l − k .In the standard basis of W n +1 ,l the element α k w corresponds, modulo W n,l , toa multiple of v ⊗ F ( k − w . To be more precise:(76) α k w = λ k Φ( v ⊗ F ( k − w ) mod W n,l where λ k = s − k q k − ( s − s − ) + c k, s − k q k − = s − n − k q l − k − − s − k q k − . (77)Thus, using the identification V n,l − = L l − k =0 F ( k ) W n,l − − k ∼ = L lk =1 W n,l − k givenby Lemma 13 we see that ψ ◦ α k acts on W n,l − k as multiplication by the nonzeroconstant λ k . Definition 15. Define a map α : V n,l − → W n +1 ,l by (78) α = l M k =1 λ − k α k . The previous discussion yields the following: Lemma 16. The map α defines a B n -equivariant splitting of the map ψ : W n +1 ,l → V n,l − . This gives a decomposition as B n -modules (79) W n +1 ,l = l M k =0 W n,l − k . Irreducibility of the Representations In this last section we wish to prove Theorem 3, namely that the highest weightrepresentations W n,l are irreducible over the fraction field L . The proof makesuse of the decompositions of the previous section and proceeds by induction on n .Notice that, in the general case, if C ⊂ W n,l is a B n -submodule, then as a B n − -module it must decompose into a direct sum of lower degree submodules followingthe decomposition W n,l = l M j =0 W n − ,j . By the induction hypothesis, each of these summands is an irreducible representa-tion of B n − so that C must be a direct sum of some collection of these W n − ,j (formore detail see the proof at the end of the section). In what follows we give explicitcomputations of the action of σ ∈ B n on certain elements of these components.These computations show that we must, in fact, have W n − ,j ⊂ C for all j , thusproving the theorem.To start, let us suppose that v ∈ V n,l , then by Lemma 13 we have v = w + F (1) w + · · · + F ( l ) w l for some w t ∈ W n,l − t . We would like to be able to describethese vectors w t in terms of v .For any t ≤ l we apply E t to v to obtain E t v = E t F ( t ) w t + E t F ( t +1) w t +1 + · · · + E t F ( l ) w l (80) = µ n,l − tt, w t + µ n,l − tt, F (1) w t +1 + · · · + µ n,l − tt,l − t F ( l − t ) w l (81)so that we can solve recursively for w t :(82) w t = 1 µ n,l − tt, (cid:16) E t v − µ n,l − tt, F (1) w t +1 − · · · − µ n,l − tt,l − t F ( l − t ) w l (cid:17) . Proceeding by induction, we see that we must have(83) w t = l − t X i =0 z n,lt,i F ( i ) E t + i v for some coefficients z n,lt,i ∈ L . We see from (82) that z n,lt, = 1 /µ n, − tt, and an inductionargument shows that, in general,(84) z n,lt,i | q =1 = ( − i z n,lt + i, | q =1 = ( − i ( s n − s − n ) − t − i . In particular, the coefficients z n,lt,i are never zero. Example 17. Let us define ν j = v j ⊗ v ⊗ ( n − ∈ V n,j . Then, as above, (85) ν j = w j, + F (1) w j, + · · · + F ( j ) w j,j for some w j,i ∈ W n,j − i . Let us use (83) to define (86) ω j def = w j, = j X i =0 z n,j ,i F ( i ) E i ν j . In other words, ω j is the first term of ν j in the decomposition V n,j = L jk =0 W n,j − k .Since E i ν j = s ( n − i ν j − i , we see from (86) that ω j = 0 for all j . Also, from (85) we see that w j,i = s ( n − i z n,ji, ω j − i . Thus, equation (85) can be written as (87) ν j = j X i =0 s ( n − i z n,ji, F ( i ) ω j − i . Lemma 18. Let us define ν j,k = v j ⊗ F ( k ) v ⊗ ( n − ∈ V n,j + k with j + k ≤ l . Then ν j,k = j + k X i =0 Γ j,k,i F ( i ) ω j + k − i where Γ j,k,i ∈ L such that (1 − s n ) l Γ j,k,i | q =1 is a Laurent polynomial in s withsmallest degree term given by (88) ((cid:0) j + k − ik − i (cid:1) s i ≤ i ≤ k ( − k − i (cid:0) ii − k (cid:1) s i +2( i − k )( n − k < i ≤ j + k. Proof. From the previous example we have ν j = ν j, so that Γ j, ,i = s ( n − i z n,ji, andit is easy to verify the lemma for the case k = 0 .In the general case, we first notice that ν j,k can be expressed as follows:(89) ν j,k = k X r =0 γ j,k,r F ( k − r ) ν j + r where the coefficients γ j,k,r are defined recursively by first setting(90) γ j, ,r = ( , if r = 0 ; , if r = 0 and then defining for k ≥ (91) γ j,k,r = q − j [ k ] q (cid:16) [ k − r ] q sγ j,k − ,r − [ j + 1] q ( s q − j − q j ) γ j +1 ,k − ,r − (cid:17) . The verification of this fact follows by an induction argument from the identity(92) ν j,k = sq − j [ k ] q F (1) ν j,k − − q − j [ j + 1] q [ k ] q ( s q − j − q j ) ν j +1 ,k − . Using the k = 0 case, equation (89) becomes ν j,k = k X r =0 j + r X t =0 γ j,k,r Γ j + r, ,t h k − r + tt i q F ( k − r + t ) ω j + r − t (93) = j + k X i =0 Γ j,k,i F ( i ) ω j + k − i (94)where Γ j,k,i def = X { r,t | k − r + t = i } γ j,k,r Γ j + r, ,t h it i q . Now, we would like to know something about these coefficients Γ j,k,i . At leastwe would like to know that they are nonzero. The relation in (91) along with aneasy induction argument show that γ j,k,r | q =1 is a polynomial in s of the form γ j,k,r | q =1 = (cid:18) j + rr (cid:19) s k − r + ( higher degree terms ) . Also, from (84) we have Γ j + r, ,t | q =1 = s ( n − t ( s n − s − n ) − t . So for each r, t with k − r + t = i we see that setting q = 1 in (1 − s n ) l γ j,k,r Γ j + r, ,t h it i q will indeed give us a Laurent polynomial in s with smallest degree term ( − t (cid:18) it (cid:19)(cid:18) j + rr (cid:19) s i +2 t ( n − . Since the degree of this term is positively related to t , the overall smallest degreeterm of Γ j,k,i will occur when t is as small as possible. For ≤ i ≤ k , the smallest t may be is 0 and in this case we also have r = k − i . For k < i ≤ j + k the smallest t may be is i − k and in this case we have r = 0 . This proves (88) and the lemma. (cid:3) Suppose we take w ∈ W n,l − k to be the basis vector given by w = Φ( v ⊗ u ) (95) = l − k X t =0 b t v t ⊗ E t − u (96)for some u ∈ V n − ,l − k − and where the coefficients b t are given as in (29). We thenhave α k w = k X j =0 l − k X t =0 c k,j b t F ( k − j ) ( v j ⊗ v t ⊗ E t − u ) . Let us (temporarily) set d h,j,t = q h ( h − / s − ( j + t ) q j − h )( t + h ) . We act on α k w by σ ∈ B n +1 and compute(97) σ ( α k w ) = ∞ X h =0 k X j =0 l − k X t =0 c k,j b t d h,j,t F ( k − j ) (cid:16) F ( h ) v t ⊗ E h v j ⊗ E t − u (cid:17) . Recall that ψ : W n +1 ,l → V n,l − is the map that first mods out by W n,l , thenprojects to A n +1 ,l , then removes the leading v component in the tensor product.So applying ψ to σ ( α k w ) , the only terms to survive are those for which h + t ≤ ,and we obtain(98) ψ ( σ ( α k w )) = k X j =0 F ( k − j ) ( η j v j ⊗ u + b κ j v j − ⊗ E − u ) . where the coefficients η j and κ j are calculated to be η j = c k,j s − ( k +1) q k , (99) κ j = s − k q k − ( s − s − ) (cid:16) c k,j + c k,j − sq − j − k (cid:17) . (100)Hence we now have formulae for the σ -action on W n,l in terms of the decom-position (79). We make use of this in the next lemma which will be our main toolin proving Theorem 3. Lemma 19. Let < k < l and consider ω l − k ∈ W n,l − k as given in (86) . Then interms of the decomposition (79) , σ ( α k ω l − k ) has nontrivial components in W n,l − r for all r = 1 , , . . . , k + 1 .Proof. Let us take u = F ( l − k − ( v ⊗ ( n − ) ∈ W n − ,l − k − and define w = l − k X t =1 b t v t ⊗ E t − u + x v ⊗ F (1) u where x = b [ l − k ] q µ n − ,l − k − ,l − k − . Then w ∈ W n,l − k , which follows by the computations found in (71) and the proofof Theorem 1, and comparing the expressions of w and ω l − k in the standard basiswe see that ω l − k is a nonzero multiple of w . Hence if we prove the lemma for w then it will also follow for ω l − k .Notice that the discussion following Lemma 16 will apply formally to w if wemake the substitutions b → x and E − → F (1) . Thus in the present case (98)becomes(101) ψ ( σ ( α k w )) = k X j =0 η j F ( k − j ) ( v j ⊗ u ) + k X j =1 x κ j F ( k − j ) ( v j − ⊗ F (1) u ) . We have F ( k − j ) ( v j − ⊗ F (1) u ) = sq − j − [ k − j + 1] q F ( k − j +1) ( v j − ⊗ u ) − sq − j − ( sq − j s − q j − )[ j ] q F ( k − j ) ( v j ⊗ u ) (102)which allows us to write (101) as(103) ψ ( σ ( α k w )) = k X j =0 Υ j F ( k − j ) ( v j ⊗ u ) where(104) Υ j = η j + x sq − j − (cid:0) κ j +1 q − [ k ] q − κ j ( sq − j − s − q j − )[ j ] q (cid:1) which makes sense for all j = 0 , , . . . , k so long as we define κ = κ k +1 = 0 . We apply Lemma 18 to obtain(105) ψ ( σ ( α k w )) = k X j =0 j + l − k − X i =0 Υ j Γ j,l − k − ,i h k − j + ii i q F ( k − j + i ) ω j + l − k − − i . Thus, if ψ ( σ ( α k w )) = P l − r =0 F ( r ) w r where w r ∈ W n,l − − r , then for all r =0 , , . . . , k we will have(106) w r = r X i =0 Υ k − r + i Γ k − r + i,l − k − ,i h ri i q ω l − − r . Thus, to complete the proof we need only show that the coefficient in (106) isnonzero for all r = 0 , , . . . , k . But we can check that s n +2( k +1) ( s n − − s − n +1 )Υ k − r + i | q =1 is a Laurent polynomial in s having smallest degree term kl − k s − ( i + k − r )(2 n +1) . Compare this to the minimum degree term of Γ k − r − i,l − k − ,i given via (88) and we seethat, after multiplying by an appropriate constant not depending on i , the smallestdegree term of Υ k − r + i Γ k − r + i,l − k − ,i is a strictly decreasing function of i (details areleft to reader). Thus, P ri =0 Υ k − r + i Γ k − r + i,l − k − ,i h ri i q must be nonzero. (cid:3) Before we prove the irreducibility of W n,l we need one last result. Lemma 20. For any w ∈ W n,l there is a polynomial P w ( x ) ∈ L [ x ] such that P w ( σ ) w ∈ L · w max where w max = Φ( v ⊗ v l − ⊗ v ⊗ ( n − ) . Proof. We give the standard basis of W n,l the following “lexicographical” ordering: w ~α < w ~γ if ( | ~α | < | ~γ | or | ~α | = | ~γ | and ~α < ~γ where by | ~α | we mean the number of components in the multi-index (that is, if ~α =( α j , . . . , α n ) then | ~α | = n − j + 1 ) and by ~α < ~γ we mean the usual lexicographicalordering on ordered tuples of integers that have the same number of components.The element w max is the maximal element in this ordering and the braid σ actson w max as the invertible constant ( − l s − l q l ( l − . Now take w ~α < w max . A simple case-by-case analysis shows that there is a polyno-mial P ~α ( x ) of degree at most 2 such that P ~α ( σ ) w ~α either belongs to L · w max or isa sum of strictly higher order terms.For instance, suppose | ~α | = n − and consider w = Φ( v ⊗ v k ⊗ u ) for some k > and some u ∈ V n − ,l − k − . Then we compute σ w = X i ≥ Φ( v ⊗ v k + i ⊗ u i ) for some u i ∈ V n − ,l − k − i − . When i = 0 we have u = z u for a nonzero constant z ∈ L so that we will have ( σ − z ) w = X i ≥ Φ( v ⊗ v k + i ⊗ u i ) . Similar arguments apply in the remaining cases.Thus, we can multiply any basis vector by a polynomial in σ and get a sum ofhigher order terms. The result now follows by induction and by commutativity ofpolynomials in σ . (cid:3) We now come to the main result of this section. Theorem 21. The B n representations W n,l are irreducible over the fraction field L .Proof. We proceed by induction on n . The base case when n = 2 is trivial since thedimension of W ,l is 1 for all l ≥ . Suppose now the theorem is true for all k < n .Suppose C ⊂ W n,l is a B n -submodule. As a B n − -module we have seen that W n,l decomposes into a direct sum W n,l = l M j =0 W n − ,j . By the induction hypothesis, each of these summands is an irreducible representa-tion of B n − and since the dimensions are different they are inequivalent. 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