Symmetry, Integrability and Geometry: Methods and Applications Sklyanin Determinant for Reflection Algebra ⋆

Reflection algebras is a class of algebras associated with integrable models with boundaries. The coefficients of Sklyanin determinant generate the center of the reflection algebra. We give a combinatorial description of Sklyanin determinant suitable for explicit computations.

In [7] E.K. Sklyanin introduced a class of algebras associated with integrable models with boundaries. Following [5], we call them reflection algebras. The family of reflection algebras B(n, l) is defined as associative algebras whose generators satisfy two types of relations: the reflection equation and the unitary condition. In [5] A.I. Molev and E. Ragoucy show that the center of B(n, l) is generated by the coefficients of an analogue of quantum determinant, called the Sklyanin determinant. In the same paper the authors develop an analogue of Drinfeld's highest weight theory for reflection algebras and give a complete description of their finitedimensional irreducible representations.
The reflection algebras B(n, l) have many common features with the twisted Yangians, introduced by G. Olshanski [6]. For example, the center of twisted Yangian is also generated by coefficients of Sklyanin determinant, which is defined similarly to the Sklyanin determinant of reflection algebras. The detailed exposition of the representation theory of twisted Yangians can be found in [1,2].
In [3] A.I. Molev gives a combinatorial formula for the Sklyanin determinant of twisted Yangians in terms of matrix elements of the matrix of generators of the twisted Yangians (see also [1]). The goal of this paper is to give analogues combinatorial description of the Sklaynin determinant of the reflection algebra.
In Section 1 we review the main definitions. In Sections 2, 3 we rewrite the definition of the Sklyanin determinant of B(n, l) in an alternative form. We describe combinatorics of the corresponding product of generating matrices 'twisted' by Jucys-Murphy elements. In Section 4, Theorem 1, we prove combinatorial formula for Sklyanin determinant in terms of matrix elements of the generating matrix of B(n, l). In Section 5 we give examples of computations by this formula.

Def initions
The following notations will be used through the paper. For a matrix X with entries (x ij ) i,j=1,...,n in an associative algebra A write ⋆ This paper is a contribution to the Proceedings of the International Workshop "Recent Advances in Quantum Integrable Systems". The full collection is available at http://www.emis.de/journals/SIGMA/RAQIS2010.html Consider the Yang matrix where P : v ⊗ w → w ⊗ v is a permutation operator in the space C n ⊗ C n . For i, j such that 1 ≤ i < j ≤ k the notations P ij and R ij (u) are used for the action P and R(u) on the i-th and j-th copies of the vector space C n in (C n ) ⊗k .
Combine the generators b and collect them into generating matrix Then the reflection equation relation is given by and the unitary condition is Remark 1. The unitary condition allows to realize the algebra B(n, l) as a subalgebra of Yangian Y (gl n (C)). The mentioned below Proposition 1 is proved in [5] through this inclusion of algebras. However, we do not need the unitary conditions anywhere for the proofs of the statements below. Hence all the results of the paper automatically become true for the extended version of the reflection algebra, which is defined the same way as the algebra B(n, l), but with the unitary condition (2) omitted.
We abbreviate R ij := R ij (2u − i − j + 2). Put This expression is as an element of End (C n ) ⊗k ⊗ B(n, l)[[u −1 ]]. Let A n be the full anti-symmetriztion operator in the space End (C n ) ⊗n : One can show [5] that A n B 1 , . . . , B n is equal to the product of the anti-symmetrizer A n and a series in u −1 with coefficients in B(n, l). This series is called Sklyanin determinant.
are algebraically independent generators of the center of B(n, l).

Alternative def inition of the Sklyanin determinant
The following proposition allows to simplify the combinatorial description of sdet B(u).
Proposition 2. For k = 1, . . . , n, define and put Proof . We use the notation A k for the antisymmetrizator in (C n ) ⊗n that acts on the first k copies of C n , and A ′ n−k for the antisymmetrizator in (C n ) ⊗n that acts on the last (n − k) copies of C n : Proof . We follow the lines of the proof of Proposition 1.6.2 in [1].
where (a, b) denotes a transposition. Therefore, The product of R-matrices R kk+1 · · · R kn is a sum of the identity operator and of monomials P ki 1 P ki 2 · · · P kim with rational coefficients of the form .
Using (4), we substitute each monomial by P ki 1 and collect the coefficients for each of i = k + 1, . . . , n. Then the first equality of the lemma follows.
In the group algebra of the symmetric group S n for any i, j, k ∈ {1, . . . , n}, such that i, j, k are pairwise distinct numbers, one has Since A ′ n−k is a linear combination of products of transpositions P ij with i > k and j > k, by (5) the antisymmetrizer A ′ n−k commutes with the sum n i=k+1 P ki , and the second equality follows.
For i ≤ m ≤ n the antisymmetrizer A ′ n−m commutes with B i (u − i + 1), and by (3) it also commutes with Π m . Using that for m = 2, 3, . . . , n, and Lemma 1, we can rewrite the product A n B 1 , . . . , B n in the definition of sdet B(u) in terms of operators Π 1 , . . . , Π n : For κ ∈ I n introduce a rational function α(κ) = α(κ, u) of a variable u defined by .
For all k = 1, . . . , n we identify P kk with Id ⊗n ∈ End (C n ) ⊗n . Then (The last operator P nkn = P nn = Id ⊗n in the end of the formula is added for a uniform presentation.) Lemma 2. For any X ∈ End(C n ) ⊗ B(n, l) Using this simple fact all permutation operators in the expression B 1 , . . . , B n can be moved in front of all elements B s (u − a).
Define also Then Here q Proof . From Lemma 2 and the equality (6) the statement of the proposition follows immediately.
For the second combinatorial description of B 1 , . . . , B n we assign to each η ∈ I n a permutation p η that is defined by the following recursive rule.

Proposition 4.
Then where p −1 η is an element of the symmetric group S n , identified with the corresponding operator acting on (C n ) ⊗n .
In Propositions 3, 4 the sum of monomials is taken over the set I n . This set has n! elements. One can ask, if the monomials can be naturally numerated by permutations σ ∈ S n .
We will need the following notion of a word restriction of a permutation. Let σ be a permutation of the elements of the set {1, . . . , n}, and let σ = Γ 1 · · · Γ l be a decomposition into non-intersecting cycles Γ i = (g i,1 , . . . , g i,s i ). Let G be a subset of {1, . . . , n}.
Definition 2. We say that the cycle Γ| G is the word restriction of the cycle Γ on the set G, if Γ| G is obtained from Γ by deleting from the word of the cycle of Γ of all the elements that do not belong to G. Then the word restriction of σ on the set G is defined as a product of word restrictions of cycles Γ 1 | G , . . . , Γ l | G on the set G: We say that a cycle Γ has an empty restriction on the set G, if it contains no elements from G. Proposition 5. For any σ ∈ S n consider a decomposition of σ into nonintersecting cycles: σ = Γ 1 · · · Γ s . Let g i be the maximal element of the cycle Γ i . Denote as G σ the set {g i } s i=1 . Then Hereᾱ the permutation σ [i] is the word restriction of σ on the set {i, . . . , n}. We identify σ −1 with the corresponding operator acting on (C n ) ⊗n .
Proof . Observe that any σ ∈ S n can be written uniquely as a product of transpositions σ = (n, k n ) · · · (1, k 1 ) for some κ = (k 1 , . . . , k n ) ∈ I n . In other words, there is a one-to-one correspondence between the elements of I n and permutations in S n . One can check that the word restriction of σ on {i, . . . , n} has the property The set of maximal elements of the cycles {g i } s i=1 in the decomposition of σ into non-intersecting cycles coincides with the set {k i | k i = i} in the decomposition (9). This implies the formula forᾱ(σ).

Computation of Sklyanin determinant
In this section we give formulas that allow to compute explicitly Sklyanin determinant in terms of matrix elements of the generating matrix B(u).
For any n-tuple η ∈ I n we will associate several quantities and sets. The n-tuple η ∈ I n can be considered as a function on the interval of integer numbers {1, . . . , n}. If η = (η 1 , . . . , η n ), then Let the image Im(η) consist of distinct numbers {N 1 , . . . , N K }, and let the pre-image η −1 (N l ) = {i | η i = N l } consist of numbers {a l,1 , . . . , a l,m l }. We choose the order of the elements so that a l,1 < · · · < a l,m l .
For each l = 1, . . . , K the sequence a l,1 < · · · < a l,m l defines a increasing cyclic permutation Γ l = (a l,1 , . . . , a l,m l ) ∈ S n . Put Thus, for every η ∈ I n we associate a permutation γ η ∈ S n that is a product of non-intersecting increasing cycles. We also denote as G ± η the sets of maximal and minimal elements of those cycles: Let S(η) be the the subgroup of S n of all permutations that act trivially on the complement of the set Im(η) in {1, . . . , n}.