Non-crossing linked partitions and multiplication of free random variables
aa r X i v : . [ m a t h . OA ] J a n NON-CROSSING LINKED PARTITIONS AND MULTIPLICATIONOF FREE RANDOM VARIABLES
MIHAI POPA
Abstract.
The material gives a new combinatorial proof of the multiplicativeproperty of the S -transform. In particular, several properties of the coefficientsof its inverse are connected to non-crossing linked partitions and planar trees.AMS subject classification: 05A10 (Enumerative Combinatorics); 46L54(FreeProbability and Free Operator Algebras). Introduction and definitions
The relation between non-crossing partitions and free probabilities has been stud-ies extensively (see [5], [8]), but the closely related non-crossing linked partition havenot received the same attention. Recently (see [2], [7]), the latest object was shownto give the recurrence for computing the coefficient of the inverse of the Voiculescu’s S -transform in a similar manner the non-crossing partitions are used for the com-putation of the R -transform (see (1) and (2) below). The present material gives anew, combinatorial proof of the multiplicative property of the S -transform using arelation between planar rooted trees and the Kreweras complement.A non-commutative probability space is a couple ( A , φ ), where A is a unital( ∗ -)algebra and φ : A −→ C is a linear mapping such that φ (1) = 1 and φ ( x ∗ x ) ≥ x ∈ A if A is a ∗ -algebra. The ( ∗ -)subalgebras {A i } i ∈ I of A are free if φ ( a a · · · a n ) = 0for any a k ∈ A ε ( k ) such that ε ( j ) = ε ( j + 1) and φ ( a k ) = 0. The elements { X i } i ∈ I from A are free if the unital ∗ -algebras they generate are free.A non-crossing partition γ of the ordered set { , , . . . , n } is a collection C , . . . , C k of subsets of { , , . . . , n } , called blocks, with the following properties:(a) k G l =1 C l = { , . . . , n } (disjoint union of sets)(b) C , . . . , C k are non-crossing, in the sense that there are no two blocks C l , C s and i < k < p < q such that i, p ∈ C l and k, q ∈ C s . Example 1 : Below is represented graphically the non-crossing partition π = (1 , , , (2 , , (5) , (7 , , (9 , , (11 , ∈ N CL (10): s s s s s s s s s s
Non-crossing partitions appear in the definition of the free cumulants, the mul-tilinear functions κ n : A n −→ C given by the recurrence ( X , . . . , X n ∈ A ):(1) φ ( X · · · X n ) = X γ ∈ NC ( n ) Y C =block in γC =( i ,...,i l ) κ l ( X i , . . . , X i l )If X = · · · = X n = X , then we write κ m ( X ) for κ n ( X , . . . , X n ). A remarkableproperty of the free cumulants is the following: Proposition 1.1. If {A i } i ∈ I is a family of unital subalgebras of A , then the fol-lowing statements are equivalent: (i) {A i } i ∈ I are free independent. (ii) For all a k ∈ A ı( k ) ( k = 1 , . . . , n and i ( k ) ∈ I we have that κ n ( a , . . . , a n ) =0 whenever there exist ≤ l, k ≤ n with i ( l ) = i ( n ) . An immediate consequence is the additive property of the Voiculescu’s R -transform(see [5], [3], [9]): Proposition 1.2.
If, for any a ∈ A , we define R a ( z ) = n X n =1 κ n ( X ) z n , then, for X, Y free, we have that R X + Y = R X + R Y . By a non-crossing linked partition π of the ordered set { , , . . . , n } we willunderstand a collection B , . . . , B k of subsets of { , , . . . , n } , called blocks, withthe following properties:(a) k [ l =1 B l = { , . . . , n } (b) B , . . . , B k are non-crossing, in the sense that there are no two blocks B l , B s and i < k < p < q such that i, p ∈ B l and k, q ∈ B s .(c) for any 1 ≤ l, s ≤ k , the intersection B l T B s is either void or contains onlyone element. If { j } = B i T B s , then | B s | , | B l | ≥ j is the minimalelement of only one of the blocks B l and B s .We will use the notation s ( π ) for the set of all 1 ≤ k ≤ n such that there areno blocks of π whose minimal element is k . A block B = i < i < · · · < i p of π will be called exterior if there is no other block D of π containing two elements l, s such that l = i or l < i < i p < s . The set of all non-crossing linked partitions on { , . . . , n } will be denoted by N CL ( n ). Example 2 : Below is represented graphically the non-crossing linked partition π = (1 , , , , (2 , , (4 , , (6 , , , (10 , , (11 , ∈ N CL (12). Its exterior blocksare (1 , , ,
9) and (10 , s s s s s s s s s s s s ✚✚ ✚✚ ✚✚ A ◦ = A \
Kerφ . Using non-crossing linked partitions, we define the t - coefficients { t n } ∞ n =0 as the mappings t n : A × ( A ◦ ) n −→ C ON-CROSSING LINKED PARTITIONS 3 given by the following recurrence: (2) φ ( X · · · X n ) = X π ∈ NCL ( n ) ( Y B =block in πB =( i ,...,i l ) t l − ( X i , . . . , X i l ) · Y k ∈ s ( π ) t ( X k )) . To simplify the writing we will use the shorthand notations t π [ X , . . . , X n ] forthe summing term of the right-hand side of (2), and t π [ X ], respectively t n ( X ) for t π ( X, . . . , X ), respectively t n ( X, . . . , X ). Remark 1.3.
The mappings t n are well-defined. Indeed, t n − ( X , . . . , X n ) appearsonly once and has a non-zero coefficient in the right hand side of (2), namely in t n [ X , . . . , X n ] = t n − ( X , . . . , X n ) n Y l =2 t ( X l )where n is the partition with a single block (1 , , . . . , n ) . Also, while the free cumulants are multilinear, the t -coefficients have the property t n ( c X , c X , . . . , c n X n ) = c t n ( X , X , . . . , X n )for all c ∈ C , c , . . . , c n ∈ C ∗ . The above relation is a immediate consequence of(2), since in each therm of the right-hand side of (2), there is exactly one factorcontaining X , on the first position in its block, and exactly two factors containing X j ( j ≥ X j in the first position.An alternate form of (2) is described below. For 1 ≤ k ≤ n and π ∈ N CL ( n ),we define t [ k,π ] ( X , . . . , X n ) = t s ( X k , X i (1) , . . . , X i ( s ) ) if k is the minimal elementof the block ( k, i (1) , . . . , i ( s )) t ( X k ) if k is not the minimal element of any block.Then t π ( X , . . . , X n ) = n Y k =1 t [ k,π ] ( X , . . . , X n ) φ ( X · · · X n ) = X π ∈ NCL ( n ) t π ( X , . . . , X n ) . As shown in [2] and [7], the T -transform, defined via T X = ∞ X n =0 t n ( X ) z n , hasthe property that T XY = T X T Y for all X, Y free elements of A ◦ . Notable is alsothe role of N CL ( n ) in defining a conditionally free version of the T -transform (see[7]). In the following sections we will discuss the lattice structure of N CL ( n ),prove a property similar to Proposition 1.1 for the t -coefficients and give a prooffor the multiplicative property of the T -transform based on the connection between N CL ( n ), the Kreweras complement on N C ( n ) and planar rooted trees.2. The lattice
N CL ( n )On N CL ( n ) we define a order relation by saying that π (cid:23) σ if for any block B of π there exist D , . . . , D s blocks of σ such that B = D ∪ · · · ∪ D s . With respectto the order relation (cid:23) , the set N CL ( n ) is a lattice. The maximal, respectively the MIHAI POPA minimal element are n = (1 , , . . . , n ) and 0 n = (1) , (2) , . . . , ( n ). Note also that N C ( n ) is a sublattice of N CL ( n ).We say that i and j are connected in π ∈ N CL ( n ) if there exist B , . . . , B s blocks of π such that i ∈ B , j ∈ B s and B k ∩ B k +1 = ∅ , 1 ≤ k ≤ s − π ∈ N CL ( n ) we assign the partition c ( π ) ∈ N C ( n ) defined as follows: i and j are in the same block of c ( π ) if and only if they are connected in π . (I. e. theblocks of c ( π ) are exactly the connected components of π .) We will use the notation[ c ( π )] = { σ ∈ N CL ( n ) : c ( σ ) = c ( π ) } . In the above Example 2, we have that 5 and 8 as well as 10 and 12 are connected.More precisely, c ( π ) = (1 , , , , , , , (2 , , (10 , , (cid:23) , we have that, for every γ ∈ N C ( n ),[ γ ] is a sublattice of N CL ( n ) and its maximal element is γ . Moreover, if γ has theblocks B , . . . , B s , each B l of cardinality k l , then we have the following ordered setisomorphism:(3) [ c ( π )] ≃ [ k ] × · · · × [ k s ]The above factorization has two immediate consequences: Proposition 2.1.
For any positive integer n and any X , . . . , X n ∈ A we have that κ n ( X , . . . , X n ) = X π ∈ [ n ] t π [ X , . . . , X n ] Proof.
For n = 1 the assertion is clear, since, by definition, φ ( X ) = κ ( X ) = t ( X ) . If n >
1, note first that X π ∈ NCL ( n ) t π [ X , . . . , X n ] = X γ ∈ NC ( n ) X π ∈ [ γ ] t γ [ X , . . . , X n ] . Also, if π ∈ N CL ( n ) has the connected components B , . . . , B k such that each B l = ( i l, , . . . , i l,s ( l ) ), then t π [ X , . . . , X n ] = k Y l =1 t π | Bl [ X i i, , . . . , X i l,s ( l ) ] , where π | B l denotes the restriction of π to the set B .Since the blocks of c ( π ) are by definition the connected components of π , therelation (2) becomes: φ ( X · · · X n ) = X γ ∈ NC ( n ) Y B =block in γB =( i ,...,i s ) X π ∈ [ γ ] t π | B [ X i , . . . , X i s ] , and the factorization (3) gives: φ ( X · · · X n ) = X γ ∈ NC ( n ) Y B =block in γB =( i ,...,i s ) X σ ∈ [ s ] t σ [ X i , . . . , X i s ] . The conclusion follows now utilizing (1) and induction on n . (cid:3) ON-CROSSING LINKED PARTITIONS 5
Proposition 2.2. (Characterization of freeness in terms of t -coefficients) If {A i } i ∈ I is a family of unital subalgebras of A , then the following statements areequivalent: (i) {A i } i ∈ I are free independent. (ii) For all a k ∈ A ı( k ) ( k = 1 , . . . , n ) and i ( k ) ∈ I we have that t n − ( a , . . . , a n ) = 0 whenever there exist ≤ l, k ≤ n with i ( l ) = i ( k ) .Proof. It suffices to show the equivalence between 2.2(ii) and 1.1(ii).Let first suppose that 2.2(ii) holds true. Choose a positive integer n and π ∈ [ n ].If not all X k are coming from the same subalgebra A j , since 1 , . . . , n are allconnected in π , there is a block B = ( i , . . . , i s ) such that { X i , . . . , X i s } con-tains elements from different subalgebras, therefore t s − ( X i , . . . , X i s ) = 0, hence t π [ X , . . . , X n ] = 0 and 2.1 implies that κ n ( x , . . . , X n ) = 0, i. e. 1.1(ii).Suppose now that 1.1(ii) holds true. Proposition 2.1implies that κ = t , so2.2(ii) is true for n = 2. An inductive argument on n will complete the proof:Proposition 2.1 also implies that κ n ( X , . . . , X n ) = t n − ( X , . . . , X n ) + X π ∈ [ n ] π = n t π [ X , . . . , X n ] . If not all X k are coming from the same subalgebra A j , then the left-hand sideof the above equation cancels from 1.1(ii) and so does the second term of the right-hand side, from the induction hypothesis, so q.e.d.. (cid:3) planar trees and the multiplicative property of the T -transform In this section we will give a combinatorial proof for the multiplicative propertyof the T -transform, that is(4) T XY = T X T Y whenever X and Y are free elements from A ◦ . The proof will consist mainly in describing certain bijections between [ n ] andthe set T ( n ) of planar trees with n vertices, respectively between N C ( n ) and theset B ( n ) of bicolor planar trees with n vertices.3.1. Non-crossing linked partitions and planar trees.
By an elementary pla-nar tree we will understand a graph with m ≥ v , v , . . . , v m , and m − v (called root ) to the vertices v , . . . , v m (called off-springs ). By convention, a single vertex (with no offsprings) will be also consideredan elementary planar tree.A planar tree will be seen as consisting in a finite number of levels , such that:- first level consists in a single elementary planar tree, whose root will be alsothe root of the planar tree;- the k -th level will consist in a set of elementary planar trees such that theirroots are among the offsprings of the k − A will be denoted by E ( A ).Below are represented graphically the elementary planar tree T and the 2-levelplanar tree T : MIHAI POPA s ✡✡✡ ss s ❏❏❏ T ss ✔✔✔ s ❚❚❚ s ✔✔✔ s ❚❚❚ T level 3level 2level 1We will need to consider on the vertices of a planar tree the order relation (similar to the “ left depth first ” order from [1]) given by:(i) roots are less than their offsprings;(ii) offsprings of the same root are ordered from left to right;(iii) if v is less that w , then all the offsprings of v are smaller than any offspringof w .Intuitively, one may understand it as the order in which the vertices are passed bywalking along the branches from the root to the right-most vertex, not countingvertices passed more than one time (see the example below). Example ss (cid:0)(cid:0) s s ❚❚ s ✔✔ s ❚❚
123 45 6We construct now the bijection Θ : [ n ] −→ T ( n ) by putting Θ( π ) be theplanar tree composed by the elementary trees of vertices numbered ( i , . . . , i s ) (withrespect to the above order relation), for each ( i , . . . , i s ) block of π .More precisely, if (1 , , i , . . . , i s ) is the block of π containing 1, then the first levelof Θ( π ) is the elementary planar tree of root numbered 1 and offsprings numbered(2 , i , . . . , i s ). The second level of Θ( π ) will be determined by the blocks (if any)having 2 , i , . . . , i s as first elements etc (see the example below)It is easy to see that Θ is well-defined and injective. To show that Θ is bijective,it suffices to observe that Θ − is given by assigning to each tree the partition havingthe blocks given by the numbers of the vertices from the constituent elementaryplanar trees. Part (a) of the definition of N CL ( n ) (see Section 1) is automaticallyverified, since the trees have exactly n vertices. Part (b) follows from the conditions(ii) and (iii) in the definition of the order relation on the vertices, and part (c) fromthe conditions (i) and (ii).If A is an elementary planar tree with n vertices and X ∈ A ◦ , we define E X ( A ) = t n − ( X ). The evaluation E X extends to the set of planar trees by E X ( A ) = Y A ∈ E ( A ) E X ( A ) . ON-CROSSING LINKED PARTITIONS 7
Consequently,(5) κ n ( X ) = X A ∈ T ( n ) E X ( A ) . The Kreweras complement and bicolor planar trees.
For γ ∈ N C ( n ),its Kreweras complement
Kr( γ ) is defined as follows. We consider the additionalnumbers 1 , . . . , n forming the ordered set1 , , , , . . . , n, n. Kr( γ ) is defined to be the biggest element γ ′ ∈ N C (1 , . . . , n ) ∼ = N C ( n ) such that γ ∪ γ ′ ∈ N C (1 , , , , . . . , n, n ) . The total number of blocks in γ and Kr( γ ) is n + 1 (see [5], [4]). The Krewerascomplement appears in the following corollary of Proposition 1.1: Proposition 3.1. If X, Y are free elements of A , then κ n ( XY ) = X γ ∈ NC ( n ) κ γ [ X ] κ Kr ( γ ) [ Y ] . Let
N C s (2 n ) be the set of all γ ∈ N C (2 n ) such that elements from the sameblock of γ have the same parity and γ + = Kr ( γ − ), where γ + = γ |{ , ,..., n } γ − = γ |{ , ,..., n − } . Denote also
N CL S (2 n ) = { π ∈ N CL (2 n ) : c ( π ) ∈ N C s (2 n ) } . With the abovenotations, the relation from Proposition 3.1 becomes: κ n ( XY ) = X γ ∈ NC s (2 n ) κ γ − [ X ] κ γ + [ Y ](6)For π ∈ N C S (2 n ), we will say that the blocks with odd elements are of color 1and the ones with even elements are of color 0. Note that π ∈ N C S (2 n ) if and onlyif π has exactly 2 exterior blocks, one of color 1 and one of color 0 and if i and i are two consecutive elements from the same block, then π | ( i +1 ,...,i − has exactlyone exterior block, of different color than the one containing i and i .We will represent blocks of color 1 by solid lines and blocks of color 0 by dashedlines:In the remaining part of this subsection we will define the set B ( n ) of bicolorplanar trees and construct a bijection Λ : N C S (2 n ) −→ B ( n ).A bicolor elementary planar tree is an elementary tree together with a mappingfrom its offsprings to { , } such that the offsprings whose image is 1 are smaller (inthe sense of Section 3.1) than the offsprings of image 0. Branches toward offspringsof color ), respectively 1, will be also said to be of color 0, respectively 1. We willrepresent by solid lines the branches of color 1 and by dashed lines the branchesof color 0. The set of all bicolor planar trees with n vertices will be denoted by EB ( n ). Below is the graphical representation of EB (4): MIHAI POPA A bicolor planar tree is a planar tree whose constituent elementary trees are allbicolor; the set of all bicolor planar trees will be denoted by B ( n ).Given π ∈ N CL S (2 n ), we construct Λ( π ) ∈ B ( n ) as follows:- If ( i , . . . , i s ) and ( j , . . . , j p are the two exterior blocks of π , then the firstlevel of Λ( π ) is an elementary tree with s − p − s − i , . . . , i s ), in this order, and the last p − j , . . . , i p ), in this order.- Suppose that i and i are consecutive elements in a block of π alreadyrepresented in an elementary tree of Λ( π ), that π has the exterior block( j , . . . , j p ) and that i id the minimal element of the block ( i , d , . . . , d r ).The the blocks B = ( j , . . . , j p ) and D = ( i , d , . . . , d r ) will have differentcolors. They will be then represented by an elementary tree of vertexcorresponding to i (the block of i and i has been already representedfrom the hypothesis), and with p − k offsprings, keeping the colors ofthe blocks B and D , the ones of color 1 placed before the ones of color 0.Note that the mapping Λ is bijective, the inverse is constructing reversing the stepsabove.Fix X, Y ∈ A ◦ , free. If B ∈ EB ( n ) has k offsprings of color 1 and n − k − ω X,Y ( B ) = t k ( X ) t n − k − ( Y ) . The functional ω X,Y extends to B ∈ B ( n ) via ω X,Y ( B ) = Y B ∈ E ( B ) ω X,Y ( B ) . For π ∈ N C S (2 n ) the definition of the mapping Λ gives(7) κ π − [ X ] κ π + [ Y ] = ω X,Y (Λ( π )) Theorem 3.2. If X, Y are free elements from A ◦ , then T XY = T X T Y .Proof. We need to show that, for all m ≥ t m ( XY ) = m X k =0 t k ( X ) t m − k ( Y ))For m = 0, the assertion is trivial. Suppose (8) true for m ≤ n −
1. Let A m bethe elementary planar tree with m vertices. In terms of planar trees, the inductionhypothesis is written as(9) E XY ( A m ) = X B ∈ EB ( n ) ω X,Y ( B ) . For example, E XY ( s s ss ✁✁ ❆❆ ) = ω XY ( s s ss ✁✁ ❆❆ ) + ω XY ( s s ss ✁✁ ) + ω XY ( s s ss ✁✁ ) + ω XY ( s s ss ) ON-CROSSING LINKED PARTITIONS 9
The relations (5) and (7) give: X A ∈ T ( n ) E XY ( A ) = κ n ( X )= X π ∈ NC S (2 n ) κ π − [ X ] κ π + [ Y ]= X π ∈ NC S (2 n ) ω X,Y (Λ( π ))= X B ∈ B ( n ) ω X,Y ( B ) . (10)All non-elementary trees from T ( n ) consists on elementary trees with less than n vertices. The relation (9) implies that the image under E XY of any such tree isthe sum of the images under ω XY of its colored versions. Hence(11) X A ∈ T ( n ) A = A n E XY ( A ) = X B ∈ B ( n ) B ∈ EB ( n ) ω X,Y ( B )Finally (11) and (10) give E XY ( A n ) = X B ∈ B ( n ) ω X,Y ( B )that is (8). (cid:3) References [1] M. Anshelevich, E. G. Effros, M. Popa. Zimmermann type cancellation in the free Fa`a diBruno algebra. J. Funct. Anal. 237 (2006), no. 1, 76–104.[2] K. Dykema. Multilinear function series and transforms in Free Probability theory. Preprint,arXiv:math.OA/0504361 v2 5 Jun 2005[3] U. Haagerup. On Voiculescu’s R - and S -transforms for Free non-commuting Random Vari-ables. Fields Institute Communications, vol. 12(1997), 127–148[4] G. Kreweras. Sur les partitions non-croisees d’un cycle. Discrete Math. 1 (1972), pp. 333-350[5] A. Nica, R. Speicher. Lectures on the Combinatorics of the Free Probability. London mathe-matical Society Lecture Note Series 335, Cambridge University Press 2006[6] M. Popa. A new proof for the multiplicative property of the boolean cumulants with appli-cations to operator-valued case arXiv:0804.2109[7] Popa, Mihai, J. C. Wang. On multiplicative conditionally free convolution, arXiv:0805.0257[8] R. Speicher. Combinatorial Theory of the Free Product with amalgamation and Operator-Valued Free Probability Theory. Mem. AMS, Vol 132, No 627 (1998)[9] D.V. Voiculescu, K. Dykema, A. Nica. Free random variables. CRM Monograph Series, 1.AMS, Providence, RI, 1992. Indiana University at Bloomington, Department of Mathematics, Rawles Hall, 831E 3rd St, Bloomington, IN 47405
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