Fusion procedure for Degenerate cyclotomic Hecke algebras
aa r X i v : . [ m a t h . R T ] D ec FUSION PROCEDURE FOR DEGENERATE CYCLOTOMIC HECKE ALGEBRAS
DEKE ZHAO AND YANBO LI
Abstract.
The primitive idempotents of the generic degenerate cycloctomic Hecke algebras are derivedby consecutive evaluations of a certain rational function. This rational function depends only on theSpecht modules and the normalization factors are the weights of the Brundan-Kleshchev trace. Introduction
In [ ] Jucys observed that the primitive idempotents of the symmetric groups canbe obtained by taking certain limit values of a rational function in several variables. Asimilar construction, now commonly referred to as the fusion procedure , was developed byCherednik [ ], while complete proofs relying on q -version of the Young symmetrizers weregiven by Nazarov [ ]. This method has already been used by Nazarov (and Tarasov)[ – ] and Grime [ ]. A simple version of the fusion procedure for the symmetric groupwas given by Molev in [ ]. Here the idempotents are obtained by consecutive evaluationsof a certain rational function. This version of the fusion procedure relies on the existenceof a maximal commutative subalgebra generated by the Jucys-Murphy elements and wasdeveloped for various algebras and groups (see e.g. [ – , , ]).Let m, n be positive integers and let W m,n be the complex reflection group of type G ( m, , n ). By [ ], W m,n has a presentation with generators τ, s , . . . , s n − where thedefining relations are τ m = 1 , s = · · · = s n − = 1 and the homogeneous relations τ s τ s = s ts τ, τ s i = s i t for i ≥ ,s i s i +1 s i = s i +1 s i s i +1 for i ≥ s i s j = s j s i , if | i − j | > . It is well-known that W m,n ∼ = ( Z /m Z ) n ⋊ S n , where S n is the symmetric group of degree n (generated by s , . . . , s n − ). The degenerate cyclotomic Hecke algebra H m,n ( Q ) withparameters Q is a natural deformation of the group algebra KW m,n over a filed K (seeDefinition 2.2 below).Let P m,n be the set of all m -multipartition of n . For λ ∈ P m,n , let S λ be the Spechtmodule H m,n ( Q ) of type λ . Then { S λ | λ ∈ P m,n } is the set of irreducible representationsof the generic cyclotomic Hecke algebras H m,n ( Q ). Let E t be the primitive idempotentof H m,n ( Q ) corresponding to the standard λ -tableau t . A complete system of pairwiseorthogonal primitive idempotents of H m,n ( Q ) is parameterized by the set of standardtableaux of the m -multipartitions of n .This paper is concerned with the fusion procedure for H m,n ( Q ). As in [ , ], we usethe Jucys-Murphy elements of H m,n ( Q ). The main result of the paper is the following: Theorem.
Let λ be an m -multipartition of n and t a standard λ -tableau. Then theprimitive idempotent E t of H m,n ( Q ) corresponding to t can be obtained by the followingconsecutive evaluations E t = Θ λ ( Q )Φ( z , · · · , z n ) (cid:12) (cid:12)(cid:12)(cid:12) z =res t (1) · · · (cid:12) (cid:12)(cid:12)(cid:12) z =res t ( n − (cid:12) (cid:12)(cid:12)(cid:12) z =res t ( n ) , where Φ( z , · · · , z n ) is a rational function in several variables with values in H m,n ( Q ) and Θ λ ( Q ) is a rational functions in variables Q . Remarkably, the coefficient Θ λ ( Q ) appearing in Theorem depends only on the m -multipartition λ . In fact, the coefficient Θ λ ( Q ) is the weight of the Brundan-Kleshchev Mathematics Subject Classification.
Primary 20C99, 16G99; Secondary 05A99, 20C15.
Key words and phrases.
Complex reflection group; Degenerate cyclotomic Hecke algebra; Jucys-Murphy elements; Schurelements; Fusion procedure.Zhao is supported by the National Natural Science Foundation of China (Grant No. 11101037).Li is supported by the Natural Science Foundation of Hebei Province, China (A2013501055). race on H m,n ( Q ) corresponding to the Specht modules S λ , that is, Θ λ ( Q ) is the inverseof the Schur element s λ ( Q ) of H m,n ( Q ) (see [ , Theorem 4.2]).In additional, the degenerate cyclotomic Hecke algebra is a cellular algebra with Jucys-Murphy element (see [ , ]). It may be surprising that the cellularity is not used inthe construction of the rational function Φ( z , · · · , z n ) appearing in Theorem. It seemsreasonable that we may develop an abstract framework for the fusion procedure for thosealgebras equipped with a family of inductively defined Jucys-Murphy elements satisfyingsome certain conditions (cf. [ ], [ ]). We hope to return this issue in future work.This paper is organized as follows. Section 2 contains definitions and notations aboutthe multipartitions, the degenerate cyclotomic Hecke algebra, the Jucys-Murphy elementsand the Baxterized elements and gives facts. The rational function Θ λ ( Q ) is introducedand investigated in Section 3, in particular, a combinatorial formulation of Θ λ ( Q ) ispresented. Finally, we define the rational function Φ( z , · · · , z n ) and prove the maintheorem in the last section. Acknowledgements.
The authors are grateful to Professor Chengming Bai for his hos-pitality during their visits to the Chern Institute of Mathematics (CIM) in Nankai Uni-versity. Part of this work was carried out while the first author was visiting CIM andthe Kavli Institute for Theoretical Physics China (KITPC) at the Chinese Academy ofSciences in Beijing. 2.
Preliminaries partition λ = ( λ , λ , · · · ) is a decreasing sequence of non-negative integers con-taining only finitely many non-zero terms. We define the length of λ to be the smallestinteger ℓ ( λ ) such that λ i = 0 for all i > ℓ ( λ ) and set | λ | := P i ≥ λ i . If | λ | = n we saythat λ is a partition of n .Let m, n be positive integers. An m -multipartition of n is an ordered m -tuple λ =( λ ; . . . ; λ m ) of partitions λ i such that n = P mi =1 | λ i | . We denote by P m,n the set of all m -multipartitions of n .The diagram of an m -multipartition λ is the set[ λ ] := { ( i, j, c ) ∈ Z > × Z > × m | ≤ j ≤ λ ci } , where m = { , . . . , m } . The elements of [ λ ] are the nodes of λ ; more generally, a node is any element of Z > × Z > × m . We may and will identity an m -multipartition with its diagram. A node α / ∈ λ is addable for λ if λ ∪ { α } is the diagram of an m -multipartition and a node β of λ is removable for λ if λ \ { β } is the diagram of an m -multipartition. We denote by A ( λ )(resp. R ( λ )) the set of addable (resp. removable) nodes of λ .A λ -tableau is a bijection t : [ λ ] → { , , . . . , n } and write Shape( t ) = λ if t is a λ -tableau. We may and will identify a λ -tableau t with an m -tuple of tableaux t =( t ; . . . ; t m ), where the c -component t c is a λ c -tableau for 1 ≤ c ≤ m . A λ -tableau is standard if in each component the entries increase along the rows and down the columns.We denote by Std( λ ) the set of all standard λ -tableaux.2.2. Definition.
Let K be a field and Q = { q , . . . , q m } ⊂ K . The degenerate cy-clotomic Hecke algebra is the unital associative K -algebra H := H m,n ( Q ) generated by t, t , . . . , t n − and subjected to relations(i) ( t − q ) . . . ( t − q m ) = 0,(ii) t ( t tt + t ) = ( t tt + t ) t and tt i = t i t for i ≥ t i = 1 for 1 ≤ i < n ,(iv) t i t i +1 t i = t i +1 t i t i +1 for 1 ≤ i < n − t i t j = t j t i for | i − j | > Jucys-Murphy elements of the algebra H are define inductively as following:(2.3) J := t and J i +1 := t i J i t i + t i , i = 1 , · · · , n − . hen t i J j = J j t i if i = j − , j and J j J k = J k J j if 1 ≤ j, k ≤ n . Furthermore, theJucys-Murphy elements J i ( i = 1 , · · · , n ) generate a maximal commutative subalgebraof H , furthermore, the center Z ( H ) of H is the algebras generated by the symmetricpolynomials of the Jucys-Murphy elements J , · · · , J n (ref. [ ]).For any distinct i, j = 1 , · · · , n −
1, we define the
Baxterized elements with spectralparameters x, y :(2.4) t ij ( x, y ) := 1 x − y + t ij , Here and in what follows t ij denotes the element t ij := t i t i +1 · · · t j for all | i − j | > K S n and satisfy the followingrelations: t i ( x, y ) t i +1 ( x, z ) t i ( y, z ) = t i +1 ( y, z ) t i ( x, z ) t i +1 ( x, y ) ,t ij ( x, y ) t ji ( y, x ) = 1 − ( x − y ) − . From now on we let f ( z ) = ( z − q )( z − q ) · · · ( z − q m ) and define the following rationalfunction with values in H :(2.5) t ( z ) := f ( z ) z − t . Then t ( z ) is a polynomial function in z . Moreover the elements t ( z ) and t ( x, y ) satisfythe following reflection equation with parameters x, y :(2.6) t ( x ) t ( x, y ) t ( y ) t + t ( x ) t ( x, y ) = t ( x, y ) t ( x ) + t t ( y ) t ( x, y ) t ( x ) . Remark.
The facts that Baxterized elements satisfy the reflection equation (2.6) willnot be used in this paper. It seems likely that we may use this fact to construct theBethe subalgebra of degenerate cyclotomic Hecke algebras (cf. [ ]). We hope to returnthis issue in the future.We shall work with a generic degenerated cyclotomic Hecke algebra, that is, q , · · · , q m are indeterminates and the algebra H over a certain localization of the ring K [ q , · · · , q m ],or in a specialization such that the following separation condition is satisfied: P H ( Q ) = n ! Y ≤ i Assume that H is generic. Let λ and µ be m -multipartitions of n andsuppose that s ∈ Std( λ ) and t ∈ Std( µ ) . (i) s = t (and λ = µ ) if and only if res s ( k ) = res t ( k ) for k = 1 , . . . , n . (ii) Suppose that λ = µ and there exists an i such that res s ( k ) = res t ( k ) for all k = i, i + 1 . Then either s = t or s = t ( i, i + 1) . conjugate of a partition λ is the partition ˆ λ = (ˆ λ ≥ ˆ λ ≥ . . . ) whose diagramis the transpose of the diagram of λ , i.e. ˆ λ i is the number of nodes in the i -th columnof the diagram of λ . Hence ˆ λ = ℓ ( λ ) and a node ( i, j ) of λ is removable if and only if j = λ i and i = ˆ λ j . ecall that the ( i, j )-th hook in λ is the collection of nodes to the right of and belowthe node ( i, j ), including the node ( i, j ) itself, and that the ( i, j )-th hook length of λ is h λi,j = λ i − i + ˆ λ j − j + 1 . Now let λ and µ be partitions. If ( i, j ) is a node of λ then the generalized hook length of the node ( i, j ) with respect to ( λ, µ ) is h λ,µi,j = λ i − i + ˆ µ j − j + 1 . Observe that if λ = µ then h λ,µi,j = h λi,j .Let λ = ( λ ; · · · ; λ m ) be an m -multipartition of n . We introduce the following functionin variables Q :(3.4) Θ λ ( Q ) = Y ≤ s ≤ m Y ( i,j ) ∈ λ s Y ≤ t ≤ m h λ s ,λ t i,j + q s − q t . Thanks to Lemma 3.2, Θ λ ( Q ) is a rational function and can be reformulated as following:Θ λ ( Q ) = Y ≤ s ≤ m Y ( i,j ) ∈ λ s (cid:0) h λ s i,j Y ≤ t ≤ m & t = s h λ s ,λ t i,j + q s − q t (cid:1) . Remark. The rational function Θ λ ( Q ) is the weight of the Brundan-Kleshchev traceon H m,n ( Q ) corresponding to the Specht module S λ , that is, it is the inverse of the Schurelement s λ ( Q ) of H m,n ( Q ) (ref. [ , Theorem 5.5] and [ , Theorem 4.2]).3.6. Lemma. Let λ and µ be partitions. Assume that α = ( ı, ) is a removable node of λ and let ν = λ −{ α } . We define the following rational function in variables x, y : Θ λ,µ ( x, y ) := Y ( i,j ) ∈ µ h λ,µi,j + x − y Y ( i,j ) ∈ λ h µ,λi,j + y − x . Then (3.7) Θ λ,µ ( x, y )Θ ν,µ ( x, y ) = Y β ∈ R ( µ ) (cid:0) res( β ) − res( α ) + y − x (cid:1) Y γ ∈ A ( µ ) (cid:0) res( γ ) − res( α ) + y − x (cid:1) − . Proof. Let z := x − y and suppose that µ = (cid:0) µ = · · · = µ i > µ i +1 = · · · = µ i > · · · · · · > µ i p − +1 = · · · = µ i p > (cid:1) . We set i = 0, µ i p +1 = 0 and let z = q c − q t . Then R ( µ ) = { ( i k , µ i k ) | k = 1 , · · · , p } ; A ( µ ) = (cid:8) ( i k + 1 , µ i k +1 + 1) | k = 0 , · · · , p (cid:9) . Therefore, the right hand side of (3.7) isRHS = Y β ∈ R ( µ ) (cid:0) res( β ) − res( α ) − z (cid:1) Y γ ∈ A ( µ ) (cid:0) res( γ ) − res( α ) − z (cid:1) − = 1 µ + ı − − z p Y k =1 µ i k − i k + ı − − zµ i k +1 − i k + ı − − z . Suppose that i q < ı ≤ i q +1 for some 0 ≤ q ≤ p + 1 where i p +1 = + ∞ . Since α = ( ı, )is a removable node of λ and ν = λ − { α } , we yield that λ i = ν i for i = ı ; ˆ λ j = ˆ ν j for j = ; λ ı = ν ı + 1 = ; ˆ λ = ˆ ν + 1 = ı. ThereforeΘ λ,µ ( x, y )Θ ν,µ ( x, y ) = 1 h µ,λı, − z Y ( i,j ) ∈ µ h ν,µi,j + zh λ,µi,j + z Y ( i,j ) ∈ ν h µ,νi,j − zh µ,λi,j − z µ ı − − z + 1 Y ( i,j ) ∈ µ ν i + ˆ µ j − i − j + 1 + zλ i + ˆ µ j − i − j + 1 + z Y ( i,j ) ∈ ν µ i + ˆ ν j − i − j + 1 − zµ i + ˆ λ j − i − j + 1 − z = 1 µ ı − − z + 1 Y ( ı,j ) ∈ µ ˆ µ j − j + − ı + z ˆ µ j − j + − ı + 1 + z Y ( i, ) ∈ ν µ i − i + ı − − zµ i − i + ı − + 1 − z = 1 µ ı − − z + 1 µ iq +1 Y j =1 ˆ µ j − j + − ı + z ˆ µ j − j + − ı + 1 + z ı − Y i =1 µ i − i + ı − − zµ i − i + ı − + 1 − z = 1 µ ı − − z + 1 Y k = p, ··· ,q +1 µ ik Y j = µ ik +1+1 ˆ µ j − j + − ı + z ˆ µ j − j + − ı + 1 + z × (cid:0) q Y k =1 i k Y i = i k − +1 µ i − i + ı − − zµ i − i + ı − + 1 − z (cid:1) ı − Y i = i q +1 µ i − i + ı − − zµ i − i + ı − + 1 − z = 1 µ i q +1 − i q + ı − − z p Y k = q +1 i k − µ i k + − ı + zi k − µ i k +1 + − ı + z q Y k =1 µ i k − i k + ı − − zµ i k − i k − + ı − − z = p Y k = q +1 µ i k − i k + ı − − zµ i k − i k − + ı − − z q Y k =1 µ i k − i k + ı − − zµ i k − i k − + ı − − z = 1 µ + ı − + z p Y k =1 µ i k − i k + ı − − zµ i k +1 − i k + ı − − z . As a consequence, we have completed the proof. (cid:3) The following fact will be used in the sequence.3.8. Lemma. Assume that λ is a partition and that α is a removable node of λ . Let µ be the subpartition of λ by removing the node α . Then Q ( i,j ) ∈ µ h µi,j Q ( i,j ) ∈ λ h λi,j = Y β ∈ R ( µ ) (cid:0) res( β ) − res( α ) (cid:1) Y α = γ ∈ A ( µ ) (cid:0) res( γ ) − res( α ) (cid:1) − = Y β ∈ R ( µ ) (cid:0) res( α ) − res( β ) (cid:1) Y α = γ ∈ A ( µ ) (cid:0) res( α ) − res( γ ) (cid:1) − . Proof. Suppose that α = ( ı, ) and that µ = (cid:0) µ = · · · = µ i > µ i +1 = · · · = µ i > · · · · · · > µ i p − +1 = · · · = µ i p > (cid:1) . We set i = 0, µ i p +1 = 0. Therefore we have R ( µ ) = { ( i k , µ i k ) | k = 1 , · · · , p } ; A ( µ ) = (cid:8) ( i k + 1 , µ i k +1 + 1) | k = 0 , · · · , p (cid:9) . Since α is removable, ı = i q + 1 and = µ i q +1 + 1 for some 0 ≤ q ≤ p + 1. Furthermore,we yield that λ i = µ i for i = ı ; ˆ λ j = ˆ µ j for j = ; λ ı = µ ı + 1 = ; ˆ λ = ˆ µ + 1 = ı. As a consequence, Q ( i,j ) ∈ µ h µi,j Q ( i,j ) ∈ λ h λi,j = Q ( i,j ) ∈ µ µ i + ˆ µ j − i − j + 1 Q ( i,j ) ∈ λ λ i + ˆ λ j − i − j + 1 Y ( i, ) ∈ µ µ i − i + ı − µ i − i + ı − + 1 Y ( ı,j ) ∈ µ ˆ µ j − j + − ı ˆ µ j − j + − ı + 1= i q Y i =1 µ i − i + ı − µ i − i + ı − + 1 µ iq +1 Y j =1 ˆ µ j − j + − ı ˆ µ j − j + − ı + 1= (cid:18) q Y k =1 i k Y i = i k − +1 µ i − i + ı − µ i − i + ı − + 1 (cid:19)(cid:18) p Y k = q +1 µ ik Y j = µ ik +1+1 ˆ µ j − j + − ı ˆ µ j − j + − ı +1 (cid:19) = q Y k =1 µ i k − i k + ı − µ i k − i k − + ı − p Y k = q +1 i k − µ i k + − ıi k − µ i k +1 + − ı = q Y k =1 µ i k − i k + ı − µ i k − i k − + ı − p Y k = q +1 µ i k − i k + ı − µ i k +1 − i k + ı − = Y β ∈ R ( µ ) (cid:0) res( β ) − res( α ) (cid:1) Y α = γ ∈ A ( µ ) (cid:0) res( γ ) − res( α ) (cid:1) − = Y β ∈ R ( µ ) (cid:0) res( α ) − res( β ) (cid:1) Y α = γ ∈ A ( µ ) (cid:0) res( α ) − res( γ ) (cid:1) − . As a consequence, we have completed the proof. (cid:3) The following result gives another combinatorial reformulation for the rational functionΘ λ ( Q ) defined in § Proposition. Assume that λ is an m -multipartition of n and that t is a standard λ -tableau with the node α containing the number n . Let µ be the shape of the subtableauof t by removing the node α . Then Θ λ ( Q )Θ µ ( Q ) − = Y β ∈ R ( µ ) (cid:0) res( β ) − res t ( n ) (cid:1) Y α = γ ∈ A ( µ ) (cid:0) res( γ ) − res t ( n ) (cid:1) − . Proof. Suppose that λ = ( λ ; . . . ; λ m ) and α = ( ı, , c ). Then α is removable and µ = ( µ ; . . . ; µ m ) = ( λ ; . . . ; λ c − ; µ c ; λ c +1 ; . . . ; λ m ) , where µ c = (cid:0) λ c , . . . , λ cı − , λ cı − , λ cı +1 , . . . , λ cℓ ( λ c ) (cid:1) . Observe that A ( µ t ) = A ( λ t ) and R ( µ t ) = R ( λ t ) for 1 ≤ t = c ≤ m .Applying the equality (3.4), we obtain thatΘ λ ( Q )Θ µ ( Q ) = Y ≤ s ≤ m Y ( i,j ) ∈ µ s Y ≤ t ≤ m (cid:0) h µ s ,µ t i,j + q s − q t (cid:1)Y ≤ s ≤ m Y ( i,j ) ∈ λ s Y ≤ t ≤ m (cid:0) h λ s ,λ t i,j + q s − q t (cid:1) = Q ( i,j ) ∈ µ c h µ c i,j Q ( i,j ) ∈ λ c h λ c i,j Y ≤ t ≤ m & t = c Θ λ c ,µ t ( q c , q t )Θ µ c ,µ t ( q c , q t )= Q β ∈ R ( µ c ) (cid:0) res( β ) − res( α ) (cid:1)Q α = γ ∈ A ( µ c ) (cid:0) res( β ) − res( α ) (cid:1) Y ≤ t ≤ m & t = c Q β ∈ R ( µ t ) (cid:0) res( β ) − res( α ) (cid:1)Q γ ∈ A ( µ t ) (cid:0) res( β ) − res( α ) (cid:1) = Y β ∈ R ( µ ) (cid:0) res( β ) − res( α ) (cid:1) Y α = γ ∈ A ( µ ) (cid:0) res( γ ) − res( α ) (cid:1) − , here the third equality follows by applying Lemmas 3.6 and 3.8. We have completedthe proof. (cid:3) Fusion formulae for primitive idempotents From now on, the following notations will be used throughout unless otherwise stated.4.1. Notations. λ = ( λ ; · · · ; λ m ) is an m -multipartition of n and t is a standard λ -tableau with the number n appearing in the node α = ( ı, , c ). We let u be the subtableauof t by removing the node α and let µ = Shape( u ). Denote by v the subtableau of u which contains the numbers 1 , · · · , n − ν = Shape( v ).Let λ be an m -multipartition of n and let S λ be the Specht module corresponding to λ . Then S λ admits the following decomposition, as a vector space, S λ = M t ∈ Std( λ ) v t , which is equipped with a Young seminorm form and the Jucys-Murphy elements actdiagonally in this basis.If H is generic then { S λ | λ ∈ P m,n } is a complete set of pairwise non-isomorphicirreducible H -modules. For a standard λ -tableau t , we denote by E t the correspondingprimitive idempotent of H . For all i = 1 , · · · , n , we have(4.2) J i E t = E t J i = res t ( i ) E t . Note that the idempotent E t can be expressed in terms of the Jucys-Murphy elements.Indeed, the inductive formula for E t in terms of Jucys-Murphy elements can be formulatedas following: E t = E u Y α = β ∈ A ( µ ) J n − res( β )res t ( n ) − res( β ) , with the initial condition E ∅ = 1, which is well-defined thanks to Lemma 3.2.On the other hand, let t , · · · , t a be the set of pairwise different standard λ -tableauxobtained for u by adding an node with number n . Then the branching properties of theYoung basis imply that E u = a X i =1 E t i and the rational function E u z − res t ( n ) z − J n in z is well-defined, which is non-singular at z =res t ( n ) according to (4.2). Furthermore, we have(4.3) E t = E u z − res t ( n ) z − J n (cid:12)(cid:12)(cid:12)(cid:12) z =res t ( n ) . We first define the following rational function in variable z :(4.4) Θ t ( z ) := z − res t ( n ) f ( z ) n − Y i =1 (cid:0) z − res t ( i ) (cid:1) (cid:0) z − res t ( i ) + 1 (cid:1)(cid:0) z − res t ( i ) − (cid:1) . Clearly if n = 1 then Θ t ( z ) = z − res t (1) f ( z ) .4.5. Lemma. Keep notations as in § Θ t ( z ) = (cid:0) z − res t ( n ) (cid:1) Y β ∈ R ( µ ) (cid:0) z − res( β ) (cid:1) Y γ ∈ A ( µ ) (cid:0) z − res( γ ) (cid:1) − . roof. We prove the lemma by induction on n . If n = 1 then µ = ∅ , R ( µ ) = ∅ , and A ( µ ) = { (1 , , i ) | ≤ i ≤ m } . Thus the lemma follows directly by using the equality(4.4) for n = 1.Now assume that the lemma holds for all standard tableaux t with n − ≥ n nodes. Suppose that thenode α = ( a, b, c ) of t contains the number n − 1. Then we have the following cases:(i) If ( a − , b, c ) , ( a, b − , c ) / ∈ R ( ν ) then R ( µ ) = R ( ν ) ∪ { α } and A ( µ ) = (cid:0) A ( ν ) ∪ { ( a + 1 , b, c ) , ( a, b + 1 , c ) } (cid:1) \ { α } . (ii) If ( a − , b, c ) ∈ R ( ν ) and ( a, b − , c ) / ∈ R ( ν ), then R ( µ ) = (cid:0) R ( ν ) ∪ { ( a, b, c ) } (cid:1) \ { ( a, b, c ) } ; A ( µ ) = (cid:0) A ( ν ) ∪ { ( a + 1 , b, c ) } (cid:1) \ { α } . (iii) If ( a − , b, c ) / ∈ R ( ν ) and ( a, b − , c ) ∈ R ( ν ), then R ( µ ) = (cid:0) R ( ν ) ∪ { α } (cid:1) \ { ( a, b − , c ) } ; A ( µ ) = (cid:0) A ( ν ) ∪ { ( a + 1 , b, c ) } (cid:1) \ { α } . (iv) If ( a − , b, c ) , ( a, b − , c ) ∈ R ( ν ), then A ( µ ) = A ( ν ) \ { α } and R ( µ ) = (cid:0) R ( ν ) ∪ { α } (cid:1) \ { ( a − , b, c ) , ( a, b − , c ) } . Now applying the induction argument, we obtainΘ t ( z ) = ( z − res t ( n ))( z − res t ( n − ( z − res t ( n ) + 1)( z − res t ( n ) − Y β ∈ R ( ν ) (cid:0) z − res( β ) (cid:1) Y γ ∈ A ( ν ) (cid:0) z − res( γ ) (cid:1) − = (cid:0) z − res t ( n ) (cid:1) Y β ∈ R ( µ ) (cid:0) z − res( β ) (cid:1) Y γ ∈ A ( µ ) (cid:0) z − res( γ ) (cid:1) − . We complete the proof. (cid:3) The following lemma establishes the relationship between the rational function Θ λ ( Q )and the rational function Θ t ( z ), which is crucial to the fusion formula.4.6. Lemma. The rational function Θ t ( z ) is non-singular at z = res t ( n ) and Θ t (cid:0) res t ( n ) (cid:1) = Θ λ ( Q )Θ µ ( Q ) − . Proof. Lemma 4.5 shows that the rational function Θ t ( z ) is non-singular at z = res t ( n ).Noticing that the node α is removable and applying Proposition 3.9, we obtain thatΘ t (cid:0) res t ( n ) (cid:1) = Y β ∈ R ( µ ) (cid:0) res t ( n ) − res( β ) (cid:1) Y α = γ ∈ A ( µ ) (cid:0) res t ( n ) − res( γ ) (cid:1) − = Θ λ ( Q )Θ µ ( Q ) − . It completes the proof. (cid:3) Let φ ( z ) = t ( z ) and define φ k +1 ( z , · · · , z k ; z ) := t k ( z, z k ) φ k ( z , · · · , z k − ; z ) t k = t k ( z, z k ) t k − ( z, z k − ) · · · t ( z, z ) t · · · t k . Lemma. Keep notations as in § (4.8) Θ t ( z ) φ n (cid:0) res t (1) , · · · , res t ( n − , z (cid:1) E u = z − res t ( n ) z − J n E u . roof. We prove the equality (4.8) by induction on n . Using (2.5), (2.3) and (4.4), weobtain z − res t (1) z − J E ∅ = z − res t (1) f ( z ) · f ( z ) z − t E ∅ = Θ t ( z ) φ ( z ) E ∅ . That is the equality holds for n = 1.Note that E v E u = E u and E u t n − = t n − E u . Then, by induction hypothesis, the lefthand side of the equality (4.8) can be rewritten asLHS =Θ t ( z ) t n − (cid:0) z, res t ( n − (cid:1) φ n − (cid:0) res t (1) , · · · , res t ( n − , z (cid:1) t n − E v E u =Θ t ( z ) t n − ( z, res t ( n − φ n − (cid:0) res t (1) , · · · , res t ( n − , z (cid:1) E v t n − E u = Θ t ( z )Θ u ( z ) t n − (cid:0) z, res t ( n − (cid:1)(cid:0) Θ u ( z ) φ n − (res t (1) , · · · , res t ( n − , z ) E v (cid:1) t n − E u =Θ t ( z )Θ u ( z ) − t n − ( z, res t ( n − z − res t ( n − z − J n − E v t n − E u =Θ t ( z )Θ u ( z ) − t n − ( z, res t ( n − z − res t ( n − z − J n − t n − E u = ( z − res t ( n − t n − ( z, res t ( n − z − res t ( n − 1) + 1)( z − res t ( n − 1) + 1) z − res t ( n ) z − J n − t n − E u = t n − (res t ( n − , z ) − z − res t ( n ) z − J n − t n − E u . Note that J n commutes with E u . Therefore, to prove the equality (4.8), it suffices toshow that t n − ( z − J n ) E u = ( z − J n − ) t n − (res t ( n − , z ) E u , which follows directly by using the equalities (2.4) and (2.3). It completes the proof. (cid:3) Define the following rational function with values in the algebra H :(4.9) Φ( z , · · · , z n ) := φ n ( z , · · · , z n − , z n ) φ n − ( z , · · · , z n − ) · · · φ ( z ) . Now we can prove the main result of this paper.4.10. Theorem. Let λ be an m -multipartition of n and t a standard λ -tableau. Then theprimitive idempotent E t of H m,n ( Q ) corresponding to t can be obtained by the followingconsecutive evaluations E t = Θ λ ( Q )Φ( z , · · · , z n ) (cid:12)(cid:12)(cid:12)(cid:12) z =res t (1) · · · (cid:12)(cid:12)(cid:12)(cid:12) z =res t ( n − (cid:12)(cid:12)(cid:12)(cid:12) z =res t ( n ) . Proof. The theorem follows, by induction on n , from (4.3) and Lemmas 4.7 and 4.6. (cid:3) We close this paper with some remarks on the study of the fusion procedure for thedegenerate cyclotomic Hecke algebras and related topics.4.11. The fusion procedure for the Yang-Baxter equation was first introduced by Kulishet.al. in [ ], which allows the construction of new solutions of the Yang-Baxter equationstarting from a given fundamental solution. As it is suggested in [ ], a “fused solution”of the Yang-Baxter equation can be investigated using the certain version of the Schur-Weyl duality (see [ ]). Note that Brundan and Kleshchev have established the Schur-Weyl duality for higher levels in [ ]. It may be interesting to use the fusion formulafor the degenerate cyclotomic Hecke algebras to obtain a family of fused solutions ofthe Yang-Baxter equation acting on finite-dimensional irreducible representations of thefinite W -algebras.It is well-known that the (degenerate) cyclotomic Hecke algebras, the (degenerate)affine Hecke algebras are closely related (ref. [ , ]). 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Toda, Finite unitary reflection groups , Canad. J. Math. (1954), 273–304.[29] D.K. Zhao, Schur elements for degenerate cyclotomic Hecke algebras , a rXiv:1110.1735.[30] D.K. Zhao, The symbolic and cancellation-free formulae for Schur elements , Monatsh. Math. DOI10.1007/s00605-013-0500-7. School of Applied Mathematics, Beijing Normal University at Zhuhai, Zhuhai, 519087, China E-mail address : [email protected] School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao, 066004,China E-mail address : [email protected]@163.com