Conditional probabilities and permutahedron

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CONDITIONAL PROBABILITIES AND PERMUTAHEDRON

^✩

PROBABILITÉS CONDITIONNELLES ET PERMUTAÈDRE

František MATÚŠ

Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodárenskou vˇeží 4, 18208 Prague, Czech Republic

Received 14 February 2002, revised 8 November 2002

ABSTRACT. – The conditional probabilities of finite conditional probability spaces are considered for points of a smooth manifold of conditional charges. A linear diffeomorphism on the manifold is constructed so that the conditional probabilities map bijectively onto a permutahedron. The facial structure of the permutahedron corresponds to the ways conditional probability spaces decompose. A new global inversion lemma is devised.

2003 Éditions scientifiques et médicales Elsevier SAS

MSC: primary 60A05; secondary 52B11, 53A07, 57R50, 52B70, 14P05, 05A18

Keywords: Conditional probability space; Permutahedron; Lattice of faces; Ordered partition;

Manifolds in Euclidean spaces; Algebraic variety; Global inverse theorem

RÉSUMÉ. – Les probabilités conditionnelles sur un espace fini sont vues comme des points d’une variété différentiable. Un difféomorphisme est construit de manière à ce que les probabilités conditionnelles soient en bijection avec un permutaèdre.

2003 Éditions scientifiques et médicales Elsevier SAS

1. Introduction

For a finite setN, letN^∗ be the set of ordered couples(i|J )whereJ ⊆N andi∈J. A real functionP onN^∗is calledc-probability onN if it is nonnegative,

i∈J

P (i|J )=1 forJ ⊆N nonempty,

✩This work was supported by the grants GA AV A1075104 and GA ˇCR 201/01/1482.

E-mail address: [email protected] (F. Matúš).

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and

P (i|K)=P (i|J )

j∈J

P (j|K) fori∈J ⊆K⊆N. (∗) The expressionP (i|J )withJ = {i, j}is simplified toP (i|i, j ).

This paper studies the family PN of all c-probabilities on N. Obviously, PN is a semialgebraic compact subset of the Euclidean space R^N^∗. Writing P (J|K) =

j∈J∩KP (j|K), any c-probability extends to a function on the set of couples(J|K) where J, K ⊆N and K is nonempty. This extension coincides with the conditional probability in a conditional probability space, see Remark 1.

Let Πd ⊆R^d be the d-dimensional permutahedron defined as convex hull of the vectors (π(0), . . . , π(d)) where π runs over all permutations of the set d = {0,1, . . . , d},d0.

THEOREM 1. – The linear mapping W =(W0, . . . ,Wd) between R^d^∗ and R^d given by

Wi(P )= d j=0 j=i

P (i|i, j ), 0id, restricts to a homeomorphism betweenPdandΠd.

This assertion follows from three lemmas. First,W is restricted to the family of trivial c-probabilities P onN = d, defined by requiringP (i|N )=xi>0 for eachi. Such a functionP is uniquely determined by the vectorx=(x0, . . . , xd)because Eq. (∗) implies P (i|J )=P (i|N )/P (J|N )wherei∈J andP (J|N )=j∈J xj is positive. This vector xhas positive coordinates satisfying^d_i₌₀xi =1, and thus belongs to the relative interior ri(Σd)of the standardd-dimensional simplex Σd. On the other hand, givenx∈^ri(Σd), the formulaP (i|J )=xi/_j_∈_Jxjobviously defines a trivialc-probabilityP. Therefore, the family of trivial c-probabilities on dis homeomorphic to ri(Σd). For a trivialP, W(P )equalsT(x)whereT =(T₀, . . . ,T_d)is given by

Ti(x)=xi

d j=0 j=i

1 xi+xj

, 0id.

LEMMA 1. – The rational mappingT is aC^∞-diffeomorphism betweenri(Σd)and ri(Πd).

Hence,W restricts to a homeomorphism between the family of trivialc-probabilities on d and ri(Πd). Our proof of Lemma 1 relies on Lemma 2 which is a topological instance of the global inversion theorems, see Section 2.

Actually, there is more structure transformed betweenPd and Πd by means ofW, than claimed in Theorem 1. This is detailed in Lemma 3 of Section 3 which relates the faces of Πd to decompositions of conditional probability spaces. Consequently, Theorem 1 is obtained and continualization of conditional probabilities is discussed in Remark 4.

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For investigation of smoothness of the familyPN in neighbourhoods of nontrivialc- probabilities, a familySNof allc-charges onN is introduced. Thec-charges are defined by omitting the nonnegativity constraints in the definition ofc-probabilities. Obviously, the familySN is an algebraic subset of the Euclidean spaceR^N^∗. Lemma 4 in Section 4 claims that, moreover,SNis aC^∞-manifold; for the manifolds and diffeomorphisms see e.g. [3].

THEOREM 2. – The mappingW restricts to aC^∞-diffeomorphism between an open neighbourhood of Pd in Sd and a neighbourhood of Πd in the hyperplane of R^d given by^d_i₌₀xi =^d⁺₂¹.

For the proof of this theorem see Section 5. By Remark 6,W is not injective on the whole familySd.

An equivalent formula forW restricted toSN is presented in Section 6.

To conclude, while the family of usual probability measures on d is obviously identified with the d-dimensional simplex Σd, the family ofc-probabilities on d is, by the above theorems, a ‘twistedd-dimensional permutahedron’, obtained fromΠd by an inversion ofW.

Remark 1. – A conditional probability space is a quadruple (Ω,A,B, Q)whereA is aσ-algebra of subsets ofΩ,B⊆Ais nonempty, andQis a nonnegative function on A×Bso that(Ω,A, Q(·|B))is a probability space withQ(B|B)=1 for everyB∈B, and

Q(A∩B|C)=Q(A|B∩C)·Q(B|C), A, B∈A, B∩C, C∈B.

Here,Qis the conditional probability of the space. Given a triple(Ω,A,B), a natural problem is to describe the family of all mappings Q that turn (Ω,A,B, Q) into a conditional probability space. The above results solve the problem when Ω =N is finite, A is the power set of N and B contains all nonempty subsets of N (note that

∅ ∈/ B in every conditional probability space). Indeed, it is not difficult to see that (N,2^N,2^N \ {∅}, Q) is a conditional probability space if and only if Q restricted to N^∗is ac-probability onN.

Backgrounds [18] of the conditional probability spaces go back to A. Rényi [15]

where a remark on p. 287 witnesses that the concept was conceived independently by A.N. Kolmogorov earlier. Further development can be found in [16,17,5]. Yet, the independent philosophical work of B. de Finetti [6,7] precedes Rényi’s recognition of axioms. For this line of research see the recent monograph [4].

Remark 2. – The permutahedron, or sometimes also permutohedron, is a classical polytope, for references see [2] and [22]. The lattice of faces appears in [2, p. 96]

and [10] where also applications to VLSI circuits emerge. Convex hulls of the points (π(1), . . . , π(n))for a set of permutationsπ related to a poset, and their facial structure, are recently studied in [19] and [1] where motivation originates from a scheduling problem. Another line of generalization leads to the generalized permutahedra of [12]

defined as subsets of the extension lattice of a poset.

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2. Global inversion lemma and proof of Lemma 1

The proof of Lemma 1 below is based on the fact that T is a local diffeomorphism on the relative interior ri(Σd)ofΣd. To prove that a local diffeomorphism is bijective, a global inversion theorem is usually evoked. The following simple lemma, tailored for this purpose in a topological setting, seems to be of more general interest.

LEMMA 2. – Let X and Y be Hausdorff spaces and B⊆X be a nonempty, open, connected, and relatively compact set. Letf:B→Y be a local homeomorphism, i.e.

eachx∈B has an open neighbourhood Ux⊆B such that f (Ux) is open inY andf restricts to a homeomorphism betweenUxandf (Ux). The two conditions

(i) there existsx∈B such thatf (x)=f (y)impliesx=yfory∈B,

(ii) if a netxλ inB converges tox∈X\B then the netf (xλ)has an accumulation point inY \f (B),

are sufficient for the mappingf to be a homeomorphism betweenBandf (B). They are also necessary whenf (B)is relatively compact.

Proof. – The set A= {x ∈B; ∀y∈B: f (x)=f (y)⇒x=y} is nonempty by (i).

If A and B\A were both open then A=B by the connectivity of B. In turn, f is injective and, since every local homeomorphism is obviously continuous and open, the mappingf makesB andf (B)homeomorphic.

IfAwere not open then there would existx∈Aand a netxλ out ofAthat converges to x. The net can be supposed to be inUx⊆B. By construction ofA, one can write f (xλ)=f (yλ) for some yλ∈B different fromxλ. Sincef is bijective onUx the net yλ is out ofUx. AsB is relatively compact there existsy∈X such that a subnetyλ of the net yλ converges to y. The continuity off atx implies f (xλ)→f (x), and since Y is Hausdorff f (x)is the unique limit point of the netf (yλ). Thusf (yλ)converges tof (x)∈f (B). From (ii) applied to the netyλ one deduces thaty∈B. By continuity off aty, one hasf (yλ)→f (y). Hencef (x)=f (y), and then x=y becausex∈A.

Thereforeyλ is eventually inUx, a contradiction.

IfB\A were not open then there would existx ∈B\Aand a netxλ out ofB\A that converges to x. The net can be supposed to be in the open set B, and hence in A. There is y ∈B different from x giving f (x)=f (y). Since X is Hausdorff x and y possess disjoint open neighbourhoods, sayx∈C and y∈D. The continuity off at x implies f (xλ)→f (x), and then the net f (xλ) is eventually in the open set f (D ).

Oncef (xλ)∈f (D )one can findyλ∈D such thatf (xλ)=f (yλ). From some moment xλ ∈B∩C and yλ ∈B ∩D, and as C and D are disjoint xλ=yλ. This contradicts xλ∈A.

For the necessity, let f:B →f (B) be a homeomorphism and f (B) be relatively compact inY. Then (i) holds trivially. If a netxλinBconverges tox∈X\B, then the net f (xλ) has at least one accumulation pointy ∈Y. Passing to a subnet if necessary, one can suppose that f (xλ)→y and xλ→x. When y∈f (B) the continuity off⁻¹ implies xλ→f⁻¹(y). SinceX is Hausdorff one has x=f⁻¹(y)∈B, a contradiction.

Thereforey∈Y \f (B)and (ii) is established. ✷

Remark 3. – There is a considerable number of global univalence and inverse function theorems that bear similarities to Lemma 2. For an overview and further references

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see [11,13,14,21]. What makes Lemma 2 different from known assertions of this type is combination of the conditions (i) and (ii).

Proof of Lemma 1. – LetH1be the hyperplane given by^d_i₌₀xi =1 andH2be the parallel hyperplane given by ^d_i₌₀xi =^d⁺₂¹. The relative interior of Σd consists of the points of H1 with all coordinates positive. It is well known that the permutahedron Πd, viewed as a polytope, is equivalently described by Πd⊆H2 and the inequalities

i∈Ixi ^|^I₂^| for I d. Then ^ri(Πd) consists of the points x ∈ H2 satisfying

i∈Ixi>^|^I₂^|for all∅ =I d.

Lemma 2 will be applied to the open, connected and bounded setB =^ri(Σd)in the spaceX=H1and the mappingf =T ofB intoY =H2. The mappingT mapsri(Σd) intori(Πd)because forx∈^ri(Σd)

i∈I

Ti(x)=^|I|₂+

i∈I

j∈ d\I

xi

xi+xj

^|I|₂, |I| ⊆ d,

with the strict inequality forI nonempty and different from d. Another consequence of this inequality to be used in the sequel is

(iii) if a sequencexⁿinri(Σd)converges toxout ofri(Σd)then the sequenceT(xⁿ) has all accumulation points out ofΠd.

Indeed, if the setI⊆ dlabels the zero coordinates of the limit pointxthen∅ =I d and each accumulation point z of T(xⁿ) satisfies _i_∈_Izi =^|^I₂^|, the double sum vanishing in limit. Thus,z∈Πd\^ri(Πd).

To show thatT is a local homeomorphism onri(Σd), observe that the Jacobi matrix ∂T_i

∂xj

d i,j=0

has d k=0 k=i

xk

(xi+xk)² and − xi

(xi +xj)²

for diagonal and off-diagonal elements, respectively. The column sums equal zero, the diagonal entries are positive, and the off-diagonal entries are negative. Each principal proper submatrix of the matrix is strictly diagonally dominant up to the transposition, and by [8, Theorem 6.1.10, p. 349] regular. Therefore, T is a local homeomorphism of ri(Σd)intoH2.

To verify (i), note that x = (_d₊¹₁, . . . ,_d₊¹₁) ∈ ^ri(Σd) is transformed to T(x) = (^d₂, . . . ,^d₂). IfT(y)=T(x)for somey∈^ri(Σd)then

0=Ti(y)−Tj(y)=yi−yj

yi+yj

+ d

k=0 k /∈{i,j}

yi

yi+yk

− yj

yj +yk

, 0i, jd.

This entailsyi=yj, and thusy=x.

The validity of (ii) follows directly from (iii).

Now, Lemma 2 implies that T is a homeomorphism of ri(Σd) onto its image.

The image is open in the topology of H2. From (iii) it follows that the closure of T(ri(Σd)) is contained in T(ri(Σd))∪ Πd \^ri(Πd). Then ri(Πd)\T(ri(Σd)), the

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relative complement of this closure in ri(Πd), is open inH2. Sinceri(Πd)is connected and T(ri(Σd)) with the complement are both open the latter must be empty. Hence,T mapsri(Σd)ontori(Πd).

Obviously, bothri(Σd)andri(Πd)areC^∞-manifolds andT is aC^∞-diffeomorphism between them, see [3, p. 67]. ✷

3. Faces of permutahedron and the mappingW

WhereN is a finite nonempty set, sayN = d, letρ be an ordered partition of N, i.e. a sequence N1, . . . , Nk, 1k|N|, of nonempty disjoint subsets, blocks, of N that coverN. Letρρfor two ordered partitions ofN if each block inρis union of Ni, Ni+1, . . . , Nj for some 1ijkand if the order of blocks inρis induced from the order of blocks inρ. The ordered partitions ofN with the relation form a poset that becomes a lattice when a bottom element is added.

This lattice is isomorphic to the lattice of faces of the permutahedron Πd. Given an ordered partition ρ ofN = d, up to symmetry with the blocks j1\j0, . . . ,jk\jk−1

where −1=j0< j1<· · ·< jk =d, j0= ∅ and 1kd +1, the corresponding nonempty faceFρ ofΠdis defined by the equalities

d i=j₎+1

xi=^d−j₂⁾, 0)k.

In fact, any face ofΠdis obtained by prescribing equalities in some of the inequalities

i∈Ixi^|^I₂^|, I⊆ d. The equalities indexed byI,J and satisfied by x from a face provide

_|I_| 2

+^|J₂^|=

i∈I∩J

xi+

j∈I∪J

xj^|I∩J₂ ^|+^|I∪J₂ ^|

what implies 0|I\J||J \I|. Thus,I andJ are in inclusion. Prescribed equalities are therefore indexed byd\j0, . . . ,d\jk−1up to symmetry.

The face Fρ can be shifted in R^d⁺¹ to coincide with Cartesian product of the permutahedra Πj₁−j₀−1, . . . , Πj_k−j_k₋₁−1. The appropriate shift moves the coordinate xi

ofx∈R^dtoxi−(d−j))forj)−1< ij), 1)k.

An ordered partition ofN can be identified within eachc-charge on N. To describe the partition, compositions and decompositions ofc-charges are discussed below. These correspond to the construction of conditional probability spaces from an ordered set of measures that goes back to [16] (on p. 57 this idea is attributed to E. Marczewski) and to [5,9].

Having twoc-chargesP ∈SN,Q∈SM, andN,Mdisjoint, ac-chargeR∈SN∪M can be defined by

R(i|I )=





Q(i|I ), I⊆M, P (i|I∩N ), i∈N,

0, i∈M, I∩N = ∅,

where(i|I )∈(N∪M)^∗. Obviously,P andQare the restrictions ofR toN^∗ andM^∗, respectively. WritingR=P Q, the operationis associative but not commutative.

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It follows from(∗) that any R∈SN can be written as P1QwithP1 equal to the restriction ofR toN₁^∗ where the setN1= {i∈N; P (i|N )=0}is not empty. Here, the c-charge P1∈SN1 is trivial in the sense that the valuesP1(i|N1),i∈N1, are nonzero.

Repeating this kind of decomposition one has R=P1· · ·Pk for 1k|N|and trivialc-chargesP1∈SN₁, . . . , Pk∈SN_k. The sequence of setsN1, . . . , Nkis an ordered partition ofN induced byR.

Given an ordered partitionρofN, the notationPN^ρ is reserved for the set of allc-probabilities RonN such thatρ is the ordered partition induced byR.

LEMMA 3. – For an ordered partitionρ of d the mapping W restricts to aC^∞- diffeomorphism betweenP^ρdand the relative interior of the faceFρ ofΠd.

Proof. – Letρhave the blocksj1\j0, . . . ,jk\jk−1as above andι^ρ be the coordinate projection of R^d^∗toR^dkeeping the coordinatesP (i|j)\j)−1)forj)−1< ij)and 1)k. Since the c-probabilities from P^ρd decompose uniquely into trivial c-pro- babilities on the blocks ofρthe projectionι^ρis aC^∞-diffeomorphism betweenP^ρdand the Cartesian product ofri(Σj₁−j₀−1), . . . ,ri(Σj_k−j_k₋₁−1), to be denoted byri(Σ_d^ρ).

ForP ∈P^ρdandj)−1< ij)one hasP (i|i, m)=0 if and only ifj)−1< md, and P (i|i, m)=1 if and only ifj)< md orm=i. Therefore

Wi(P )=(d−j))+

j₎

m=j₎₋₁+1 m=i

P (i|i, m)=(d−j))+T^ρ_iι^ρ(P )

whereT^ρ is the mapping fromri(Σ_d^ρ)toR^dgiven by T^ρ_i(x)=xi

j₎

m=j₎₋₁+1 m=i

1 xi+xm

.

By Lemma 1, T^ρ is a C^∞-diffeomorphism between ri(Σ_d^ρ) and the Cartesian product ofri(Πj₁−j₀−1), . . . ,ri(Πj_k−j_k₋₁−1), to be denoted byri( Π_d^ρ). Then the compositionT^ρι^ρ is aC^∞-diffeomorphism betweenP^ρd andri(Π_d^ρ). SinceW differs from this mapping only by the shift that movesri(Π_d^ρ)tori(Fρ), the assertion follows. ✷

Proof of Theorem 1. – The set Pd partitions into P^ρd and the permutahedron Πd

partitions into ri(Fρ) where ρ runs in both cases over the ordered partitions of d. By Lemma 3, W maps injectively P^ρd onto ri(Fρ). Thus, W is a bijection between Pd andΠd. Obviously, W is continuous and closed on the compactPd, and hence a homeomorphism. ✷

As a consequence, the family of trivialc-probabilities onN is dense inPN.

Remark 4. – In [20] a sophisticated homeomorphismφ ofri(Σd) intoR^d was constructed so that the closure of U =φ(ri(Σd)) is homeomorphic to a d-dimensional simplex, and for any i∈J ⊆ d the real functionfi|J(y)=φ⁻¹(y)i/_j_∈_J φ⁻¹(y)j, y ∈U, is uniformly continuous. This result, interpreted as a simultaneous continual- ization of conditional probabilities, is a simple consequence of Theorem 1. In fact, the

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choiceφ=T leads toU=^ri(Πd)with its closureΠdobviously homeomorphic to a simplex. Fory∈Uone hasy=T(x)for a uniquex∈^ri(Σd)and then a uniquec-probability P exists such thatxi=P (i| d) >0 for each 0id. In turn,W(P )=T(x)=yand thus the number fi|J(y)=xi/_j_∈_J xj =P (i|J )is a coordinate of W⁻¹(y). By The- orem 1, W⁻¹ is continuous on the compact Πd and thus uniformly continuous on U.

Consequently, each functionfi|J is uniformly continuous onU. 4. Charts of the manifoldSN

First, a new construction ofc-charges is presented. Givent=(t1, . . . , td)∈R^d, let si,j,k(t)=

j m=i+1

tm

k n=j+1

(1−tn), 0ijkd,

and let Vd⊆R^d be the set oft’s rendering _i_∈_Is_∧I,i,∨I(t)=0 for every∅ =I ⊆ d. Here, ∧I is the smallest element and ∨I the greatest element ofI. Let a mapping Z from the open setVdintoR^d^∗be given by

Zt (i|J )= s_∧J,i,∨J(t)

j∈Js_∧J,j,∨J(t), i∈J ⊆ d.

Then Zt is a c-charge on d for every t ∈Vd. In fact, _i_∈_JZt (i|J )=1 and for i∈J ⊆K⊆ d

Zt (i|J )·Zt (J|K)= s_∧J,i,∨J(t)

j∈Js_∧J,j,∨J(t)·

j∈Js_∧K,j,∨K(t)

k∈Ks_∧K,k,∨K(t)=Zt (i|K),

using the obvious identitys_∧J,i,∨J·s_∧K,j,∨K=s_∧J,j,∨J·s_∧K,i,∨K. Due toZt (i|i, i−1)= ti, 1id, the mappingZone-to-one.

LEMMA 4. – The familySdofc-charges on dis aC^∞-manifold of dimensiond. Proof. – The idea is to compose the mapping Z with the isometries of R^d^∗ that transform P to (P (π(i)|π(J )); (i|J )∈ N^∗) where π is a permutation of d and π(J )= {π(j ); j∈J}. These compositions will provide an atlas of charts.

The cased=0 is trivial. Letd1 andP be ac-charge on d. The vectorx∈R^d⁺¹ with the coordinates xi=P (i| d)is nonzero and, due to(∗), solves the homogeneous linear equationsxi⊕1=P (i⊕1|i⊕1, i)(xi⊕1+xi), 0id. Here,⊕is the addition modulod+1. Hence the determinant

P (1|1,0) −P (0|1,0) 0 ... 0 0 P (2|2,1) −P (1|2,1) ... 0

... ... ... ... ...

−P (d|0,d) 0 0 ... P (0|0,d)

=

d i=0

P (i|i, i1)− d i=0

P (i|i, i⊕1)

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vanishes. This implies

P (d|0, d) _d

m=1

tm+ d n=1

(1−tn)

= d m=1

tm, whereti=P (i|i, i−1), 1id. Along the same guidelines

P (j|i, j ) _j

m=i+1

tm+ j n=i+1

(1−tn)

= j m=i+1

tm, 0i < jd. (†) Since the vector x solves also the d +1 equations xi =P (i|i, i−1)(xi +xi−1), 1id, and^d_i₌₀xi=1, by Cramer’s rule

xi

P (1|1,0) −P (0|1,0) 0 ... 0 0

0 P (2|2,1) −P (1|2,1) ... 0 0

... ... ... ... ... ...

0 0 0 ... P (d|d,d−1) −P (d−1|d,d−1)

1 1 1 ... 1 1

= i m=1

tm

d n=i+1

(1−tn)

and thus

Pi| d· d k=0

s0,k,d(t)=s0,i,d(t).

The same argument withJ = {i0, . . . , i)}(wherei0<· · ·< i)and 1)d) implies P (ik|J )

) j=0

j m=1

P (im|im, im−1) ) n=j+1

P (in−1|in−1, in)

= k m=1

P (im|im, im−1) ) n=k+1

P (in−1|in−1, in) for 0k). EliminatingP (j|i, j )by means of (†)

P (i|J )·

j∈J

s_∧J,j,∨J(t)=s_∧J,i,∨J(t), i∈J ⊆ d. (‡) Let the c-charge P be decomposed into P =P1· · ·Pk, 1kd +1, with all components trivial. Up to a permutation of d, the underlying ordered partition has the blocks j1 \j0, . . . ,jk \jk−1, see Section 3. Then all numbers ti =P (i|i, i−1), 1i d, differ from 1 and therefore s_∧J,∧J,∨J(t) does not vanish for nonempty J. On account of (‡), _j_∈_Js_∧J,j,∨J(t) in nonzero. Hence, t∈Vd,P =Zt and Z maps a neighbourhood oft onto a neighbourhood ofP inSd. This proves thatZVd is a chart ofSdand everyc-charge belongs to the chart up to a permutation.

Remark 5. – The above arguments imply that eachc-chargeP is uniquely determined by its valuesP (i|i, j )(forc-probabilities this follows independently from Theorem 1).

Thus,P ∈S 2 is determined by the triple (P (1|0,1), P (2|1,2), P (0|2,0)). Projecting

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the familyS2to these coordinates one obtains an unbounded surface inR³. In a neighbourhood of the origin the surface can be depicted as

The six white segments in the figure correspond to the edges of the unit cube which belong to the surface. They border a twisted hexagon which is the projection of the familyP2, homeomorphic to the hexagonΠ2by Theorem 1. It is not difficult to show thatS2has four connected components, each oneC^∞-diffeomorphic toR².

5. Proof of Theorem 2

The cased=0 is trivial. First, let us show that ford1 eachc-probabilityP ∈Pd

has a neighbourhood UP inSd such thatW is aC^∞-diffeomorphism betweenUP and an open subset of the hyperplaneH2.

For a trivial c-probability P one can take for UP the family of all trivial c-pro- babilities. Then W maps UP ontori(Πd)by Lemma 3. LetP be ac-probability which is not trivial. Up to a permutation, P can be decomposed and expressed as P =Zt where t ∈R^d has components 0ti <1, 1id (see end of the previous proof).

To conclude thatP has a desired neighbourhood, it is necessary and sufficient to verify that the Jacobi matrixCof the compositionWZat the pointt has rankd. An induction argument on the dimensiond will be applied.

Using the computations

∂si,j,k

∂t)

= ∂

∂t)

j m=i+1

tm

k n=j+1

(1−tn)=





0, )ior)k+1,

si,)−1,)−1·s),j,j·sj,k,k, i+1)j,

−si,j,j·sj,j,)−1·s),),k, j+1)k, and

(WZ)_i =

i−1

k=0

sk,i,i

sk,k,i+sk,i,i

+ d k=i+1

si,i,k

si,i,k+si,k,k

, 0id,

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Table 1

P (0| 2) P (1| 2) P (2| 2) P (0|0,1) P (1|0,1) P (0|0,2) P (2|0,2) P (1|1,2) P (2|1,2)

−1 −1 3 ¹₂ ¹₂ −¹₂ ³₂ −¹₂ ³₂

0 −1 2 0 1 0 1 −1 2

the matrixC has the elements ci,j =∂(WZ)i

∂tj =sj,i,i·sj,j,i j−1

k=0

sk,j−1,j−1·sk,k,j−1

[sk,k,i+sk,i,i,]² , 1jid, and

ci,j=∂(WZ)_i

∂tj

= −si,i,j−1·si,j−1,j−1

d k=j

sj,j,k·sj,k,k

[si,i,k+si,k,k]², 0i < jd.

Since P is not trivial t) =0 for some 1)d. Then the element ci,j vanishes if 1j < )id or 0i < ) < j d. In other words, the nonzero elements of C belong to the submatrix Aindexed by 0i < )and 1j < ), or to the submatrix B indexed by )id and ) < j d, or to the )th column of C. If ) >1 then, in the dimension)−1< d,Acoincides with the Jacobi matrix of the compositionWZat the point(t1, . . . , t)−1). By induction,Ahas rank)−1. Analogous argument shows that the rank ofBisd−)when) < d. Since not allti’s are positive everysi,j,k(t)is nonnegative and every si,i,k(t) positive. Now,ci,)0 for 0i < )−1 and c)−1,)−1. In turn, (c0,), . . . , c)−1,)) cannot be a linear combination of the columns ofA as each column sums to zero. Therefore the first)columns ofCare linearly independent. Remembering that the rank of B isd−), the Jacobi matrix C has rankd. This means that for every P ∈Pdthe mappingW is a homeomorphism, and in turn aC^∞-diffeomorphism, of an open setUP ⊆SdintoH2.

Having the desired neighbourhoods UP, suppose, by contradiction, that W is not injective on every neighbourhood ofPd inSd. By compactness ofPd, inSdthere exist two convergent sequences Pn →P and Qn→Q with limits in Pd such that W(Pn)=W(Qn)and Pn=Qn. Due to the continuity ofW, one has W(P )=W(Q), and thusP =Q, by Theorem 1. Then, the sequencesPnandQnare eventually in the set UP whereW is injective, a contradiction. This closes the proof of Theorem 2.

Remark 6. – The mappingWis not injective on the whole familyS2. In fact, the two c-charges given by rows of Table 1 are mapped byW to the single point(0,0,3) /∈Π2.

6. Alternative formula forW restricted toSN

The mappingW can be written, modulo the polynomials defining SN, in a different form that seems to be interesting per se. To this end, let S be the permutation group of the set d. Forπ ∈S and P:R^d^∗→R the symbol P (_π(k...d)^π(k) ) abbreviates P (π(k)|{π(k), . . ., π(d)}), 0kd.

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LEMMA 5. – For everyc-chargeP on d W_i(P )=d−

π∈S

π⁻¹(i) d k=0

P

π(k) π(k . . . d)

, 0id.

Proof. – Let

α^m_i,j,k^→ⁿ=

π∈S π(m)=n

P

n π(k . . . d)

^j−1 )=i

P

π()) π() . . . d)

, 0ijkmd, 0nd.

It is straightforward thatα_i,j,k^m^→ⁿ=α_i,j,k^d^→ⁿ. Hence, keepingnfixed, the superindices can be omitted. The above formula forαi,j+1,kwithj < kcontainsP (_{π(j ...d)}^{π(j )} )as the last term of a product. By symmetry, this term can be replaced byP (_{π(j ...d)}^π()) )forj ) < kand hence

αi,j+1,k=

π∈S π(d)=n

P

n π(k . . . d)

1 k−jP

π(j . . . k−1) π(j . . . d)

j−1 )=i

P

π()) π() . . . d)

.

UsingP (^{π(j ...k}_{π(j ...d)}⁻¹⁾)=1−P (^π(k...d)_{π(j ...d)})and(∗) αi,j+1,k= 1

k−j[αi,j,k−αi,j,j].

This recurrence is used to prove αi,j,j =

j−i

)=0

(−1)^j⁻ⁱ⁻⁾ αi,i,i+)

)!(j−i−))!

by induction onj−i=0,1, . . . , d. In fact, the casei=j is obvious. Forj+1d αi,j+1,j+1=αi,j,j+1−αi,j,j = · · · = αi,i,j+1

(j+1−i)!−

j−i

k=0

αi,j−k,j−k

(k+1)! whereαi,j−k,j−k is rewritten by the induction hypothesis

αi,j+1,j+1= αi,i,j+1

(j+1−i)!−

j−i

k=0

1 (k+1)!

j−k−i )=0

(−1)^j⁻^k⁻ⁱ⁻⁾ αi,i,i+)

)!(j−k−i−))!. After changing the order of summations and manipulations

αi,j+1,j+1= αi,i,j+1

(j +1−i)!+

j−i

)=0

(−1)^j⁺¹⁻ⁱ⁻⁾ αi,i,i+)

)!(j+1−i−))!

j−i−) k=0

(−1)^k^j+1−i−)_k₊₁ which proves the induction step.