Interacting particle models and the Pieri-type formulas : the symplectic case with non equal weights

We have introduced recently a particles model with blocking and pushing interactions which is related to a Pieri type formula for the orthogonal group. This model has a symplectic version presented here.


introduction
Let us recall that the Pieri's formula describes the product of a Schur polynomial by a complete symmetric function.It is a specific case of the Littlewood-Richardson rules for decomposing the tensor products of representations of the unitary group into irreducible components.In that specific case, irreducible components of the tensor products have a multiplicity equal to one.
In [5] an interacting particle model has been introduced and proved to be related to a Pieri-type formula for the orthogonal group.Particles of the model can move to the left or to the right and are submitted to blocking and pushing interactions.Moreover, they are constrained to stay on the non-negative real axis.Such a model is said to be with a wall.One can find for instance in [2] and [8] other examples of models with a wall.In these last two references, models only differ by the behavior of the particles near the wall : in the first one, these particles are reflected by the wall, in the second one, they are blocked by the wall.Actually the first one, connected to models of [5], is strongly related to representations of the orthogonal group whereas the second one involves representations of the symplectic compact group.Pursuing the study of models with a wall in the same way as in [5] we construct here a new interacting particle model with a wall depending on parameters.Two particular values of the parameters lead on the one hand to a model studied in [8] on the other hand to a random matrix model of [3].
A fundamental difference between models considered here and models usually considered in the literature is that they are related to tensor products of representations which aren't decomposed into irreducible components with multiplicity one.Besides, the case with non equal weights studied here is of particular interest since it exposes connections between symplectic Schur functions and the random particle models.For these reasons, this paper could hopefully provide a step towards a generalization.
The paper is organized as follows.In the second section, I shall recall the definitions of the symplectic Gelfand-Tsetlin patterns and Schur functions.In the third one I will describe the interacting particle model studied in the paper.In the fourth section I'll recall some properties about tensor product of particular representations of the symplectic compact group, which naturally leads to some Markov kernels involved in the interacting particle model.These Markov kernels are defined in the fifth section.Section six is devoted to a random matrix model related to the particle model.Results of the paper are stated in section seven.I will sketch the proofs in the last section.

Symplectic Gelfand-Tsetlin patterns, symplectic Schur functions
Here and elsewhere N stands for the set of nonnegative integers.For n ∈ N * and x, y ∈ R n such that x n ≤ • • • ≤ x 1 and y n ≤ • • • ≤ y 1 , we write x y if x and y are interlaced, i.e.
x n ≤ y When x ∈ R n and y ∈ R n+1 we add the relation y n+1 ≤ x n .We denote by |x| the sum of the coordinates n i=1 x i .Definition 2.1.Let k be a positive integer.
(1) We denote by GT k the set of symplectic Gelfand-Tsetlin patterns defined by Usually, a symplectic Gelfand-Tsetlin pattern is represented by a triangular array as indicated at figure 1 for k = 2r.Actually the more right an entry is, the largest it is.Thus the interlacing property satisfied by the coordinates of a symplectic Gelfand-Tsetlin pattern is graphically represented.Note that the zero and the minus indicate that the coordinates are positive.For x a symplectic Gelfand-Tsetlin pattern of GT k and (q i ) i≥1 a sequence of positive real numbers one defines w k x (q 1 , . . ., q [ k+1 2 ] ) by recursion by letting w 1 x (q 1 ) = q 1 , and x (q 1 , . . ., q i , q i+1 ) = w 2i x (q 1 , . . ., q i ) q , we denote by s k λ the symplectic Schur function defined by A symplectic Gelfand-Tsetlin pattern of GT 2r

Interacting particle models
In this section we construct two processes evolving on the set GT k of symplectic Gelfand-Tsetlin patterns.These processes can be viewed as interacting particle models.For this, we associate to a symplectic Gelfand-Tsetlin pattern x = (x 1 , . . ., x k ), a configuration of particles on the integer lattice Z 2 putting one particle labeled by (i, j) at point (x i j , k−i) of Z 2 for i ∈ {1, . . ., k}, j ∈ {1, . . ., [ i+1 2 ]}.Several particles can be located at the same point.In the sequel we will say "particle x i j " instead of saying "particle labeled by (i, j) located at point (x i j , k − i)".
At time n + 1/2 : All particles try to jump to the left one after another in the lexicographic order pushing the other particles in order to stay in the set of symplectic Gelfand-Tsetlin patterns and being blocked by the initial configuration X(n) of the particles: • Particle X 1 1 (n) tries to move to the left being blocked by 0, i.e.
2 ) • Particle X 3 1 (n) tries to move to the left being blocked by X 2 1 (n) : Particle X3 2 (n) tries to move to the left being blocked by 0, i.e Suppose now that rows 1 through l − 1 have been updated for some l > 1. Particles (n) try to jump to the left being blocked as follows by the initial position X(n) of the particles.For i = 1, . . ., [ l+1 2 ], with the convention that X l−1 l+1 2 (n) = 0 when l is odd.
At time n + 1 : All particles try to jump to the right one after another in the lexicographic order pushing particles in order to stay in the set of symplectic Gelfand-Tsetlin patterns and being blocked by the initial configuration X(n + 1 2 ) of the particles.The first three rows are updated as follows.
• Particle X 3 2 (n + 1 2 ) tries to move to the right being blocked by X 2 1 (n + 1 2 ).Particle X3 1 (n + 1 2 ) moves to the right.That is )) Suppose rows 1 through l − 1 have been updated for some l > 1.Then particles of row l are pushed to intermediate positions with the convention X l−1 l+1 2 (n + 1) = 0 when l is odd.Then particles Xl 2 ) try to jump to the right being blocked by the initial position of the particles as follows.For i = 1, . . ., [ l+1 2 ], 3.2.Exponential waiting times.The interacting particle model described now has been introduced in [8].In this model particles evolve on Z 2 and jump on their own volition by one rightwards or leftwards after an exponentially distributed waiting time.The evolution of the particles is described by a random process (Y (t)) t≥0 on GT k .As in the previous model, at time t ≥ 0 there is one particle labeled by (i, j) at point (Y i j (t), k − i) of the integer lattice, for i = 1, . . ., k, j = 1, . . ., [ i+1 2 ].Particle labeled by (2i, j) tries to jump to the left by one after an exponentially distributed waiting time with mean q i or to the right by one after an exponentially distributed waiting time with mean q −1 i .Particle labeled by (2i−1, j) tries to jump to the left by one after an exponentially distributed waiting time with mean q −1 i or to the right by one after an exponentially distributed waiting time with mean q i .Waiting times are all independent.When a particle tries to jump, all particles are pushed and blocked according to the same rules as previously: particles above push and block particles below.That is if particle labeled by (i, j) wants to jump to the right at time t ≥ 0 then (2) else particles (i, j), (i + 1, j), . . ., (i + l, j) jump to the right by one for l the largest integer such that Y i+l j If particle labeled by (i, j) wants to jump to the left at time t ≥ 0 then (1) if i is odd, j = (i + 1)/2 and Y i j (t − ) = 0 then particle labeled by (i, j) doesn't move. (2 if i is even or j = (i + 1)/2, and Y i j (t − ) = Y i−1 j (t − ) then particles don't move.
(4) if i is even or j = (i + 1)/2, and Y i j (t − ) > Y i−1 j (t − ) then particles (i, j), (i + 1, j + 1), . . ., (i + l, j + l) jump to the left by one for l the largest integer such that Y i+l j+l Actually, process (Y (t), t ≥ 0) is obtained by letting α go to zero in the previous model.More precisely we get the following proposition.
Proof.The proposition is obtained by replacing q by α in Lemma 8.9 of [5].

A Pieri type formula for the symplectic group
Let r be a positive integer.One recalls some usual properties of the finite dimensional representations of the compact symplectic group Sp 2r (see for instance [6] for more details).The set of finite dimensional representations of Sp 2r is indexed by the set For λ ∈ W 2r , using standard notations, we denote by V λ the so called irreducible representation with highest weight λ of Sp 2r .
Let m be an integer and λ an element of W 2r .Consider the irreducible representations V λ and V γm of Sp 2r , with γ m = (m, 0, • • • , 0).The decomposition of the tensor product V λ ⊗ V γm into irreducible components is given by a Pieri-type formula for the symplectic group.It has been recalled in [3].We have where the direct sum is over all β ∈ W 2r such that there exists c ∈ W 2r which satisfies In addition, the multiplicity M λ,γm (β) of the irreducible representation with highest weight β is the number of c ∈ W 2r satisfying these relations.Note that V λ ⊗ V γm is not free multiplicity if m / ∈ {0, 1}.
Lemma 5.1 implies in particular that the measure ν is a probability measure.Thus one defines a Markov kernel P 2r on W 2r by letting Proof.The follows immediately from the tensor product rules recalled for the decomposition (1).
We let W 2r−1 = W 2r .For c 0 , λ, c, β ∈ W k , we let when k = 2r and The main purpose of this paper is to show that S k describes the evolution of the k th row of the random symplectic Gelfand-Tsetlin patterns (X(t), t ≥ 0).Note that Proposition 5.2 ensures that S 2r defines a Markov kernel on W 2r × W 2r .There isn't such an argument for S 2r−1 .Anyway as Λ k and Q k defined in section 8 are Markov kernels, Proposition 8.1 ensures that S k is a Markov kernel in both the odd and the even cases.Thus one defines also a Markov kernel P 2r−1 on W 2r−1 by letting Actually P 2r−1 (λ, .) is the image of the measure S 2r−1 ((c 0 , λ), (., .)),for any arbitrary c 0 ∈ W 2r , by the map We'll see that the image measure P k describes the evolution of the k th row of the random symplectic Gelfand-Tsetlin patterns observed at integer times.This Markov kernel is relevant for the understanding of the relation between the particle model and the random matrix model of the next section.

Random matrices
We denote by H the set of quaternions.For us, a quaternion is just a 2 × 2 matrix Z with complex entries which can be written as where a, b ∈ C. Its conjugate Z * is the usual adjoint of the complex matrix Z.Let us denote by M r,m the real vector space of r × m matrices with entries in H and by P r the set of r × r Hermitian matrices with entries in iH.Since a matrix in P r is a 2r × 2r Hermitian complex matrix, it has real eigenvalues We put on M r,m the Euclidean structure defined by the scalar product, Let C r be the subset of R r defined by Theorem 4.5 of [3] and Proposition 4.8 of [3] imply the following proposition.
Proposition 6.1.Let r be a positive integer and (M (n), n ≥ 0), be a discrete process on P r defined by where the Y l 's are independent standard Gaussian variables in M r,1 .For n ∈ N, Then the process (Λ(n), n ≥ 0), is a Markov process with a transition densities p r defined by x i , x, y ∈ C r .

results
We have introduced a Markov process (X(t), t ≥ 0) on the set of symplectic Gelfand-Tsetlin patterns.We will show that if only one row of the pattern is considered by itself, it found to be a Markov process too.Similar results have been proved in [8].The main specificity of our model is that coordinates of particles are not upgraded in the same way at every times.We are mainly interested in the process (X k (n), n ≥ 0) but intermediate states (X k (n + 1  2 ), n ≥ 0) are considered for the proofs.Actually these intermediate states come from the fact that the tensor product of ( 1) is not free-multiplicity.We let If P k is the Markov kernel defined in ( 2) and ( 5) then Theorem 7.1 implies immediately the following theorem which is our main result.[1] implies that the process converges towards a Markov process with infinitesimal generator equal to A as α goes to zero.The convergence stated in Proposition 3.1 achieves the proof.
If (Λ(n), n ≥ 0) is the process of eigenvalues considered in Proposition 6.1 then the following corollary holds.
Proof.The Weyl dimension formula (see Knapp [6], Thm V.5.84) for the symplectic groups gives Thus the corollary follows immediately from Theorem 7.2 and Proposition 6.1.

proofs
The proof of Theorem 7.1 rests on the same ingredients as the proof of Proposition 8.8 of [5].It will follow from an intertwining property stated in Proposition 8.1 and an application of a criterion established in [7] by Pitman and Rogers who give a simple condition sufficient to ensure that a function of a Markov process is again a Markov process.For λ ∈ W k we consider the measure M λ on GT k (λ) defined by where δ x is the dirac measure at x, and the measure m λ defined as the image of the measure M λ by the map As (Z k (n), n ≥ 0) is conditionally independent of (Z i (t), t ≥ 0, i = 1, . . ., k − 2) given (Z k−1 (t), t ≥ 0), Theorem 7.1 may be proved by induction on k.The theorem is true for k = 1.Suppose that the process (Z k−1 (n), n ≥ 0) is Markovian with transition kernel S k−1 .The dynamic of the model implies that (Z k−1 (n), Z k (n), n ≥ 0) is also Markovian.As for any λ ∈ W k−1 , S k−1 ((c 0 , λ), (., .))doesn't depend on c 0 , it implies that the process ((X k−1 (n), Z k (n)), n ≥ 0) is also Markovian.Let us denote by Q k its transition kernel.Proposition 8.1 claims that it satisfies an intertwining property, which implies, using the Rogers and Pitman criterion of [7], that the process (Z k (n), n ≥ 0) is Markovian with transition kernel S k .Thus the theorem is true for the integer k.
We get the intertwining summing over u first and over v after, using respectively identities (4) and ( 5) of Lemma 8.3 of [5].