Fusion Procedure for Cyclotomic Hecke Algebras

A complete system of primitive pairwise orthogonal idempotents for cyclotomic Hecke algebras is constructed by consecutive evaluations of a rational function in several variables on quantum contents of multi-tableaux. This function is a product of two terms, one of which depends only on the shape of the multi-tableau and is proportional to the inverse of the corresponding Schur element.


Introduction
This article is a continuation of the article [10] on the fusion procedure for the complex reflection groups G(m, 1, n). The cyclotomic Hecke algebra H(m, 1, n), introduced in [1,2,3], is a natural flat deformation of the group ring of the complex reflection group G(m, 1, n).
In [10], a fusion procedure, in the spirit of [8], for the complex reflection groups G(m, 1, n) is suggested: a complete system of primitive pairwise orthogonal idempotents for the groups G(m, 1, n) is obtained by consecutive evaluations of a rational function in several variables with values in the group ring CG(m, 1, n). This approach to the fusion procedure relies on the existence of a maximal commutative set of elements of CG(m, 1, n) formed by the Jucys-Murphy elements.
Jucys-Murphy elements for the cyclotomic Hecke algebra H(m, 1, n) were introduced in [1] and were used in [9] to develop an inductive approach to the representation theory of the chain of the algebras H(m, 1, n). In the generic setting or under certain restrictions on the parameters of the algebra H(m, 1, n) (see Section 2 for precise definitions), the Jucys-Murphy elements form a maximal commutative set in the algebra H(m, 1, n).
A complete system of primitive pairwise orthogonal idempotents of the algebra H(m, 1, n) is indexed by the set of standard m-tableaux of size n. We formulate here the main result of the article. Let λ (m) be an m-partition of size n and T be a standard m-tableau of shape λ (m) .
Theorem. The idempotent E T of H(m, 1, n) corresponding to the standard m-tableau T of shape λ (m) can be obtained by the following consecutive evaluations (1) Here Φ(u 1 , . . . , u n ) is a rational function with values in the algebra H(m, 1, n), F λ (m) is an element of the base ring and c 1 , . . . , c n are the quantum contents of the m-nodes of T .
The classical limit of our fusion procedure for algebras H(m, 1, n) reproduces the fusion procedure of [10] for the complex reflection groups G(m, 1, n). For CG(m, 1, n), the variables of the rational function are split into two parts, one is related to the position of the m-node (its place in the m-tuple) and the other one -to the classical content of the m-node. The position variables can be evaluated simultaneously while the classical content variables have then to be evaluated consequently from 1 to n. For the algebra H(m, 1, n), the information about positions and classical contents is fully contained in the quantum contents, and now the function Φ depends on only one set of variables.
Remarkably, the coefficient F λ (m) appearing in (1) depends only on the shape λ (m) of the standard m-tableau T (cf. with the more delicate fusion procedure for the Birman-Murakami-Wenzl algebra [6]). In the classical limit, this coefficient depends only on the usual hook length, see [10]. However, in the deformed situation, the calculation of F λ (m) needs a non-trivial generalization of the hook length. It appears that the coefficient F λ (m) is proportional to the inverse of the Schur element of the algebra H(m, 1, n) corresponding to the m-partition λ (m) (see [4] for an expression of the Schur elements of H(m, 1, n) in terms of generalized hook lengths); the proportionality factor is a unit of the ring C[q, q −1 , v 1 , . . . , v m ], where q, v 1 , . . . , v m are the parameters of H(m, 1, n) (see Section 2 for precise definitions).
For m = 1, the cyclotomic Hecke algebra H(1, 1, n) is the Hecke algebra of type A and our fusion procedure reduces to the fusion procedure for the Hecke algebra in [5]. The factors in the rational function are arranged in [5] in such a way that there is a product of "Baxterized" generators on one side and a product of non-Baxterized generators on the other side. For m > 1 a rearrangement, as for the type A, of the rational function appearing in (1) is no more possible.
The additional, with respect to H(1, 1, n), generator of H(m, 1, n) satisfies the reflection equation whose "Baxterization" is known [7]. But -and this is maybe surprising -the full Baxterized form is not used in the construction of the rational function in (1). The rational expression involving the additional generator satisfies only a certain limit of the reflection equation with spectral parameters.
The Hecke algebra of type A is the natural quotient of the Birman-Murakami-Wenzl algebra. The fusion procedure, developed in [6], for the Birman-Murakami-Wenzl algebra provides a one-parameter family of fusion procedures for the Hecke algebra of type A. We think that for m > 1 the fusion procedure (1) can be included into a one-parameter family as well.
We shall work with a generic cyclotomic Hecke algebra (that is, q, v 1 , . . . , v m are indeterminates and we consider the algebra H(m, 1, n + 1) over a certain localization of the ring C[q, q −1 , v 1 , . . . , v m ]), or in a specialization such that the following conditions are satisfied 1 + q 2 + · · · + q 2N = 0 for N such that N < n , q 2i v j − v k = 0 for i, j, k such that j = k and − n < i < n , q = 0 , v j = 0 for j = 1, . . . , m . (3) Note that the restrictions (3) for the parameters of H(m, 1, n + 1) imply the corresponding restrictions for the parameters of H(m, 1, n).
Define, for i = 1, . . . , n, the Baxterized elements, with spectral parameters α and β: These Baxterized elements satisfy the Yang-Baxter equation with spectral parameters: The following formula will be used later: We also define the following rational function with values in H(m, 1, n + 1): Remarks. (i) One can rewrite τ (ρ) as a polynomial, in ρ, function. Indeed, let a 0 , a 1 , . . . , a m be the polynomials in v 1 , . . . , v m defined by The polynomials a i (ρ), i = 0, . . . , m, are given explicitly by It is straightforward to verify that It follows from (6) and (8) that For example, for m = 1, we have (ii) The elements τ (ρ) and σ 1 (α, β) satisfy a certain form of a reflection equation with spectral parameters: Indeed, due to (5) and (6), the equality (10) is equivalent to which is proved by a straightforward calculation. The equation (10) is a certain (we leave the details to the reader) limit of the usual reflection equation with spectral parameters.
An m-partition, or a Young m-diagram, of size n + 1 is an m-tuple of partitions such that the sum of their sizes equals n + 1; e. g. the Young 3-diagram ( , , ) represents the 3-partition (2), (1), (1) of size 4.
An m-node α (m) is a pair (α, k) consisting of a usual node α and an integer k = 1, . . . , m, indicating to which diagram in the m-tuple the node belongs. The integer k will be called position of the m-node, and we set pos(α (m) ) := k.
For an m-partition λ (m) , an m-node α (m) of λ (m) is called removable if the set of m-nodes obtained from λ (m) by removing α (m) is still an m-partition. An m-node β (m) not in λ (m) is called addable if the set of m-nodes obtained from λ (m) by adding β (m) is still an m-partition. For an m-partition λ (m) , we denote by E − (λ (m) ) the set of removable m-nodes of λ (m) and by E + (λ (m) ) the set of addable m-nodes of λ (m) . For example, the removable/addable m-nodes (marked with -/+) for the 3-partition ( , , ) are is obtained by placing the numbers 1, . . . , n + 1 in the m-nodes of the diagrams of λ (m) in such a way that the numbers in the nodes ascend along rows and columns in every diagram.
For a standard m-tableau T of shape λ (m) let α where {ξ 1 , ξ 2 , ξ 3 } is the set of all third roots of unity, ordered arbitrarily.
Generalized hook length. The hook of a node α of a partition λ is the set of nodes of λ consisting of the node α and the nodes which lie either under α in the same column or to the right of α in the same row; the hook length h λ (α) of α is the cardinality of the hook of α. We extend this definition to m-nodes. For an m-node α (m) = (α, k) of an m-partition λ (m) , the hook length of α (m) in λ (m) , which we denote by h λ (m) (α (m) ), is the hook length of the node α in the k-th partition of λ (m) .
Let λ (m) be an m-partition. For j = 1, . . . , m, let l λ (m) ,x,j be the number of nodes in the line x of the j-th diagram of λ (m) , and c λ (m) ,y,j be the number of nodes in the column y of the j-th diagram of λ (m) . The hook length of an m-node α (m) lying in the line x and the column y of the k-th diagram of λ (m) can be rewritten as Define a generalized hook length (see also [4]) by where α (m) is the m-node lying in the line x and the column y of the k-th diagram of λ (m) (in particular, h For an m-partition λ (m) of size n, we define .

Idempotents and Jucys-Murphy elements of H(m, 1, n + 1)
The Jucys-Murphy elements J i , i = 1, . . . , n + 1, of the algebra H(m, 1, n + 1) are defined by the following initial condition and recursion: We recall that, under the restrictions (3), the elements J i , i = 1, . . . , n+1, form a maximal commutative set of H(m, 1, n + 1) [1,9]. Recall also that The irreducible representations of H(m, 1, n + 1) are labelled by the m-partitions of size n + 1. The basis vectors of the irreducible representation of H(m, 1, n + 1) labelled by the m-partition λ (m) are parameterized by the standard m-tableaux of shape λ (m) . The Jucys-Murphy elements are diagonal in this basis. For a standard m-tableau T , denote by E T the corresponding primitive idempotent of H(m, 1, n + 1). We have, for all i = 1, . . . , n + 1, where c i := c(T |i), i = 1, . . . , n+1. Due to the maximality of the commutative set formed by the Jucys-Murphy elements, the idempotent E T can be expressed in terms of the elements J i , i = 1, . . . , n + 1. Let γ (m) be the m-node of T containing the number n + 1. As the m-tableau T is standard, the m-node γ (m) of λ (m) is removable. Let U be the standard m-tableau obtained from T by removing the m-node γ (m) , and let µ (m) be the shape of U . The inductive formula for E T in terms of the Jucys-Murphy elements reads: with the initial condition: E U 0 = 1 for the unique m-tableau U 0 of size 0. Here E U is considered as an element of the algebra H(m, 1, n + 1). Note that, due to the restrictions (3), we have c(β (m) ) = c(γ (m) ) for any β (m) ∈ E + (µ (m) ) such that β (m) = γ (m) .
Let {T 1 , . . . , T a } be the set of pairwise different standard m-tableaux which can be obtained from U by adding an m-node with number n + 1. The formula with (15), implies that the rational function is non-singular at u = c n+1 and moreover, due to the restrictions (3), 4. Fusion formula for the algebra H(m, 1, n + 1) Let φ 1 (u) := τ (u) and, for k = 1, . . . , n, Define the following rational function with values in the algebra H(m, 1, n + 1): Let λ (m) be an m-partition of size n + 1 and T a standard m-tableau of shape λ (m) . For i = 1, . . . , n + 1, we set c i := c(T |i). Theorem 1. The idempotent E T corresponding to the standard m-tableau T of shape λ (m) can be obtained by the following consecutive evaluations (21) Proof. The Theorem 1 follows, by induction on n, from (18) and the Propositions 2 and 5 below.
Till the end of the text, γ (m) and δ (m) denote the m-nodes of T containing the numbers n + 1 and n respectively; U is the standard m-tableau obtained from T by removing γ (m) , and µ (m) is the shape of U ; also, W is the standard m-tableau obtained from U by removing the m-node δ (m) and ν (m) is the shape of W.
For a standard m-tableau V of size N , we define by convention, for N = 1, the product in the right hand side is equal to 1.

Proposition 2. We have
Proof. We prove (23) by induction on n. As J 1 = τ , we have by (6) u which verifies the basis of induction (n = 0).
We have: E W E U = E U and E W commutes with σ n . Rewrite the left hand side of (23) as By the induction hypothesis we have for the left hand side of (23): Since J n+1 commutes with E U , the equality (23) is equivalent to (the inverse of σ n (u, c n ) is calculated with the help of (5)). By (22), Therefore, to prove (24), it remains to show that Replacing J n+1 by σ n J n σ n , we write the left hand side of (26) in the form As J n E U = c n E U , the right hand side of (26) is and thus coincides with (27).
Lemma 3. We have The proof is by induction on n. For n = 0, we have which is equal to the right hand side of (29). Now, for n > 0, we write Using the induction hypothesis, we obtain In each case, it follows that the right hand side of (30) is equal to which establishes the formula (29).

Lemma 4. We have
Proof. 1. The definition (11), for a usual partition λ, reduces to The Lemma 4 for a usual partition λ is established in [5], Lemma 3.2.
We set p 0 := 0, p s+1 := +∞ and µ p s+1 := 0. Assume that the m-node γ (m) lies in the line x and column y. The left hand side of (35) is equal to The factors in the product (33) correspond to the m-nodes of an m-partition. The m-nodes lying neither in the column y of the k-th diagrams (of λ (m) or µ (m) ) nor in the line x of the j-th diagrams do not contribute to the right hand side of (35). Let t ∈ {0, . . . , s} be such that p t < x ≤ p t+1 . The contribution from the m-nodes in the column y and lines 1, . . . , p t of the k-th diagrams is: the contribution from the m-nodes in the column y and lines p t + 1, . . . , x of the k-th diagrams is: The contribution from the m-nodes lying in the line x of the j-th diagrams is: After straightforward simplifications, we obtain for the right hand side of (35) The comparison of (36) and (37) concludes the proof of the formula (35).
3. The assertion of the Lemma is a consequence of the formulas (34), (35) together with the part 1 of the proof.
Proposition 5. The rational function F T (u) is non-singular at u = c n+1 , and moreover Proof. The formula (29) shows that the rational function F T (u) is non-singular at u = c n+1 , and moreover We use the Lemma 4 to conclude the proof of the Proposition.
The Baxterized elements (41) have been used in [10] for a fusion procedure for the complex reflection group G(m, 1, n + 1).

It is immediate that
The classical limit of F λ (m) is proportional to f λ (m) . More precisely, we have The formula (44) is obtained directly from (12) since m i = 1 i = k (ξ k − ξ i ) = m/ξ k for k = 1, . . . , m.

4.
Using formulas (40), (42) and (44), it is straightforward to check that the classical limit of the fusion procedure for H(m, 1, n + 1) given by the Theorem 1 leads to the fusion procedure [10] for the group G(m, 1, n + 1). Also, for m = 1 one reobtains the fusion procedure [5] for the Hecke algebra and, in the classical limit, the fusion procedure [8] for the symmetric group.