A $q$-deformation of the symplectic Schur functions and the Berele insertion algorithm

A randomisation of the Berele insertion algorithm is proposed, where the insertion of a letter to a symplectic Young tableau leads to a distribution over the set of symplectic Young tableaux. Berele's algorithm provides a bijection between words from an alphabet and a symplectic Young tableau along with a recording oscillating tableau. The randomised version of the algorithm is achieved by introducing a parameter $0<q<1$. The classic Berele algorithm corresponds to letting the parameter $q \to 0$. The new version provides a probabilistic framework that allows to prove Littlewood-type identities for a $q$-deformation of the symplectic Schur functions. These functions correspond to multilevel extensions of the continuous $q$-Hermite polynomials. Finally, we show that when both the original and the $q$-modified insertion algorithms are applied to a random word then the shape of the symplectic Young tableau evolves as a Markov chain on the set of partitions.


Introduction
It is well known (see [Rob38,Sce61,Sta01]) that there is a one-to-one mapping, namely the Robinson-Schensted correspondence, between the set of words of length m in the alphabet {1, ..., n} and the set of pairs of Young tableaux (P, Q), where P is a semistandard Young tableau with entries from the given alphabet and Q is a standard tableau with entries {1, 2, ..., m}. The two tableaux have the same shape which is a partition of the length of the word. If the given word is a permutation of {1, ..., n}, then P, Q are both standard Young tableaux with the same shape. Hence the Robinson-Schensted algorithm provides a bijection between the symmetric group S n and pairs of standard Young tableaux with the same shape, which implies the fundamental identity λ∈Λn |{P : standard Young tableau of shape λ}| 2 = n! = |S n | where Λ n denotes the set of partitions of length at most n.
Berele in [Ber86] proved that similarly to the Robinson-Schensted algorithm, there exists a one-to-one correspondence between words of length m from the alphabet {1,1, ..., n,n} and the sets of pairs (P, f ), where P is a symplectic tableau and f = (f 0 , ..., f m ) is a recording sequence of oscillating diagrams such that the shape of P is given by f m . The Berele correspondence can be used to prove Littlewood-type identities associated with the symplectic group. For example, Berele's scheme gives a combinatorial proof for the following identity ( [Sun90b]) (a 1 + a −1 1 + ... + a n + a −1 λ (a 1 , ..., a n ) where we denote by Sp (n) λ the symplectic Schur function, namely the character of an irreducible finite-dimensional representation of the symplectic group Sp(2n), parametrized by λ and we write Q λ m (n) for the number of oscillating sequences of diagrams f 0 = ∅, f 1 , ..., f m = λ with length at most n. In the thesis of Sundaram [Sun86] more results for the symplectic Schur functions are proved using Berele's bijection.
In this paper we will propose a randomisation of Berele's algorithm introducing a tuning parameter q ∈ (0, 1). This will enable us to derive a generalisation of identity (1) where the symplectic Schur functions are replaced by a multivariate extension of the continuous q-Hermite polynomials (for an extensive review on the properties of the continuous q-Hermite polynomials and their connetctions with other basic hypergeometric polynomials we refer the reader to [KLS10]), which we denote by P (n) λ (a 1 , ..., a n ; q). In Corollary 6.5 we present the following identity (a 1 + a −1 1 + ... + a n + a −1 n ) m = λ∈Λn Q λ m (n; q)P (n) λ (a 1 , ..., a n ; q).
where the function Q λ m (n; q) is given as the sum over sequences of diagrams f 0 = ∅, f 1 , ..., f m = λ with length at most n where each such sequence has a weight that depends on the parameter q. As q → 0, the quantity Q λ m (n; q) converges to Q λ m (n). Later we will also see that the symplectic Schur function Sp (n) λ (a 1 , ..., a n ) corresponds to the polynomial P (n) λ (a 1 , ..., a n ; q) with q = 0, recovering identity (1). We remark that although the left hand sides of identity (1) and the q-deformed identity are the same and their right hand sides consist of the same number of terms, these terms do not coincide with each other.
A randomisation of the column insertion version of the Robinson-Schensted algorithm was previously proposed by O'Connell-Pei in [OCP13], where the insertion process is formulated using insertion paths. A randomisation of the Robinson-Schensted algorithm in the framework of Gelfand-Tsetlin patterns can be found in the work of Borodin-Petrov [BP13] and Matveev-Petrov [MP16]. In our work, we follow the latter approach.
If the Robinson-Schensted algorithm is applied to a random word from the alphabet {1, ..., n} then O'Connell proved in [O'C03] that the shape of the Young tableau evolves as a Markov chain on the set of partitions with length at most n. The transition probabilities are given in terms of the Schur functions. Similarly, the q-weighted version by O'Connell-Pei leads to a random walk on Λ n with transition probabilities involving q-Whittaker functions instead. In this paper we will show that if we insert a random word from the alphabet {1,1, ..., n,n} according to the Berele insertion algorithm or its q-deformation proposed, then the shape of the symplectic Young tableau evolves as a Markov chain on Λ n with transition probabilities given in terms of the symplectic Schur functions and the functions P (n) (· ; q) respectively.
The outline of the paper is the following. In section 2 we introduce all the relevant notation. In section 3 we present the insertion algorithm proposed by Berele and study the evolution of the shape of a Young tableau when we insert randomly chosen letters. In section 4 we describe the algorithm using Gelfand-Tsetlin patterns instead of Young tableau, which helps us to q-modify the insertion process in section 5. In section 6 we state the main results obtained using an intertwining argument. In section 7 we study the shape of the symplectic Young tableau when the q-deformation of the Berele insertion algorithm is applied to a random word. Finally, the proof of the main intertwining identity can be found in section 8.

Partitions and Young diagrams
A partition is a sequence λ = (λ 1 , λ 2 , ...) of integers satisfying λ 1 ≥ λ 2 ≥ ... ≥ 0 with finitely many non-zero terms. We call each term λ i a part of λ and the number of non-zero parts the length of the partition λ, denoted by l(λ). Moreover we call |λ| := i≥1 λ i , the weight of the partition λ. If |λ| = k, then we say that λ partitions k and we write λ k. The set of partitions of length at most n is denoted by Λ n .
We define the natural ordering on the space of partitions called the dominance order. For two partitions λ, µ we write λ ≥ µ if and only if We also define the notion of interlacing of partitions. Let λ and µ be two partitions with λ ≥ µ. The partitions λ, µ are said to be interlaced, and we write µ λ if and only if A partition λ can be represented in the plane by an arrangement of boxes called a Young diagram. This arrangement is top and left justified with λ i boxes in the i-th row. We then say that the shape of the diagram, denoted by sh, is λ. For example the partition λ = (3, 2, 2, 1) can be represented as in figure 1. Given two partitions/diagrams λ and µ such that µ ⊂ λ (as a set of boxes), we call the set difference λ \ µ a skew Young diagram. A skew Young diagram λ \ µ is a horizontal strip if in each column it has at most one box.
Let us now consider a filling of a Young diagram, namely the symplectic Young tableau, with entries from an alphabet [n,n] := {1 <1 < ... < n < n}. For the rest of this paper, we will refer to the relative position of a letter l in the alphabet [n,n] as the order of the letter and write order(l).
Definition 2.1 ([Kin71]). A symplectic Young tableau is a filling of a Young diagram with entries from the alphabet [n,n] satisfying the following conditions. (S1) The entries are weakly increasing along the rows; (S2) The entries are strictly increasing down the columns; (S3) No entry < i occurs in row i, for any i. We denote the set of symplectic tableaux with entries from [n,n] by T [n,n] .
Example 2.2. The filling of the diagram of shape (4, 3) is a symplectic is not since there exists an entry1 at the second row violating condition (S3).
We will now define an oscillating tableau, which plays the role of a recording tableau.
Definition 2.3. An oscillating tableau of length m is a sequence of partitions f = (f 0 , ..., f m ), with f 0 = ∅ such that any two consecutive partitions differ by exactly one box, i.e. for 1 ≤ i ≤ m, it holds that f i \ f i−1 = (1) or f i−1 \ f i = (1). We say that an oscillating tableau f of length m has shape λ if f m = λ.
We will denote the set of oscillating tableaux of length m by O m and the set of oscillating tableau f ∈ O m , which moreover satisfy the property that for each 1 ≤ i ≤ m, f i ∈ Λ n by O m (n). Finally, for λ ∈ Λ n we write Q λ m (n) for the number of oscillating tableau f ∈ O m (n) of shape λ.

The Berele insertion algorithm
We recall that the Robinson-Schensted algorithm has two versions; the row insertion and the column insertion. The Berele algorithm follows the row insertion version. Before we describe the insertion algorithm we need to introduce a sliding algorithm called the jeu de taquin algorithm.
Definition 3.1. A punctured tableau of shape z is a Young diagram of shape z in which every box except one is filled. We will refer to this special box as the empty box.
Definition 3.2 (jeu de taquin). Let T be a punctured tableau with (α, β)entry, where α denotes the row and β the column, t αβ and with empty box in position (i, j). We consider the transformation jdt : T → jdt(T ) defined as follows • if T is an ordinary tableau then jdt(T ) = T ; • while T is a punctured tableau T → T switching the empty box and t i,j+1 , if t i,j+1 < t i+1,j T switching the empty box and t i+1,j , if t i,j+1 ≥ t i+1,j .
Here we will use the convention that if the empty box has only one right/down neighbouring box (α, β), then T → T switching the empty box and t α,β . We are now ready to describe the Berele insertion algorithm. To insert a letter i from the alphabet [n,n] to a symplectic tableau P , we begin by trying to place the letter at the end of the first row. If the result is a symplectic tableau we are done. Otherwise, the smallest entry which is larger than i is bumped and we proceed by inserting the bumped letter to the second row and so on. Suppose now that at some instance of the insertion process condition (S3) in Definition 2.1 is violated. This means that we tried to insert a letter l to the l-th row of the tableau, for some 1 ≤ l ≤ n, and bumped anl letter. Since we cannot insertl to the (l + 1)-th row we erase both l andl, leaving the box formerly occupied by thel as an empty box. We then apply jeu de taquin algorithm. The insertion algorithm can be applied to a word w = w 1 , ..., w m , with w i ∈ [n,n], starting with an empty tableau. The output, denoted by B(w), is a symplectic Young tableau along with an oscillating tableau f = (f 0 , ..., f m ) that records the shapes of the symplectic tableau for all the intermediate steps. More specifically, if we denote by P (i) the tableau after the insertion of the i-th letter then shP (i) = f i for i = 1, .., m.  O'Connell observed, in [O'C03], that if the Robinson-Schensted algorithm is applied to a random word, then the shape of the output tableau evolves as a Markov chain on the set of partitions with transition kernel given in terms of Schur functions.
We will devote the rest of this section to the study of the evolution of the shape of the symplectic tableau when we Berele-insert a random word from the alphabet [n,n]. We first need to introduce some notation.
Fix n ≥ 1 and a = (a 1 , ..., a n ) ∈ R n >0 . If w = w 1 , ..., w m is a word from [n,n], we write a w = a w 1 ...a wm where we use the convention that al = a −1 l , for 1 ≤ l ≤ n.
For a symplectic Young tableau P with entries from [n,n], we write We can now give a combinatorial definition for the symplectic Schur functions.
Definition 3.7. ([Kin71]) Let λ ∈ Λ n be a partition with length at most n. The symplectic Schur function parametrized by λ is given by The symplectic Schur function satisfies the Pieri rule, i.e for every λ ∈ Λ n and a ∈ R n >0 the following identity holds ( [Sun90a]) Let us now consider a sequence of independent random variables {w k , k ≥ 1} taking values in [n,n] with common distribution given, for 1 ≤ l ≤ n, by . (3) be the pair of tableau obtained when we apply the Berele algorithm to the random word w 1 , ..., w m from [n,n] with distribution ρ. Then, due to the fact that the Berele insertion algorithm is a bijection between words and pairs of tableaux, we conclude that for (4) where we recall that Q λ m (n) is the number of oscillating tableaux f ∈ O m (n) with shape λ. Using the fact that λ∈Λn P[shP (m) = λ] = 1 we may recover the following well-known character identity ( If on the other hand we sum (4) over P ∈ T [n,n] we obtain a distribution for the whole evolution of shapes of P , i.e. we conclude that for Note that, by the definition of conditional probability, we have that where the symbol µ λ means that the partition λ can be obtained from µ by either an addition or a deletion of a box.
The last observation summarises to the following result.
Theorem 3.8. We fix n ≥ 1 and a = (a 1 , ..., a n ) ∈ R n >0 . Let P (0) be the empty tableau, w 1 , w 2 , ... be a sequence of independent random variables chosen independently from ρ = {ρ(w), w ∈ [n,n]}, given in (3), and {P (m), m ≥ 1} be the sequence of symplectic tableaux obtained from the empty tableau after sequentially inserting the randomly chosen letters w 1 , w 2 , ..., i.e. for m ≥ 1, P (m) is given by Then the shape {shP (m), m ≥ 0} is a Markov chain on the set of partitions Λ n , started from the empty partition, with transition kernel Let us consider the n-dimensional Markov chain X = (X l (m), 1 ≤ l ≤ n; m ≥ 0) where for each 1 ≤ l ≤ n, X l (0) = 0 and for m > 0 where w 1 , w 2 , ... are letters from the alphabet [n,n] chosen independently with distribution ρ. Then X is a n-dimensional random walk, started from the origin, in which the l-th component jumps to the right with probability ρ(l) and to the left with probability ρ(l). Let us denote byρ the transition kernel of X killed upon leaving Λ n .
From the Pieri identity (2) for the symplectic Schur function it is easy to check that the function is harmonic forρ. Moreover, we observe that we may re-write the transition kernel Π as follows therefore we may identify the shape of a Young tableau obtained from sequentially inserting, randomly chosen with distribution ρ, letters from the alphabet [n,n] as the Doob h-transform of X killed upon leaving Λ n .

The Berele insertion algorithm for Gelfand-Tsetlin patterns
As we already mentioned, a different way to represent a symplectic Young tableau is via a symplectic Gelfand-Tsetlin pattern. Let us make this statement more precise. For a symplectic tableau P with entries from [n,n] and k ∈ [n,n] let z order(k) = shP k , where order(k) is the order of k in the alphabet [n,n] and P k is the sub-tableau of P that contains only entries of order less or equal to k. For example, if P = 11 2 22 22 then z 1 = shP 1 = (1), z 2 = shP1 = (2), z 3 = shP 2 = (4, 0) and z 4 = shP2 = (5, 2). By the definition of the symplectic Young tableau it follows that z k−1 z k , for 1 ≤ k ≤ N . This observation motivates the definition of the symplectic Gelfand-Tsetlin pattern.
Definition 4.1. Let N be a positive integer. A (symplectic) Gelfand-Tsetlin pattern Z = (z 1 , ..., z N ) with N levels is a collection of partitions with , which satisfy the interlacing conditions We denote the set of Gelfand-Tsetlin patterns with N levels by K N .
Schematically, a symplectic Gelfand-Tsetlin pattern is represented as in figure 2. In this paper we will only focus on patterns with even number of levels N = 2n. Before we describe the effect of Berele's insertion algorithm on the particles of a GT pattern, let us "translate" the changes that may occur to a Young tableau in the language of particles. Suppose that a letter of order k is added at the i-th row of a Young tableau, this means that z k i will increase by 1. In the language of particles this increase corresponds to a right jump of the particle at position z k i by one step. Similarly, if a letter of order k is removed from the i-th row of a Young tableau then z k i will decrease by 1 which represents a left jump for the corresponding particle.
Inserting a letter l ∈ [n,n] leads to an attempted right jump of the particle z k 1 , where k = order(l) is the order of the letter l, at the corresponding Gelfand-Tsetlin pattern.
If a particle z k i , for some i the jump is performed and z k+1 i is pushed one step to the right. This means that the letter of order k is either added at the end of the i-th row or causes a letter of order greater than k + 1 being bumped from the i-th row of the corresponding tableau; with k odd, the jump is suppressed and z k+1 i is pulled to the left instead. The transition described here corresponds to the cancellation step in Berele's algorithm; b. for all the other particles, the jump is performed and the particle z k+1 i+1 is pulled to the right. This means that a letter of order k is added at the i-th row of the corresponding tableau and bumps another letter of order k + 1 to the row beneath.
If z k i performs a left jump, then it triggers the leftward move of exactly one of its nearest lower neighbours; the left, z k+1 i+1 , if z k i = z k+1 i+1 and therefore the jump would lead to violation of the interlacing condition and the right, z k+1 i , otherwise. The transition we describe here corresponds to the jeu de taquin step of Berele's algorithm. Let us explain why. We assume without loss of generality that k = 2l and z 2l i , for some 1 ≤ i ≤ l, jumps to the left. This means that a letter ≤l is removed from a box in the i-th row of the tableau. Note that the empty box can now have only letters ≥ l + 1 to its right. If z 2l i = z 2l+1 i+1 , then beneath the empty box there is a letter ≤ l + 1, therefore according to the jeu de taquin algorithm we should swap the empty box with the box beneath. Therefore, the (i + 1)-th row now contains one less box with entry ≤ l+1, causing z 2l+1 i+1 to jump to the left.
Let us close this section by giving an example.
Example 4.2. Suppose n = 2. Inserting a letter1, in step a) the particle z 2 1 performs a rightward jump. Since z 2 1 < z 3 1 , z 2 1 triggers the move of z 3 2 . In b) z 3 2 attempts to jump to the right, but since z 3 2 < z 4 2 , the jump is suppressed. Finally, in step c) since the jump of the particle z 3 2 was suppressed, z 4 2 performs a leftward jump instead.

A q-deformation of Berele's insertion algorithm
In this section we consider a generalization of Berele's insertion algorithm that depends on a parameter q ∈ (0, 1). The q-deformed algorithm can be thought as a randomization of the usual algorithm, as inserting a letter to a given tableau results to a distribution over a set of tableaux. The proposed q-deformation has the property that the shape of the Young tableau again evolves as a Markov chain on Λ n .
Since a symplectic tableau is equivalent to a symplectic Gelfand-Tsetlin pattern we will present the algorithm in terms of dynamics on the pattern.
For q ∈ (0, 1) and (x, y) ∈ Λ n × Λ n or (x, y) ∈ Λ n−1 × Λ n satisfying the interlacing condition x y, we define the quantities for 1 ≤ i ≤ n, with the convention that if x = (x 1 , ..., x n ) ∈ Λ n , we set x n+1 ≡ 0 and x 0 ≡ ∞ (similarly for y) and the quantities l i (y; x), r i (y; x) are modified accordingly. A version of the quantities r i , l i first appeared in the work of Borodin-Petrov [BP13].
Inserting a letter l ∈ [n,n] leads to an attempted right jump of the particle z k 1 , where k denotes the order of the letter l, at the corresponding Gelfand-Tsetlin pattern.
If a particle z k i attempts a rightward jump, then i) if i = k+1 2 , with k odd, then with probability r i (z k+1 ; z k ), the jump is performed and z k+1 i is pushed to the right. Otherwise, with probability 1 − r i (z k+1 ; z k ) the jump is suppressed and z k+1 i is pulled to the left instead; ii) for all the other particles, the jump is performed and either the particle z k+1 i is pushed to the right with probability r i (z k+1 ; z k ) or the particle z k+1 i+1 is pulled to the right with probability 1 − r i (z k+1 ; z k ). If z k i performs a left jump, then it triggers the leftward move of exactly one of its nearest lower neighbours; the left, z k+1 i+1 , with probability l i (z k+1 ; z k ) and the right, z k+1 i , with probability 1 − l i (z k+1 ; z k ).
By the description of the algorithm, we see that any transition or attempted transition propagates all the way to the end of the pattern. Multiplying all the probabilities obtained till we reach the bottom of the pattern we obtain a probability distribution over the set K 2n of symplectic Gelfand-Tsetlin patterns with 2n levels. Let us denote by I l (Z,Z) the probability of obtaining a Gelfand-Tsetlin patternZ for Z after inserting the letter l ∈ [n,n].
The q-insertion Berele algorithm can be applied to a word w = w 1 ...w m ∈ [n,n] m starting from a Gelfand-Tsetlin pattern initialised at the origin, i.e. a Gelfand-Tsetlin pattern with all its coordinates equal to zero, which we denote by 0. Successively inserting the letters w 1 , ..., w m we obtain a distribution of weights φ w (·, ·) on K 2n × O m (n) recursively as follows. Set where f = (f 0 , ..., f m ).

Main Results
Before we present the main results, let us introduce some more notation. The q-Pochhammer symbol is written as (q; q) n and defined via the product The q-factorial is written as n! q and is defined as The q-binomial coefficients are defined in terms of q-factorials as follows We finally record some properties of the q-binomial coefficient that we will use later In this paper we will only work with a = e iθ ∈ R >0 .
We now need to define some special Laurent polynomials that generalise the continuous q-Hermite polynomials to higher dimensions. Let n ≥ 1, then for a = (a 1 , ..., a n ) ∈ R n >0 and each symplectic Gelfand-Tsetlin pattern Z ∈ K 2n , we define For a partition λ = (λ 1 ≥ ... ≥ λ n ) we consider the function where K 2n [λ] denotes the set of Gelfand-Tsetlin patterns with shape z 2n = λ.
Let us consider the kernel operator L n defined as We also define the kernel operators K n and M n by where al = a −1 l for 1 ≤ l ≤ n. The operator L n acts on partitions, whereas the operator M n acts on Young tableaux. The kernel K n can be used as a link between a Young tableau and its shape as follows.
Theorem 6.2. The following intertwining relation holds The proof of Theorem 6.2 can be found in section 8. The above intertwining relation implies that the polynomials P (n) λ (a; q) are eigenfunction of the operator L n . More specifically the following result holds.
Proposition 6.3. The following identity holds for all λ ∈ Λ n and a ∈ R n >0 .
Proof. We observe that the function P (n) λ (a; q) can be re-written using the kernel K n as follows therefore we have the following using the intertwining identity from Theorem 6.2 the last quantity equals Z∈K 2n We recall that I l (Z, Z) denotes the probability of obtaining the Gelfand-Tsetlin pattern Z fromZ after inserting the letter l, therefore for each l ∈ [n,n] andZ ∈ K 2n , we have Z∈K 2n I l (Z, Z) = 1 and hence we conclude that as required.
The intertwining relation also leads to an expression for the distribution over K 2n × O m (n) when we average over all the words of length m.
where we recall that a w = m i=1 a w i with al = a −1 l , for 1 ≤ l ≤ n.
Proof. Let us denote by Z(i) the Gelfand-Tsetlin pattern obtained after inserting the i-th letter to the corresponding Young tableau.
Thus, for the summation over Z(1) ∈ K 2n such that z 2n (1) = f 1 we have that where for the last equality we used the intertwining relation from Theorem 6.2 along with the fact that inside the summation for Z(2) we have that z 2n (2) = f 2 . Applying repeatedly the intertwining relation we conclude that which concludes the result observing that L n (∅, f 1 ) = 1. Summing (13) over Z ∈ K 2n and f ∈ O m (n) such that z 2n = f m gives the following therefore the following result holds.

A Markov chain on Gelfand-Tsetlin patterns
In section 5 we proposed a q-deformation of the Berele insertion algorithm on Gelfand-Tsetlin patterns. Let us now assume that the inserted letters are chosen randomly. More specifically, for a word w = w 1 , ..., w m we assume that w i is chosen independently of the other letters from [n,n] with distribution ρ, defined in (3). The probability of obtaining a pair (Z, f ) ∈ K 2n × O m (n) is given by where a w = m i=1 a w i with al = a −1 l . Using identity (13) we conclude that and summing over Z ∈ K 2n with z 2n = f m we obtain a distribution over the sequence of shapes for all the intermediate steps We moreover have L n (f m−1 , f m ) therefore the sequence of shapes {f m , m ≥ 0} evolves as a Markov chain on Λ n with transition kernel We summarise the results of this section to the following theorem which generalises Theorem 7.1 for q ∈ (0, 1).
Theorem 7.1. When applying the q-version of the Berele insertion algorithm to a random word w 1 , w 2 , ..., where each w i is chosen independently at random from [n,n] with distribution ρ defined in The sequence of shapes f m = z 2n (m) evolves as a Markov chain on Λ n with transition kernel The conditional law of Z(m), given {f 1 , ..., f m ; f m = λ} is .
The distribution of f m is given by λ (a; q)Q λ m (n; q).
Remark 7.2. The quantity ν(λ) defines a probability measure on Λ n , therefore we obtain the following identity for the function P
We will prove that the kernelsM n and L n , defined in (10), are intertwined via the kernelK n . This relation involves only a part of the Gelfand-Tsetlin pattern and hence proving it will be substantially easier to handle than the general intertwining relation.
We observe that both sides of the intertwining relation vanish unlessz = z ± e i for some 1 ≤ i ≤ n hence we only need to confirm the equality of the two sides of the intertwining relation for z ∈ Λ n , (x,ỹ,z) ∈ T n such that z = z ± e i , for 1 ≤ i ≤ n. Let us start with the casez = z + e 1 for which we provide full details of the calculations. The other cases can be checked in a similar manner. In this case the summation in (16) consists of the following terms.
Let us now proceed to the proof of Theorem 6.2. We will prove the result by induction on n.
General Case: In order to avoid heavy notation we will write (x, y, z) instead of (z 2n−2 , z 2n−1 , z 2n ) for the bottom three levels of a Gelfand-Tsetlin pattern Z ∈ K 2n . Let us assume that the intertwining relation holds for n − 1, i.e.
Regarding the kernel M n we have Inserting a letter n orn affects only the bottom two levels of the Gelfand-Tsetlin pattern and the second summation equals    a n r 1 (z; y) if (ỹ,z) = (y + e 1 , z + e 1 ) a n (1 − r 1 (z; y)) if (ỹ,z) = (y + e 1 , z + e 2 ) a −1 n if (ỹ,z) = (y, z − e 1 ) which is equal toM n ((x, y, z), (x,ỹ,z))1x =x . Let us now denote by P x→x (ỹ,z)|(y, z) the probability that the bottom two levels of the Gelfand-Tsetlin pattern performed the transition (y, z) → (ỹ,z) as a result of the transition of the particles at level 2n − 2 from x tõ x. It is easy to see that ifx = x then P x→x (ỹ,z)|(y, z) =M n ((x, y, z), (x,ỹ,z)) L n−1 (x,x) .