Fusion coefficients and random walks in alcoves
Abstract.
We point out a connection between fusion coefficients and random walks in a fixed level alcove associated to the root system of an affine Lie algebra and use this connection to solve completely the Dirichlet problem on such an alcove for a large class of simple random walks. We establish a correspondence between the hypergroup of conjugacy classes of a compact Lie group and the fusion hypergroup. We prove that a random walk in an alcove, obtained with the help of fusion coefficients, converges, after a proper normalization, towards the radial part of a Brownian motion on a compact Lie group.
1. Introduction
In the early nineties Ph. Biane pointed out relations between representation theory of semisimple complex Lie algebras and random walks in a Weyl chamber associated to a root system of such an algebra (see for instance [2]). Actually, random walks in a Weyl chamber are obtained considering the hypergroup of characters of a semisimple complex Lie algebra, with structure constants given by the LittlewoodRichardson coefficients. A Weyl chamber is a fundamental domain for the action of a Weyl group associated to a root system. If we consider an affine Lie algebra, which is an infinite dimensional KacMoody algebra, a fundamental domain for the action of the Weyl group associated to its (infinite) root system is a collection of level alcoves, . Thus it is a natural question to ask if random walks in alcoves are related to representation theory of infinite dimensional Lie algebras. There are several ways to answer. A first one could be to consider tensor products of highest weight representations of an affine Lie algebra. One would obtain random walks in alcoves with increasing level at each time. This approach has to be related to the very recent paper [16]. A second one is to consider the socalled fusion product. In that case, one obtains random walks living in an alcove with a fixed level. This is this approach that we develop in this paper. Fusion coefficients can be seen as the structure constants of the hypergroup of the discretized characters of irreducible representations of a semisimple Lie algebra (see [21] and references therein). Following an idea of Ph. Bougerol^{1}^{1}1Private communication. we point out that random walks in an alcove are related to such an hypergroup. Thus one answers positively to the question explicitly formulated in [11] : does it exist a link between representation theory and random walks in alcoves ? In particular one can completely solve the discrete Dirichlet problem on an alcove, for a large class of simple random walks, as P. H. Berard did in [1] in a continuous setting, which is important to obtain, for instance, precise asymptotic results. Thus we get a very natural new integrable probabilistic model, i.e a probabilistic object which can ”be viewed as a projection of a much more powerful object whose origins lie in representation theory” [4]. We obtain in addition a better understanding of some previous results concerning random walks in alcoves. Actually, the restriction to a classical alcove of the Markov kernel of most of reflectable random walks considered in [11] is given by fusion coefficients. This is due to the fact that these reflectable random walks are mostly related to minuscule representations of classical compact Lie groups and that in these cases fusion coefficients give the number of walks remaining in an alcove. In [11] Grabiner is interested in a class of reflectable walks, for which Gessel and Zeilberger have shown a KarlinMacGregor type formula in [10]. In our perspective, this formula has to be related to a KarlinMacGregor type formula which holds for the socalled fusion coefficients.
A random walk on a Weyl chamber converges after a proper normalization towards a Brownian motion on a Weyl chamber, which can also be realized as the radial part of a Brownian motion in a semisimple complex Lie algebra. It is maybe enlightening to notice that the orbit method of Kirillov provides a kind of intermediate between the discrete and the continuous objects. It establishes in particular a relation between convolution on a Lie algebra and tensor product of its representations. Taking an appropriate sequence of convolutions on a Lie algebra one obtains by a classical central limit theorem a chain of correspondences between random walks in a Weyl chamber, tensor product of representations, convolution on a Lie algebra and Brownian motion in this Lie algebra. We establish that convolution on a connected compact Lie group involves fusion product of irreducible representations. We prove that a random walk obtained considering the fusion hypergroup converges after a proper normalization towards the radial part of a Brownian motion in a compact Lie group. Thus, the paper should be read keeping in mind the following informal chain of correspondences.
Random walk in  Fusion  Random walk in  Brownian motion  
an alcove  product  a compact group  in a compact group. 
The paper is organized as follows. Basic definitions and notations related to representation theory of semisimple complex Lie algebras are introduced in section 3. The fusion coefficients are defined in section 4. We define in section 5 random walks in an alcove considering the hypergroup of the socalled discretized characters of irreducible representations of a semisimple complex Lie algebra, with structure constants given by fusion coefficients. Moreover we show how the discretized characters provide a complete solution to a Dirichlet problem in an alcove for a large class of simple random walks. We indicate precisely in section 6 how most of simple random walks considered in [11] and [15] appear naturally in this framework. We explain in section 7 how the fusion product is related to convolution on a compact Lie group. We established in section 8 a convergence towards the radial part of a Brownian motion in a compact Lie group.
Note that a discrete Laplacian on Weyl alcoves has been introduced in [18] in a more general framework of double affine Hecke algebras. The Bethe Ansatz method is employed to find eigenfunctions, which are proved to be the periodic Macdonald spherical functions. Even if the underlying Markov processes are the same as ours, his approach is quite different. We hope that ours, which explicitly involves the fusion hypergroup, is enlightening in a sense that fusion coefficients are proved to play the same role for random walks in an alcove as the Littlewood Richardson coefficients for random walks in a Weyl chamber.
Acknowlegments: The author would like to thank Ph. Bougerol for having made her know the fusion product and its beautiful probabilistic interpretation.
2. The case of
In order to facilitate the lecture of the paper we first begin to detail how the simplest example of random walk in an alcove has to be related to fusion coefficients. Let and . We consider the simple random walk on with transition kernel defined by , for . For , we let . The discrete Dirichlet problem consists in finding eigenvalues and eigenfunctions defined on satisfying
where . It is a consequence of the PerronFrobenius theorem that the smallest eigenvalue is positive, simple and that the corresponding eigenfunction can be chosen positive on . Such a function is said to be a PerronFrobenius eigenfunction. The eigenfunctions corresponding to the other eigenvalues change of sign on . An easy computation shows that the eigenvalues of the Dirichlet problem are , for , with corresponding eigenfunctions defined by , , where
For , one gets a PerronFrobenius eigenfunction. Actually the ’s are the socalled discretized character of the Lie algebra . The fact that they provide a solution to the Dirichlet problem comes from the fact that here the restriction of the Markov kernel P to is the substochastic matrix where the ’s are level fusion coefficients of type . Let us say how the asymptotic for the number of walks in the alcoves obtained in [15] by Krattenthaler using the explicit formulas of Grabiner, follows immediately in our framework. Classically, we define a Markov kernel letting
As is supposed to be bounded, there exists a unique invariant probability measure on each communication class of and the solution of the Dirichlet problem leads in particular to an estimation of the number of walks with initial state , remaining in and ending at after steps for large . Actually one can show that the measure defined on by
, is a invariant probability measure. As the simple random walk is irreducible with period equals , one obtains the following estimation for large
where when and otherwise.
3. Basic notations and definitions
Let be a simple, connected and compact Lie group with Lie algebra and complexified Lie algebra . We choose a maximal torus of and denote by its Lie algebra. We consider the set of real roots
We choose the set of simple roots of and denote by the set of positive roots. The half sum of positive roots is denoted by . The dual coxeter number denoted by is equal to , where is the highest root. Letting for ,
the coroot of is defined to be the only vector of in such that . We denote respectively by and the root and the coroot lattice. The weight lattice is denoted by . We equip with a invariant inner product , normalized such that . The linear isomorphism
identifies and . We still denote by the induced inner product on . Note that the normalization implies . The irreducible representations of are parametrized by the set of dominant weights , where is the Weyl chamber . Let be the irreducible representation of with highest weight and be the character of this representation. It is defined by
where is defined on by for , and is the dimension of the weight space of . We denote by the dimension of the representation , i.e. . We have the following Weyl dimension formula (see for instance [12]).
The Weyl character formula states that for any ,
where is the Weyl group i.e. the subgroup of generated by fundamental reflections , , defined by , . When is not dominant, we let if for a dominant weight. The Weyl character formula remains obviously true for a nondominant weight.
The LittlewoodRichardson coefficients , for , are defined to be the unique integers such that for every
(1) 
4. Fusion coefficients
For every , we write for the translation defined on by , . For , we consider the group generated by and the translation . Actually is the semidirect product , where and . Thus for , one can define as the determinant of the linear component of . The fundamental domain for the action of on is
Let us introduce the subset of defined by
and the subset of defined by
is called the level alcove. The level fusion coefficients , for , are defined to be the unique non negative integers such that
(2) 
where is the level discretized character, which is defined by
The Weyl character formula shows that for any and
(3) 
which implies in particular that if is on a wall for some , or on the wall . Unicity of the fusion coefficients follows from the fact  proved for instance in [13]  that the vectors are orthogonal with respect to the measure defined in proposition 5.6. The non negativity of the fusion coefficients is not clear from this definition, which is the one given in [13]. Nevertheless, fusion coefficients can be seen as multiplicities in the decomposition of some ”modified products” of representations : the truncated Kronecker product, appearing in the framework of representations of quantum groups, and the fusion product, defined in the framework of representations of affine Lie algebras. In these frameworks, the non negativity of the fusion coefficients follows from the definition (see for instance [9]). Moreover, they are proved to satisfy the following inequality, which we’ll be useful for the last section. For any ,
(4) 
It follows for instance from identities (16.44) and (16.90) in [6]. Note that we have also the following inequality
It follows for instance from the Littelmann path model for tensor product of irreducible representations (see [17]).
5. Markov chains on an alcove
Let . From a probabilistic point of view, discretized characters provide, by definition of the fusion coefficients, a basis of eigenvectors of the substochastic matrix
Actually for , is an eigenvalue with a corresponding eigenvector For , is a non negative real number. Actually we have the following formula (see for instance [13]).
(5) 
The quantity is the socalled asymptotic dimension, which appears naturally in the framework of highest weight representations of affine Lie algebras. Let . We define a Markov kernel on by letting
(6) 
In other words is defined by the formula
(7) 
Definition 5.1.
For , a random walk in the level alcove, with increment , is defined as a Markov process in , with Markov kernel .
The definition of the Markov kernel implies that for , is an eigenvalue of , with a corresponding eigenvector . Thus for any positive integer , one has for
which is equivalent to say that for any ,
where the coefficients , for , are the unique integers satisfying
for any . We denote by the dimension of the weight space of , i.e.
(8) 
Let us consider a random walk on the weight lattice , whose transition kernel is defined by
We consider the subset of of weights of , i.e. . In the case when is minuscule is and the random walk is a simple random walk with uniformly distributed steps on . The following proposition states that in that case fusion coefficients give the number of ways for the walk to go from a point to another, remaining in .
Proposition 5.2.
Let and be a minuscule weight in . Then for any , is the number of walks with steps in , initial state , remaining in and ending at after steps.
Proof.
The following formula is known as the BrauerKlimyk rule. It is an immediate consequence of the Weyl character formula. For it says that
The highest weight being minuscule for every such that . Thus and for every . As when or for some simple root , we obtain that
As is minuscule . Thus
which implies the proposition. ∎
Proposition 5.2 implies that when is minuscule, the substochastic matrix
is the restriction of to the alcove . As noticed after identity (3), the discretized characters are null on the boundary of the bounded domain , which is . Thus one obtains, when is minuscule, the following important corollary.
Corollary 5.3.
Let us consider for a discrete Dirichlet problem, which consists in finding eigenvalues and eigenfunctions defined on , satisfying
where . If is minuscule then

for ,
is an eigenvalue, with a corresponding eigenfunction defined by

the eigenfunction is a PerronFrobenius eigenfunction. In particular, the random walk in a level alcove with increment is a Doobtransformed transition kernel of .
Proposition 5.2 remains true in the framework of Littelmann paths. In that framework, it includes the case of standard representation of type . In the following a path defined on , for , is a continuous function from to such that . If is a path defined on we write (resp. ) if (resp. for every . For two paths and respectively defined on and , we write for the usual concatenation of and . Note that is a path defined on . For , we denote by the dominant path defined on by and by the Littelmann module generated by . More details about the Littelmann paths model for representation theory of KacMoody algebras can be found in [17]. The important fact for us is that for any dominant and one has
Let us recall that a weight is said to be quasiminuscule if .
Proposition 5.4.
Let and be a minuscule weight or a quasiminuscule weight such that for every weights of the representation . Then for any , is the number of paths in ending on and remaining in .
Proof.
Littelmann theory implies that
When is a minuscule weight, the Littelmann module is . When is quasiminuscule every paths in the Littelmann module are of the form for or are defined by , for . Thus, if one has for every , . If then . As for all , implies for every . One obtains, ,
∎
The first formula of the following proposition is well known for . It is a consequence of the KacWalton formula (see [19]). For , proposition 5.2 implies that when is minuscule, it turns to be the KarlinMacGregor type formula obtained for affine Weyl group by Gessel and Zeilberger in [10] in the framework of reflectable walks. The second formula can be found as an exercise in chapter of [13].
Proposition 5.5.
Let be dominant weights in the alcove . Then


, where is the highest weight of the dual representation .
Proof.
The proof rests on the Weyl character formula. We let for any . We have
The Weyl character formula implies
which is an extension of the BrauerKlimyk rule. For , it exists such that . If then is on a wall for some and . If then and . If it exists two distinct such that then and . Finally if it exists a single such that and we get that
which proves the first identity. Let us prove the second one. The affine Weyl group being the semidirect product , the first identity for implies
∎
In the following proposition is the cardinal of the quotient space .
Proposition 5.6.
The measure defined on by
for any , is a invariant probability measure.
Proof.
Let us consider the measure defined on by , which is proportional to the measure . Let us show that is invariant. We have
Thus
As the longest element of send onto , , and is invariant. For a proof of the fact the is a probability measure, see for instance theorem 13.8 in [13]. ∎
Note that the probability measure is not reversible in general. It is the case when and its dual representation are isomorphic.
Classical results on convergence of Markov chain toward the invariant probability measure provides asymptotic approximation of the fusion coefficients. We let for ,
Note that the Markov kernel is not necessary irreducible and aperiodic. As all Markov chains that we’ll consider in section 6 are irreducible, we suppose that is irreducible in the following proposition.
Proposition 5.7.
Suppose that is irreducible with period . Let and be dominant weights in the alcove . Let be an integer in defined by for some integer such that . Then,

implies

Proof.
The application is non negative on . For and , we have the following equivalence
Thus the first assertion comes from usual properties of periodic Markov chains. As is a invariant probability measure, classical results on finite state space periodic Markov chains also implies
which is equivalent to the second assertion. ∎
Proposition 5.8.
Let . Suppose that be a minuscule weight or a quasiminuscule weight such that for every weight of the representation . Suppose that is irreducible with period . Then for every , the number of paths of ending on and remaining in is equivalent to
where is an integer in defined by for some integer such that .
6. Applications
In this section we explicit which fusion products have to be considered to recover reflectable random walks studied in [11]. Moreover, we explain how to get without no additional work the asymptotics obtained by Krattenthaler in [15] for the number of walks between two points remaining in an alcove. Actually our model for the type with standard steps differs slightly from the one considered by Grabiner. Moreover our models don’t include random walks with diagonal steps in an alcove of type studied in [11].
The results presented in this section only use the knowledge of the PerronFrobenius eigenfunction given by the corollary 5.3. It would be interested to consider whole the solution of the Dirichlet problem in order to study more precisely asymptotic behaviors of the conditioned chain.
Let be the standard basis of which is endowed with the standard euclidean structure denote by . The inner product identifies and its dual. In the following we consider a random walk on with standard positive steps : its steps are uniformly distributed on the set , a random walk on with standard steps : its steps are uniformly distributed on the set and a random walk , whose steps are uniformly distributed on the set of diagonal steps . The Markov kernels of and are respectively denoted by S and D.
6.1. Alcove of type A
When is the unitary group , we have , , , , and .
Positive standard steps.
The random walk can be decomposed into a deterministic walk and a random walk on the hyperplane as follows.
where and . The random walk is a deterministic random walk and is a random walk with uniformly distributed steps on . Let us denote by its Markov kernel. The standard representation is a minuscule representation. Thus by proposition 5.2, for , the Markov Kernel defined by (7) is , which is the set of weights of the standard representation of type
where
(9) 
The weights lattice is generated by is irreducible with period equals to . Let and be in . If then , where . We define the integer by . Thus proposition 5.7 implies the following asymptotic for large . . The Markov kernel
Proposition 6.1.
For large , , the number of walks with steps in to and remaining in , after steps, is equivalent (up to a multiplicative constant which doesn’t depend on ) to , going from
Diagonal steps.
The random walk can be decomposed as the previous one.
For , the th exterior power of standard representation is a minuscule representation with highest weight