Central measures on multiplicative graphs, representations of Lie algebras and weight polytopes

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Central measures on multiplicative graphs,

representations of Lie algebras and weight polytopes

Cedric Lecouvey, Pierre Tarrago

To cite this version:

Cedric Lecouvey, Pierre Tarrago. Central measures on multiplicative graphs, representations of Lie algebras and weight polytopes. 2016. �hal-01358385v3�

REPRESENTATIONS OF LIE ALGEBRAS AND WEIGHT POLYTOPES

C´EDRIC LECOUVEY AND PIERRE TARRAGO

Abstract. To each finite-dimensional representation of a simple Lie algebra is associated a multiplicative graph in the sense of Kerov and Vershik defined from the decomposition of its tensor powers into irreducible components. It was shown in [14] and [15] that the con- ditioning of natural random Littelmann paths to stay in their corresponding Weyl chamber is controlled by central measures on this type of graphs. Using the K-theory of associ- atedC^∗-algebras, Handelman [10] established a homeomorphism between the set of central measures on these multiplicative graphs and the weight polytope of the underlying repre- sentation. In the present paper, we make explicit this homeomorphism independently of Handelman’s results by using Littelmann’s path model. As a by-product we also get an ex- plicit parametrization of the weight polytope in terms of drifts of random Littelmann paths.

This explicit parametrization yields a complete description of harmonic andc-harmonic func- tions for the Littelmann paths model describing the iterated tensor product of an irreducible representation.

1. Introduction

Consider a simple finite-dimensional Lie algebra g of rank d over C and its root system in R^d. Let P be the corresponding weight lattice and fix ∆ a dominant Weyl chamber.

Then P+ = P ∩∆ is the cone of dominant weights of g. Denote by S = {α1, . . . , α_d} the underlying set of simple roots. To each dominant weight δ ∈ P₊ corresponds a finite- dimensional representation V(δ) of g of highest weight δ. In [17] Littelmann associated to every representationV(δ) a setB(δ) of paths inR^d with length 1 starting at 0 and ending in the set Π_δof weights ofV(δ). Random Littelmann paths can then be defined first by endowing B(δ) with a suitable probability distribution, next by considering random concatenations of paths in B(δ). In [14] and [15] distributions on the set B(δ) are defined as morphisms from P to R_>0. This is equivalent to associating to each simple root α_i a real t_i in ]0,+∞[. It is then shown that these random paths and their conditioning to stay in the Weyl chamber

∆ are controlled by the representation theory of g. In fact, one obtains particular central distributions on the set Γ^R_n^d of paths of any length n ≥ 1 (obtained by concatenating n paths in B(δ)). By central distributions we mean that the probability of a finite path only depends on its length and its end. Equivalently, we get a central measure on the set of infinite concatenations Γ^R of paths inB(δ) (see Section 2).

Write H(R^d) for the set of central measures on Γ^R and H(∆) for the subset of H(R^d) of central measures on Γ^∆, the set of infinite trajectories remaining in ∆. By Choquet’s Theorem both sets H(R^d) and H(∆) are simplices so they are essentially determined by their minimal boundaries ∂H(R^d) and ∂H(∆). Write K(δ) for the convex hull of Π_δ and set K(δ)⁺ = ∆∩K(δ). For walks in the Weyl chambers, the characterization of the sets

∂H(R^d) and ∂H(∆) has been obtained by Handelman in [10] and [11] using an important work of Price [20, 21], by proving that they are respectively homeomorphic to K(δ) and K(δ)⁺. Nevertheless, Handelman did not explicit the homeomorphisms. Their existence is established by considering the central measures as traces on certain C^∗-algebras and then

Date: December 2016.

1

using analytic tools. In particular, a central element of the proof is the extension of traces on C^∗-algebras using K-theory (a short explanation of these arguments is given in Section 3.4).

The goal of this paper is essentially threefold: first we make explicit both homeomorphisms by using the Weyl characters of g (see Theorem 3.1), next we give an algebraic proof of Handelman’s results and finally we connect them with more recent works on conditioned random walks or Brownian motions, generalizations of the Pitman transform and asymptotic Young tableaux (see [19], [1], [14, 15, 16], [5], [6], [24]). As a corollary of these results, we describe the set of harmonic and c-harmonic functions corresponding to the aforementioned random walks. Finally, we get a law of large numbers for random walks distributed according the central measures we obtain. Our two last results seem quite disconnected from the initial algebraic setting in representation theory, and we conjecture that they still hold for a very broad class of random paths. Our approach extends that of Kerov and Vershik to which it essentially reduces when V(δ) is the defining representation of g = sl_n. Nevertheless, numerous difficulties arise when considering the general case of dominant weights of any simple algebra g, which explains the involved machinery used in the proof of Handelman.

Our methods to determine ∂H(R^d) and ∂H(∆) are quite similar. So we will now give its main steps only in the case of∂H(∆).

We first need to show that the characterization of ∂H(∆) is equivalent to that of the extremal harmonic functions on the growth graph G(∆) associated with Γ^∆. This growth graph is rooted, graded and multiplicative: its vertices label the basis B = {(sλ, n) | V(λ) irreducible component of V(δ)^⊗n and n ≥ 1} of a commutative algebra ˆT_δ⁺ (here s_λ is the Weyl character ofV(λ)). We then establish that the extremal nonnegative harmonic functions on G(∆) are in bijection with the algebra morphisms from ˆT_δ⁺ to Rthat are nonnegative on B. Next, we prove that all these morphisms are obtained by associating to each simple root αi, i = 1, . . . , n a real in [0,1]. The difficulty here comes from the fact that two such associations can yield the same morphism. So to obtain a genuine parametrization, we need to restrict ourselves to a subset [0,1]^d_δ (see (10) for a precise definition) of [0,1]^d whose combinatorial description is in terms of the δ-admissible subsets of S introduced in [23].

Finally, in Proposition 6.4, we show that our set [0,1]^d_δ also parametrizes the simplex K(δ)⁺ by considering, for each d-tuple in [0,1]^d_δ, the drift of the corresponding random Littelmann path appearing in the construction of [14] and [15].

The paper is organized as follows. In Section 2, we recall some background on random paths and central measures on multiplicative graphs. We also apply the Ring Theorem of Kerov and Vershik to relate extremal harmonic functions on a multiplicative graph to nonnegative morphisms of the underlying algebra. The main result is written down in Section 3 where we also introduce the algebras ˆT_δ and ˆT_δ⁺; a sketch of Handelman’s arguments is proposed at the end of Section 3. Section 4 gives the description of∂H(R^d). Here, we define our set [0,1]^d_δ and relate it to the geometry of the polytope K(δ). The description of ∂H(∆) is deduced from that of ∂H(R^d) in Section 5. It is worth noticing that we need here (as in the result of Kerov and Vershik) a classical theorem relating polynomials with non positive roots to totally positive sequences. Another important ingredient in the proof is the use of certain plethyms of Schur and Weyl characters of g. Finally, Section 6 relates both descriptions of

∂H(R^d) and ∂H(∆) to the drift of random Littelmann paths. Notably it explains how the polytope K(δ) can be simply parametrized by using the set [0,1]^d_δ.

2. General probabilistic framework

We present here a general probabilistic model of random paths in a domain, which is well suited to study probabilistic aspects of Littelmann paths and their asymptotics. We introduce first a discrete version of paths in a vector space.

2.1. Random paths on a lattice. Let d≥0 and let Λ be a lattice ofR^d with rankd. We shall denote by #S the cardinality of any setS.

Definition 2.1.

(1) Let n≥0. A path π on Λ of lengthn is a piecewise linear function π : [0, n]−→R^d with π(0) = 0, π(i) ∈Λ for all i∈ {0, ..., n}, and π(x)∈ Q^d for all x on which π is not differentiable. The path π is called infinitesimal if n= 1.

(2) An infinite path on Λ is a piecewise linear function π: [0,+∞[−→R^d withπ(0) = 0, π(i)∈Λ for all i∈N, and π(x)∈Q^d for all x on which π is not differentiable.

The length of the path π on Λ is denoted by l(π) ∈ N∪ {+∞}. We write π.τ for the concatenation of two finite paths π and τ. Let X be a countable set of infinitesimal paths and let Ω be a domain of R^d such that 0 ∈ Ω; from now on, the set X is fixed and is not mentioned in the various notations. A path π is called X-valued if π is the concatenation of infinitesimal paths coming from X: equivalently, π_|[i,i+1]−π(i)

∈ X for all i ≥ 1. In the sequel, any path is always considered as X-valued. The set of infinite X-valued paths (resp. finite X-valued paths and X-valued paths of length n ≥1) whose image is included in Ω is denoted by Γ^Ω (resp. by Γ^Ω_fin and Γ^Ω_n). For x, y∈Λ, we denote by Γ^Ω(x, y) the set of infinitesimal paths π∈X such thatπ(1) =y−x and x+π⊂Ω, and we writexր y when

#Γ^Ω(x, y)6= 0. Finally, we denote by Γ^Ω_n(y) the set of finite paths of lengthn ending aty.

In order to consider random infinite paths in Ω, we need to define aσ-algebra on Γ^Ω. Let τ be a finite path of length n, and let Γ^Ω(τ) be the set {π ∈ Γ^Ω|l(π) ≥n, π_|[0,n] =τ}. We consider the coarsest σ-algebra containing all the sets Γ^Ω(τ) for τ ∈ Γ^Ω_fin. The set M1(Γ^Ω) of probability measures on Γ^Ω is considered with the initial topology with respect to the evaluation maps on the sets Γ^Ω(τ), τ ∈Γ^Ω_fin. By Tychonov’s Theorem,M1(Γ^Ω) is a compact set with respect to this topology.

2.2. Central random paths.

Definition 2.2. A probability measure P on Γ^Ω is called central if there is a function p : Λ×N−→R⁺ such that

P Γ^Ω(π)

=p(y, n),

for all π ∈ Γ^Ω_n(y) with y ∈ Λ, n ≥ 0. A random path in Γ^Ω is called central if the induced measure is central.

Similarly, we could have defined central measure on Γ^Ω_fin by similar means. It is then easily seen that any central measure supported on Γ_fin(Ω) is a convex combination of uniform measures on the sets Γ^Ω_n(y) withy∈Λ andn≥1. Therefore, the main interesting phenomena arise for central measures supported on Γ^Ω.

The set of central measures supported on Γ^Ω is denoted byH(Ω).

LetP∈ H(Ω). Then, by Definition 2.2 there exists a function p: Λ×N−→R⁺ such that P(Γ^Ω(π)) =p(y, n),

for all π ∈ Γ^Ω_n(y) with y ∈Λ and n≥ 1. Letx ∈ Λ, and suppose that π is a finite path in Γ^Ω_n(x). A path τ of length n+ 1 ending aty ∈Λ satisfiesτ_|[0,n]=π if and only if τ_|[0,n]=π and τ_[n,n+1]is an infinitesimal path joining xto y. Therefore, Γ^Ω(π) can be decomposed as

Γ^Ω(π) = a

y∈Λ

a

τ∈Γ^Ω(x,y)

Γ^Ω(π.τ).

Thus,

P Γ^Ω(π)

=X

y∈Λ

X

τ∈Γ^Ω(x,y)

P Γ^Ω(π.τ) ,

which translates into the relation

(1) p(x, n) =X

y∈Λ

#Γ^Ω(x, y)p(y, n+ 1)

forx∈Λ∩Ω such that Γ^Ω_n(x)6=∅. Describing the set of solutions to (1) is very complicated in general. It is known however thatH(Ω) is a convex subset ofM₁(Γ^Ω) and even a Choquet simplex.

Definition 2.3. The minimal boundary of Γ^Ω is the unique subset∂H(Ω)⊂ H(Ω), such that any central measure P₀ in H(Ω)admits a unique integral representation

P₀ = Z

∂H(Ω)

Pdµ(P),

where µ is a probability measure on the set ∂H(Ω).

2.3. Central measures and Doob’s conditioning. We establish here some connections between central measures on random paths and random walks on lattices. Indeed, any random path π ∈ Γ^Ω following a central measure P ∈ H(Ω) yields a random walk Z = (π(0) = 0, π(1), . . .) on the lattice Λ∩Ω. The family of Markov kernels (Q_n)n≥0 ofZ can be explicitly given from the function p: Λ×N−→R⁺ associated to to the central measure P.

Indeed one can show that

Q_n(x, y) =1_p(x,n)6=0#Γ^Ω(x, y)p(y, n+ 1) p(x, n) . By the equality p(x, n) = P

y∈Λ#Γ^Ω(x, y)p(y, n+ 1), Q_n is a well-defined Markov kernel.

Note that this random walk is generally not homogeneous in time, since the kernelQ_ndepends on nthrough p.

Doob’s conditioning is a standard way to produce random walks on Λ∩Ω coming from central measures. Let Z be the random walk onR^d starting at 0 and with Markov kernel

P(Z_n+1 =y|Zn=x) = Γ^R(x, y)

#X

for any x, y ∈ Λ. Note that this random walk actually comes from the central measure P whose value is

P(Γ^R(π)) = 1 (#X)ⁿ for all π∈Γ^R_n^d with n≥0.

Definition 2.4. Let c > 0. A function h : Ω∩Λ −→ R⁺ is a c-harmonic function for the random path Z killed when exiting Ω if and only if

h(x) = 1 c(#X)

X

x+π⊂Ωπ∈X

h(x+π(1)) = 1 c(#X)

X

y∈Λ∩Ω

#Γ^Ω(x, y)h(y).

Letc >0 and assume thathis a c-harmonic function forZ. Then, the Doob conditioning Z^h ofZ in Ω is given by the Markov kernel

P(Z_n+1^h =y|Z_n^h=x) =1_h(x)>0 1 c(#X)

#Γ^Ω(x, y)h(y) h(x) ,

for y ∈ Λ∩Ω. The random walk Z^h is well-defined because h is c-harmonic and is time homogeneous. This random walk comes from the central measureP^h whose value is

P^h(Γ^Ω(π)) = 1

c(#X) n

h(y)

forπ∈Γ^Ω_n(y) withn≥0 andy∈Λ∩Ω.

Conversely, suppose that P is a central measure with an associated function p such that p(x, n) =aⁿh(x) for some functionh: Λ∩Ω−→R⁺ and a >0. Then, (1) yields

aⁿh(x) =X

y∈Λ

#Γ^Ω(x, y)aⁿ⁺¹h(y), which is equivalent to the relation h(x) = aP

y∈Λ#Γ^Ω(x, y)h(y). Thus, the function h is (a#X)⁻¹-harmonic.

Hence, the set ofc-harmonic functions forZ is homeomorphic to the set of central measures P ∈ H(Ω) whose associated functions p have the form p(x, n) =

1

#X·c

n

h(x) with h : Λ∩Ω−→R⁺.

For any real c >0, denote by Hc(Ω) the set of central measures coming fromc-harmonic functions. A quick computation shows that the random walkZ induced by a central measure Pis time homogeneous if and only if P∈ Hc(Ω) for somec >0.

It is easily seen that Hc(Ω) is a convex subset ofH(Ω) and we denote by∂Hc(Ω) the set of extreme points of Hc(Ω). To the best of our knowledge, there is no general proof that

∂H_c(Ω) =∂H(Ω)∩ H_c(Ω); in particular, an answer to the following problem yields a nice description of∂H(Ω).

Problem. Do we have the decomposition ∂H(Ω) =⊔c>0∂Hc(Ω) ?

In our case of study, this equality is proven by explicitly describing both sets (see Sec- tion 3.3). As we shall explain in the following we do need here to consider the closure of

⊔_c>0∂H_c(Ω).

2.4. Central measures on multiplicative graphs. We investigate here the general solu- tion of (1) when the path concatenation on Γ^Ω_fin encodes the multiplicative structure of an algebra. Let us first give a brief overview of the general theory in the setting of graded graphs before applying it to our situation.

So consider a rooted graded graph G = {∗} ⊔`

n≥1Gn where Gn is the set of vertices in level n (G0 = {∗} by convention). For any n ≥ 0, we can only have directed weighted arrows between vertices λ ∈ Gn and µ ∈ Gn+1 with weight e(λ, µ). Such a graph is called multiplicative if there exists a commutative algebraA and an injective mapι:G −→A such that ι(λ)ι(∗) = P

λրµe(λ, µ)ι(µ). Here λ ր µ means we consider all the neighbors µ of the vertex λ. We suppose that the graph is connected, which means that for all µ∈ G, the number of paths between the root and µ is positive. The weight e(π) of a path π between the root and a vertex µ is the product of all the weights of the edges of π.

Let K be the positive cone spanned by ι(G), and let AG be the unital subalgebra of A generated by K. Denote by Mult⁺(A_G) ⊂ A^∗_G the set of multiplicative functions on A_G which are nonnegative on K and equal to 1 on ι(∗). Note that ι:G −→A_G induces a map ι^∗ :A^∗_G −→F(G,R) such thatι^∗(φ) = φ◦ιfor any linear map φ:A_G → R. Now denote by H(G) the set of functions p:G −→R⁺ such that

(2)

p(∗) = 1,

p(λ) =P

λրµe(λ, µ)p(µ) for any λ∈ G.

We can characterize the setδH(G) of extremal points in H(G):

Proposition 2.5. Suppose that K.K ⊂ K. Then, the map ι^∗ yields an homeomorphism between Mult⁺(AG) and the set δH(G).

The proof of this proposition is an application of the Ring Theorem of Kerov and Vershik.

Theorem 2.6. [8, Section 8.4]Let B be a unital commutative algebra over R andK ⊂B a convex cone satisfying the following conditions:

• K−K =B (K generates B).

• K.K ⊂K (K is stable by multiplication).

• K is spanned by a countable set of elements.

• For all a∈B, there exists ǫ >0 such that 1−ǫa∈K.

If L denotes the convex set of linear forms on B which are nonnegative on K and map 1_B to 1, then φ is an extreme point of L if and only if φ is multiplicative (meaning that φ(ab) =φ(a)φ(b) for all a, b∈B).

We give now the proof of Proposition 2.5.

Proof. Let B = A_G/hι(∗) = 1i and let pr : A_G −→ B be the canonical projection; denote by ˜K the projection of the cone R⁺Id +K in B. Since K.K ⊂ K and {1, K} spans A_G, K.˜ K˜ ⊂ K˜ and ˜K spans B. Since G has a countable set of vertices, ˜K is spanned by a countable set of elements. Note that there is a bijection between the elements of H(G) and the linear forms on B which are nonnegative on ˜K and equal to 1 on 1: indeed h ∈ H(G) if and only if h(µ) = P

µրνe(µ, ν)h(ν). Thus, for f ∈ A^∗_G, ι^∗(f) ∈ H(G) if and only if f(ι(∗)ι(µ)) = f(ι(µ)); equivalently, this means that f factors through B. The fact ι^∗(f) is nonnegative on G is then equivalent to the fact it is nonnegative on ˜K. We also have [ι^∗(f)](∗) = 1 if and only if f(pr◦ι(∗)) =f(1) = 1.

Let a∈B, and let us show that there exists ǫ such that 1−ǫa ∈K. Since ˜˜ K−K˜ =B, and 1−b ∈ K˜ for all b ∈ −K, we can suppose without loss of generality that˜ a ∈ K˜. It is thus enough to prove that for µ ∈ G, there exists ǫ such that 1−ǫpr◦ι(µ) ∈ K. Suppose that µ has rank n. Since the graph is connected, there exists a path π₀ of weight e(π₀) between ∗and µ. By iteration of the relation coming from the multiplicative structure of G, ι(∗)ⁿ = P

ν∈G rk(µ)=n

(P

π:∗→µe(π))ι(ν). Thus ι(∗)ⁿ − e(π₀)ι(µ) belongs to K. Since pr(ι(∗)ⁿ) = 1, 1−e(π₀) pr◦ι(µ) belongs to ˜K. Therefore, we can apply Theorem 2.6 to (B,K), which yields that the extreme linear maps among the set of linear maps on˜ B which are nonnegative on ˜Kand equal to 1 on 1 are the multiplicative ones. Since there is a bijection between multiplicative maps on B which are nonnegative on ˜K and multiplicative maps on AG which are nonnegative onK and equal to 1 onι(∗), the proof is complete.

In order to apply the previous result to central random paths on Λ∩Ω, we need to relate Γ^Ω to a graded graph.

Definition 2.7. The growth graph ofΓ^Ω is the rooted graded graph G(Ω)with

• set of vertices of rank n: the set Gn(Ω) ={(x, n)∈Λ∩Ω×N,Γ^Ω_n(x)6=∅},

• a directed weighted edge between (x, n)∈ Gn(Ω)and(y, n+ 1)∈ Gn+1(Ω)with weight

#Γ^Ω(x, y).

It is readily seen that the weighted sum on paths inG(Ω) between the root (0,0) and (y, n) is equal to #Γ^Ω_n(y). Moreover, the sets H(Ω) and H(G(Ω)) are canonically homeomorphic through the equivalence between (1) and (2).

3. Littelmann paths in Weyl chambers

We describe a class of random paths coming from the representation theory of semi-simple Lie algebras.

3.1. Background. We consider a simple Lie groupGoverCand its Lie algebrag. LetR ⊂V be the set of roots ofgregarded as a finite subset of the Euclidean vector spaceV with scalar product h·,·i. We fix R+ a subset of positive roots and S = {α1, α2, . . . , α_d} ⊂ R+ a basis of simple roots in R. The Weyl group of g is denoted by W. This is the Coxeter group generated by the reflections s_αi associated to the simple roots. Thus for any x∈E and any α∈S, we have

(3) s_α(x) =x−2hα, xi

hα, αi α Denote by ℓthe length function on W defined fromS.

WriteP for the weight lattice ofgandω1, . . . , ω_dfor its fundamental weights so that we have P =

d

M

i=1

Zω_i.

Let us denote by ≤ the dominant order on P such that γ ≤ γ^′ if and only if γ^′ −γ is a sum of simple roots. Let ∆ be the fundamental Weyl chamber ofg with respect toS, which corresponds to the positive orthant on the weight space L_d

i=1Rω_i. The cone of dominant weights is then

P⁺=P∩∆ =

d

M

i=1

Z_≥0ω_i.

WriteQ⁺the subset ofP spanned by linear combinations of the simple roots with nonnegative coefficients. We denote by R[P] the ring group of P over R with basis{e^β |β ∈P}, and by R[Q⁺] the subalgebra of R[P] generated byQ⁺. Then

R^W[P] ={u∈R[P]|w(u) =u, w∈W}

is the character ring of g. To each λ∈P⁺ corresponds a simple finite-dimensional represen- tation of gwe denote by V(λ). The Weyl character of V(λ) is

s_λ =X

γ∈P

K_λ,γe^γ

where K_λ,γ is the dimension of the weight space γ in V(λ). For t ∈ R^d_≥0 and γ ∈ P with γ = Pd

i=1γiωi, set t^γ = Q

1≤i≤dexp(γilog(ti)), with the convention t^γ = +∞ when there exists 1≤i≤dsuch thatt_i = 0 andγ_i <0. It is then possible to evaluates_λont∈(R⁺)^das s_λ(t) =P

γ∈PK_λ,γt^γ. Hence, with our convention, s_λ(t) = +∞as soon as someti vanishes, since for any λ∈P⁺ and any 1≤i≤d, there exists γ ∈P such that K_λ,γ 6= 0 and γ_i <0.

Forµ≥λ, denote byS_λ,µ the function

(4) S_λ,µ=e^−µs_λ= X

γ∈P

K_λ,γe^γ−µ

where for any γ such that K_λ,γ >0, γ−µ is a linear combination of the simple roots with nonpositive coefficients; forµ=λ, we simply writeS_λ, instead of S_λ,λ. By setting T_i =e^−αi we thus obtain that S_λ,µ = S_λ,µ(T₁, . . . , T_d) is polynomial in the variables T₁, . . . , T_d with nonnegative integer coefficients. Recall also the Weyl dimension formula

dim(V(λ)) = Y

α∈R+

(λ+ρ, α) (ρ, α) .

In particular, dim(V(λ)) is polynomial in the coordinates of λon the basis of fundamental weights.

3.2. Random Littelmann paths. Now, fix a dominant weight δ ∈ P⁺ and denote by Π_δ the set of weights of the irreducible representation V(δ). Let P_δ be the sublattice of P generated by Π_δ. This defines subalgebras

R[P_δ] ={e^β |β ∈P_δ} ⊂R[P] and R^W[P_δ] ={u∈R[P_δ]|w(u) =u} ⊂R^W[P].

Finally write T_δ⁺ the subset of P⁺ of weights λ such that V(λ) appears as an irreducible component in a tensor power V(δ)^⊗n, n≥0. Givenλand µ inT_δ⁺,we clearly have λ+µin T_δ⁺.

Now letAδ be the subalgebra ofR^W[P] generated by the Weyl character s_λ withλ∈T_δ⁺. We have the inclusions

Aδ ⊂R^W[P_δ]⊂R[P_δ]⊂R[P].

We denote by K(δ) the convex hull of the set Π_δ: K(δ) is a polytope whose extreme points are the elements w(δ) for w ∈ W. The intersection of K(δ) with the Weyl chamber ∆ is denoted by K(δ)⁺. By Littelmann’s paths theory, there is a set B(δ) ={πi}1≤i≤dimV(δ) of infinitesimal paths onP_δ, with the following properties:

• π_i(1)∈Π_δ for all 1≤i≤dimV(δ),

• the multiplicity of the weight µinV(δ)^⊗n is equal to #Γ^R_n^d(µ),

• the multiplicity of the irreducible representation V(ν) in V(µ) ⊗V(δ) is equal to

#Γ^∆(µ, ν) and the multiplicity of the irreducible representation V(ν) in V(δ)^⊗n is equal to #Γ^∆_n(ν) for all µ, ν ∈P⁺ and n≥0.

The set of infinite paths we are interested in is the set of infinite paths starting at 0 with set of infinitesimal paths B(δ).

3.3. Statements of the results. We recall that we consider the space of probability mea- sures on each Γ^∆with the initial topology with respect to the evaluation maps on the cylinders Γ^∆(τ), τ ∈ Γ^∆_fin. We give an algebraic proof of the identification of the minimal boundaries for random paths in Γ^R and Γ^∆ with the topological spaces K(δ) and K(δ)⁺, respectively.

In both cases, the homeomorphism can be made explicit by the introduction of a natural parametrization

t:K(δ)−→[0,1]^d×W

of K(δ) such that t(K(δ)⁺) ⊂[0,1]^d×IdW (this parametrization is explained in Section 5).

For m∈K(δ), we set t(m) = (t_m, w_m). The main result of the paper is summarized in the following theorem:

Theorem 3.1.

(1) The map P:

K(δ) −→ ∂H(R^d)

m 7→ P_m with P_m(Γ^R(π)) = t^nδ−w_m ^m(λ)

S_δ(t_m) for n≥0 and π ∈Γ^R_n^d(λ) is a homeomorphism between the set of extremal measures ∂H(R^d) andK(δ).

(2) The map P⁺:

K(δ)⁺ −→ ∂H(∆)

m 7→ P⁺_m with P⁺_m(Γ^∆(π)) = S_λ,nδ(tm)

S_δ(t_m)ⁿ for n≥0 and π∈Γ^∆_n(λ) is homeomorphism between the set of extremal measures ∂H(∆) and K(δ)⁺.

It is easy to see that the measuresP_m and P⁺_m are indeed central. Note moreover that for m∈K(δ)⁺, Littelmann’s theory yields that forπ ∈Γ^∆_n(y),

X

˜

π∈Γ^∆_n+1,˜π_|[0,n]=π

P⁺_m(Γ^∆(˜π)) = 1 S_δ(t_m)ⁿ⁺¹

X

µ∈B(δ),π.µ∈Γ^∆_n+1(y)

Sπ(n)+µ(1),nδ+x(t_m)

= 1

S_δ(t_m)ⁿ⁺¹S_π(n),nδ(t_m)S_δ(t_m) = S_π(n),nδ(t_m)

S_δ(t_m)ⁿ =P⁺_m(Γ^∆(π)), so that P⁺_m is a well defined probability measure on Γ^∆. The main point of the result is to prove that Pand P⁺ are bijective.

Remark 3.2. In type A_d, when δ=ω₁ is the first fundamental weight,V(δ) can be regarded as the defining representation ofsl_d+1 or more conveniently, ofgl_d+1. The set∂H(∆) is then homeomorphic to

K(δ)⁺={(p₁, . . . , p_n+1)∈R^d+1 |p₁ ≥ · · · ≥p_n+1≥0 andp₁+· · ·+p_n+1= 1}

and we recover the finite-dimensional version of the Thoma simplex.

As a corollary of Theorem 3.1, we get the complete characterization ofc-harmonic measures killed when exiting ∆. Define the function ˆs_δ:∂H(∆)−→R⁺∪ {∞} by ˆs_δ(P⁺_m) =s_δ(t_m).

Corollary 3.3. Forc >0, the set ∂H_c(∆) is equal tosˆ⁻¹_δ ({cdimV(δ)}). In particular,

• H1(∆) = ∂H1(∆) is a singleton corresponding to P⁺0,

• and for c <1, Hc(∆) =∅.

This corollary gives a positive answer to the our Problem 2.3. We prove Corollary 3.3 in Section 6.3. We discuss here a possible generalization of the latter result. Let X be an arbitrary multiset of infinitesimal paths (or alternatively a weight set of such paths). Set Z(t) :=ˆ P

π∈Xt^π(1) and z= ˆZ(1). Finally, fix a cone C centered at 0 and denote by K_C the set of elements t∈R^d such thatP

π∈Xt^π(1)π(1)∈ C.

A function f isc-harmonic for these paths if and only if f(x) = X

x+π⊂Cπ∈X

1

czwt(π)f(x+π(1)).

We can use the same notation as in the case whereX=B(δ) is the a set of Littelmann paths associated to δ. Then, we conjecture that the following general result holds:

Conjecture 3.4. For c > 0, the set ∂Hc(∆) is homeomorphic to Zˆ⁻¹({cz}) ∩ K_C. In particular,

• for u= min_KCZˆ, H_u/z(∆) is a singleton.

• and for c < u/z, Hc(∆) =∅.

This conjecture is a generalization of the conjecture of Raschel [22, Conjecture 1] for two dimensional random walks with bounded increments, which asserts that such a random walk admits a unique harmonic function killed on the boundary of a quarter plane. This special situation can be seen in the above conjecture, in which case the minimum of ˆZ is exactly z.

3.4. The approach of Handelman and Price. The existence of the homeomorphisms of Theorem 3.1 can also be deduced from the main results of [10, 11], themselves based on fundamental results of [20, 21]. We review here their approach, and the reader could read the aforementioned articles and references therein for a detailed proof.

Let n denote the dimension of V(δ), and consider the adjoint representation ρ : G −→

GL(M_n(C)) which is defined byρ(g)(M) =u_δ(g)M u_δ(g)⁻¹, where u_δ is the irreducible rep- resentation associated withδ. Form the infinite tensor productA:=N

Mn(C) as an inductive limit of the sequence of finite-dimensional C^∗-algebras (M_n(C)^⊗k)_k≥1, where M_n(C)^⊗k em- beds in M_n(C)^⊗k+1 with the map X 7→ X⊗Id_n. We can canonically associate a structure of C^∗-algebra to this inductive limit of C^∗-algebras. Then, G acts continuously on each Mn(C)^⊗k and on A with the map ˜ρ(g) :=N

ρ(g) (which means that g acts asρ(g) on each component of the tensor product), and we can therefore consider the C^∗-algebra A^δ (resp.

A^δ_k) of elements ofA(resp. M_n(C)^⊗k) fixed byρ. The algebraA^δis the inductive limit of the finite-dimensional C^∗-algebras (A^δ_k)_k≥1, and the Bratteli diagram of this inductive limit is exactly the growth graph of Γ^∆. Therefore, the set of central measures on Γ^∆is in bijection with the set of traces on A^δ.

Doing the same construction for the restriction of the representation δ to the maximal torus T ⊂G, we get another sequence of finite dimensionalC^∗-algebras (A^T_k)_k≥1, whose inductive limit is denoted by A^T. Similarly, the Bratteli diagram ofA^T is exactly the growth graph of Γ^R, and the set of central measures on Γ^R is in bijection with the set of traces onA^T. Note that we have the natural inclusion of C^∗-algebras A^δ ⊂A^T. The main result of [10] is that any extremal trace onA^δ extends to an extremal trace onA^T. To prove this, the author uses the bijection between the set of traces on an approximately finiteC^∗-algebraA and the set of states on its associated dimension group K₀(A). Let us quickly explain the nature of K0(A): a dimension group is a group with a notion of positive cone. By considering equiv- alence classes of projections on the ∗-algebra L

k≥1M_k(A), one can canonically associate a dimension group K₀(A) to each C^∗-algebra A; this dimension group is always a ring in our case. An important fact is that an inclusion of C^∗-algebras induces an inclusion of the asso- ciated dimension groups, and therefore the problem reduces to extend any state on K₀(A^δ) to a state on K₀(A^T). Handelman managed to prove this in [10], and the main ingredient of the proof is the non-trivial property thatK₀(A^T) is a finitely generated K₀(A^δ)-module.

Once proven that any trace onA^δextends to a trace onA^T, the problem amounts to describe the set of traces on A^T. In [11], the author achieves this by proving that the set of faithful traces on A^T is in bijection with the interior of K(δ). Then, the identification of the set of faithful traces on A^δ with the interior of K(δ)⁺ is done thanks to a result of [21], which asserts that the Weyl group W acts transitively on the set of traces extending a particular faithful trace onA^T. Finally, the case of non-faithful traces is done by considering parabolic subgroups of G.

3.5. The extended algebra of characters. Our proof of Theorem 3.1 will mainly use algebraic properties of the representations of the Lie algebra g. We define the extended algebra of characters ˆA_δ as follows:

• Aˆδ is isomorphic to Aδ×R[T] as a vector space; for x ∈ Aδ, we simply denote by (x, n) the element (x, Tⁿ). A basis of ˆAδ is given by the set B = {(sλ, n)}_n≥0,λ∈T⁺

δ

and the multiplicative structure of ˆAδ is defined onB by the product (s_λ, n)×(s_µ, m) = (s_λs_µ, n+m).

• We also denote by ˆT_δ⁺ the subalgebra of ˆAδ spanned by {(sλ, n)|λ ∈ T_δ,n⁺} where T_δ,n⁺ is the set of dominant weights λ such that V(λ) is an irreducible component of V(δ)^⊗n.

Likewise, we define the extended algebra of weights ˆP_δ as follows

• Pˆ_δ is isomorphic to R[P_δ]×R[T] as a vector space. A basis of ˆP_δ is given by the set {(e^γ, n)|n≥0, γ ∈P_δ}. The multiplicative structure of ˆP_δ is defined by the product

(e^γ, n)×(e^γ′, m) = (e^γ+γ′, n+m).

WriteT_δ,nfor the set of weightsγ appearing with nonzero multiplicity in the representation V(δ)^⊗n. We shall also need the algebra ˆT_δ defined as follows

• Tˆ_δ the is subalgebra of ˆP_δ spanned by the elements {(e^γ, n)|n≥1, γ∈T_δ,n}.

Note that the inclusion A_δ ⊂ R[P_δ] translates naturally into the inclusion ˆA_δ ⊂ Pˆ_δ and Tˆ_δ⁺⊂Tˆ_δ.

We can write the multiset of weights of δ in ˆT_δ as Π_δ = {(e^γ1,1), . . . ,(e^γN,1)} where each weight appears a number of times equal to its multiplicity. For any k = 0, . . . , N, let e_k(X₁, . . . , X_N) be the k-th elementary symmetric function in the variables X₁, . . . , X_N. Define the polynomial Φ(X)∈Tˆ_δ[X] by

Φ(X) = Y

γ∈Πδ

(X+ (e^γ,1)).

Proposition 3.5. We have

(5) Φ(X) =

N

X

k=0

(e_k(e^γ1, . . . , e^γN), k)X^N−k

and for anyk= 0, . . . , N,the expression(e_k(e^γ1, . . . , e^γN), k)decomposes as a sum of elements (s_λ, n)∈Tˆ_δ⁺ with positive integer coefficients. In particular, we have Φ(X)∈Tˆ_δ⁺[X].

Proof. Recall thate_k((e^γ1,1), . . . ,(e^γN,1)) is thek-th elementary symmetric function in mi- nus the roots of the polynomial Φ. Hence, e_k(e^γ1, . . . , e^γN) is thus the plethysm of the elementary symmetric functione_k by s_δ. This means that

e_k(e^γ1, . . . , e^γN) = char

k

^V(δ)

!

is the character of the k-th exterior power of the representation V(δ). Since

k

VV(δ) is a submodule ofV(δ)^⊗k, its character indeed decomposes as a sum of characters in{sλ |V(λ)∈

V(δ)^⊗k} with positive integer coefficients.

Corollary 3.6. Tˆ_δ⁺ is integrally closed in Tˆ_δ.

Proof. Let ˆT_δ⁺ denote the integral closure of ˆT_δ⁺ in ˆT_δ. We have ˆT_δ⁺ ⊂ Tˆ_δ by definition.

Conversely, since ˆT_δ⁺ is a ring and ˆT_δ is generated by the monomials (e^γ,1) with γ ∈ Π_δ, it suffices to prove that each such (e^γ,1) belongs to ˆT_δ⁺. But−(e^γ,1) is a root of Φ(X) which is, by the previous proposition, a monic polynomial with coefficients in ˆT_δ⁺. Therefore −(e^γ,1) and (e^γ,1) are integers over ˆT_δ⁺ and thus belong to ˆT_δ⁺.

4. Minimal boundary of Γ^R

4.1. Algebraic description of the growth graph. LetG(R^d) be the growth graph of Γ^R and G(∆) be the one of Γ^∆. Namely, the setGn(R^d) of vertices of ranknof the graph G(R^d) are pairs (γ, n) where γ is a weight of P_δ such that Γ^R_n^d(γ) 6=∅, and the weight of the edge between (γ, n) and (γ^′, n+ 1) is #Γ^R(γ, γ^′). From the graph embedding of Section 2.4, the set of extreme central measures on Γ^R is in bijection with the set of extreme points of the

convex set ∂H(G(R^d)) of nonnegative harmonic functionsp:G(R^d)−→R⁺ with p(0,0) = 1 and the same holds for G(∆). An important feature of G(R^d) is that this graded graph is multiplicative: it is related to the algebra ˆT_δ as follows.

Proposition 4.1. G(R^d) is a multiplicative graph associated with the algebra Pˆ_δ with the injective map

ι:







G(R^d) −→ Pˆ_δ (γ, n) 7→ (e^γ, n), n≥1

7→ (s_δ,1) ,

and ι(G(R^d)) = ˆT_δ. In particular, ∂H(G(R^d)) is homeomorphic to Mult( ˆT_δ)⁺ through the map

ι^∗ :







Mult( ˆT_δ)⁺ −→ ∂H(G(R^d))

f 7→ f ◦ι

Proof. Since #Γ^R(γ, γ^′) =K_δ,γ^′_−γ, the following equality holds for (γ, n)∈ Gn(R^d):

ι(γ, n)ι(∗) = (e^γ, n)(X

κ∈Πδ

K_δ,κe^κ,1) = X

κ∈Πδ

K_δ,κ(e^γ+κ, n+ 1)

= X

γ^′∈Pδ,γ^′−γ∈Πδ

K_δ,γ^′_−γ(e^γ′, n+ 1)

= X

γ^′∈Pδ

#Γ^R(γ, γ^′)ι(γ^′, n+ 1).

Thus, G(R^d) is a multiplicative graph associated with ˆP_δ through the map ι. Note that by construction, the sub-algebra of ˆP_δ generated by the elements {ι(γ, n)}_(γ,n)∈G(R^d₎is precisely Tˆ_δ: the last part of the proposition is deduced from Proposition 2.5.

4.2. Characterization of the multiplicative maps on Tˆ_δ. The set of extreme central measures on G(R^d) is thus given by the set of positive morphisms from ˆT_δ to R which take the value 1 on (s_δ,1). We will prove in this subsection the following result:

Proposition 4.2. Let f ∈Mult( ˆT_δ)⁺. There exists a multiplicative map φ:R[Q⁺]−→ R⁺ and an element w∈W such that

f(e^γ, n) = 1

φ(S_δ)ⁿφ(e^nδ−w(γ)), for all (e^γ, n)∈Tˆ_δ.

Note that the elementφ(e^nδ−w(γ)) is well-defined: indeed, if (e^γ, n)∈Tˆ_δ, then the weight γ appears in the representation V(δ)^⊗n andw(γ) is thus smaller thannδ with respect to the roots order relative to the set of simple rootsS. Therefore, nδ−w(γ)∈Q⁺.

Letf be a multiplicative map on ˆT_δ. Sincef is multiplicative and ˆT_δ is generated by the set ˜Π_δ := {(e^γ,1), γ ∈ Π_δ}, f is completely determined by its values on ˜Π_δ. We can also extend naturally the action ofW by setting w.(e^γ, n) = (e^w(γ), n). We suppose from now on thatf ∈Mult( ˆT_δ)⁺. Let

(6) M_f = X

γ∈Πδ

K_δ,γf(γ,1)γ.

The vector M_f belongs toR^d, thus there exists w ∈W such that w(M_f) ∈∆. Replacing f by f◦w⁻¹ gives another multiplicative map on ˆT_δ such that

M_f◦w⁻¹ = X

γ∈Πδ

K_δ,γ(f◦w⁻¹)(e^γ,1)γ ∈∆ and we have f = (f ◦w⁻¹)◦w.

Lemma 4.3. Assume that M_f ∈∆ and letα∈S. For allγ ∈Π_δ such that f(e^γ,1) = 0 =⇒f(e^γ−α,1) = 0.

In particular, f(e^δ,1)6= 0.

Proof. It is a classical result in the representation theory ofg that for any weightγ ∈Π_δ and any simple root α ∈S such that γ −α /∈ Π_δ, we must have hγ, αi ≤ 0. Also if both γ and γ−α belong to Π_δ butγ−2α does not, one hashγ−α, αi<0.

Now let α ∈ S, and suppose that there exists γ ∈ Π_δ such that γ −α ∈ Π_δ, f(e^γ,1) = 0 and f(e^γ−α,1) 6= 0. If γ^′ is another weight of Π_δ such that f(e^γ′,1) 6= 0, then necessarily γ^′−α6∈Π_δ: indeed, ifγ^′−α∈Π_δ, then

f(e^γ′−α,1)f(e^γ,1) =f(e^{γ+γ′−α},2) =f(e^γ−α,1)f(e^γ′,1)6= 0,

which contradicts the fact that f(e^γ,1) = 0. For all γ^′ ∈ Π_δ, γ^′ −α 6∈ Π_δ implies that hγ^′, αi ≤0: thus,f(e^γ′,1)6= 0 implies that hγ^′, αi ≤0. We get

h X

γ^′6=γ−α

K_δ,γ^′f(e^γ′,1)γ^′, αi ≤0.

Since f(e^γ−α,1) 6= 0 and from the previous argument, we get γ −2α 6∈ Π_δ. This impose hγ−α, αi<0 as claimed below. Finally,

hMf, αi=K_δ,γ−αf(e^γ−α,1)hγ−α, αi+ X

γ^′6=γ−α

K_δ,γ^′f(e^γ′,1)hγ^′, αi<0,

which contradicts the fact thatM_f ∈∆. Letγ ∈Π_δ be such thatf(e^γ,1)6= 0. Sinceγ ∈Π_δ, there exists a finite sequence (xi)1≤i≤r in S such that δ −Pj

i=1xi ∈ Π_δ for all 1 ≤ j ≤ r and δ−Pr

i=1x_i = γ. Thus, from the first part of the lemma, f(e^δ−Pji=1xi,1) 6= 0 for all 1 ≤ j ≤ r; in particular, f(e^δ−x1,1) 6= 0, and applying again the first part of the lemma

yields thatf(e^δ,1)6= 0.

We can now prove Proposition 4.2:

Proof of Proposition 4.2. Let f ∈ Mult( ˆT_δ)⁺ be such that M_f ∈ ∆. Let α ∈ S. If for all γ ∈Π_δsuch thatf(e^γ,1)6= 0 we have γ−α6∈Π_δ, then setφ(e^α) = 0. Otherwise, let γ∈Π_δ be such thatf(e^γ,1)6= 0 and such thatγ−α∈Π_δ, and setφ(e^α) = ^f(e_f(e^γ−αγ,1)^,1). Then,φ(e^α) is independent of the choice ofγ. Indeed, ifγ^′is another weight satisfying the same hypothesis, then

f(e^γ,1)f(e^γ′−α,1) =f(e^{γ+γ′−α},2) =f(e^γ−α,1)f(e^γ′,1), so that finally,

f(e^γ−α,1)

f(e^γ,1) = f(e^γ′−α,1) f(e^γ′,1) .

Note that we have in particular proven that for all γ ∈ Π_δ such that γ +α ∈ Π_δ and f(e^γ+α,1)6= 0, we have

(7) f(e^γ,1)

f(e^γ+α,1) =φ(e^α).

Let φ:R[Q⁺]−→ R⁺ be the multiplicative map obtained by extending multiplicatively the map φ defined on {e^α, α ∈ S} and by specifying the value φ(1) = 1. Recall the dominant order ≤ on the weight lattice P: γ ≤γ^′ if and only if γ^′−γ is a sum of simple roots. Let us prove by induction on the dominant order that f(e^γ,1) = f(e^δ,1)φ(e^δ−γ) for γ ∈ Π_δ. For γ = δ the result is straightforward. Let γ ∈ Π_δ and suppose that the result is true for all γ^′ > γ. There exists α ∈ S such that γ +α ∈ Π_δ. If f(e^γ+α,1) = 0, then f(e^γ,1) = 0 by Lemma 4.3; in particular, f(e^γ,1) = φ(e^α)f(e^γ+α,1). By the induction hypothesis, f(e^γ+α,1) =f(e^δ,1)φ(e^δ−(γ+α)), and finally,

f(e^γ,1) =φ(e^α)f(e^δ,1)φ(e^δ−(γ+α)) =f(e^δ,1)φ(e^δ−γ).

Iff(e^γ+α,1) 6= 0, then by (7) and by the induction hypothesis,

f(e^γ,1) =φ(e^α)f(e^γ+α,1) =φ(e^α)f(e^δ,1)φ(e^δ−(γ+α)) =f(e^δ,1)φ(e^δ−γ).

Let (γ, n) ∈Tˆ_δ, and letγ₁, . . . , γ_n∈Π_δ such that γ =P_n

i=1γ_i. Then, by multiplicativity of f and the result above, we have

f(e^γ, n) =f(e^Pni=1γi, n) =

n

Y

i=1

f(e^γi,1) =

n

Y

i=1

f(e^δ,1)φ(e^δ−γi)

=f(e^δ,1)ⁿφ(e^{nδ−Pni=1γi}) =f(e^δ,1)ⁿφ(e^nδ−γ) Since f(s_δ,1) = 1, we have on the one hand

X

γ∈Πδ

K_δ,γf(e^γ,1) = 1.

On the other hand, from the previous result, X

γ∈Πδ

K_δ,γf(e^γ,1) = X

γ∈Πδ

K_δ,γf(e^δ,1)φ(e^δ−γ) =f(e^δ,1)φ(S_δ).

Thus,f(e^δ,1) = _φ(S¹

δ), which ends the proof of the proposition in the case M_f ∈∆.

Now assume that f is a general nonnegative multiplicative function on ˆT_δ. Let w ∈ W be such that M_f◦w−1 ∈ ∆. By the first part of the proof, there exists φ ∈ Mult(R[Q+])⁺ such that f ◦w⁻¹(e^γ, n) = _φ(S¹

δ)ⁿφ(e^nδ−γ). Thus, composing f ◦w⁻¹ with w yields that f(e^γ, n) = _φ(S¹

δ)ⁿφ(e^nδ−w(γ)) for (γ, n)∈Tˆ_δ.

Remark 4.4. Suppose that φ(e^α) 6= 0 for all α ∈ S. Then, the map φ extends to a homo- morphism φ:R[P]−→R⁺ with the formula

φ(e^γ) = Y

α∈S

φ(e^α)^rα for γ =P

α∈Sr_αα withr_α∈Qfor any α∈S. In this case, f(e^γ, n) =f(e^δ,1)ⁿφ(e^λ−nδ) =

f(e^δ,1) φ(e^δ)

n

φ(e^γ).

Since,f(s_δ,1) = 1, ^f_φ(e^(eδδ^,1)) =φ(s_δ)⁻¹. Hence, when φ(e^α)>0for all α ∈S, f can be written on Tˆ_δ as

f(e^γ, n) = φ(e^γ) φ(s_δ)ⁿ, withφ:P −→R⁺ a multiplicative map.

In this case, for (s_λ, n)∈Tˆ_δ⁺, we also have

f(s_λ, n) = φ(s_λ) φ(s_δ)ⁿ,

where the restriction φ:Aδ−→R⁺ is again a multiplicative map.

Let us reformulate the results of this§by introducing the map Φ : Mult(R[Q⁺])⁺×W −→

Mult( ˆT_δ)⁺ defined by

(8) Φ(φ, w)(e^γ, n) = 1

φ(S_δ)φ(e^nδ−w(γ)).

Proposition 4.2 yields that the map Φ is surjective. Since R[Q⁺] is the free commutative algebra generated by {e^α, α ∈ S}, Mult(R[Q⁺])⁺ is isomorphic to (R⁺)^d through the map θ: Mult(R[Q⁺])⁺−→(R⁺)^d given by

(9) θ(φ) = (φ(e^αi))_1≤i≤d

forφ∈Mult(R[Q⁺])⁺. The composition of Φ with θ⁻¹ yields thus a surjective map (R⁺)^d× W −→ Mult( ˆT_δ)⁺. Since Φ is not necessarily injective, the latter map is not bijective. The lack of injectivity comes from two facts: first, if M_f lies at the intersection of two Weyl chambers, thenM_f◦w−1 ∈∆ for severalw∈W. Next, some degeneracy may occur whenδ is orthogonal to some simple roots. The goal of the next subsection is to overcome the second problem.

4.3. Dominant faces of the weight polytope. Let f ∈Mult( ˆT_δ)⁺ such that M_f ∈∆; it is possible to give a geometric description of the set Π_δ(f) := {γ ∈ Π_δ |f(e^γ,1) 6= 0)}. A face of the polytope K(δ) is the intersection of K(δ) with a supporting hyperplane (that is K(δ) is contained in one of the two half-spaces defined by this hyperplane). A dominant face F is a face of the polytope K(δ) such thatF ∩∆6= 0. We denote by ΠF the intersection of Π_δ withF. By an indecomposable component of a subsetS^′ ⊂S we mean a maximal subset of S^′ consisting of mutually non-orthogonal simple roots.

Definition 4.5. A subset S^′ ⊂ S of simple roots is δ-admissible if each indecomposable component of S^′ contains a simple root which is not orthogonal toδ.

Observe that, according to this definition, the empty set is a δ-admissible subset, since it has no indecomposable component. Write hS^′i for the linear span of S^′. For each subset S^′ ⊂S, denote by W_S^′ the Weyl group generated by the elements s_α^′, α^′ ∈S^′ (where W_∅ is simply{Id}). We will use the following results which comes from [23].

Theorem 4.6. Assigning to each δ-admissible subset S^′ ⊂S the polytope F_S^′ = Conv(w^′δ | w^′ ∈W_S^′)yields a one-to-one correspondence betweenδ-admissible subsets ofSand dominant faces of the polytope K(δ). Moreover, the set Π_FS′ = Π_δ∩F_S^′ coincides with the set (δ+ hS^′i)∩Π_δ and dimF_S^′ = #S^′

Let us denote by S_F the δ-admissible subset of simple roots associated to the dominant faceF by the previous theorem. We get the following characterization Π_δ(f).