Combinatorics of Free Cumulants

We derive a formula for expressing free cumulants whose entries are products of random variables in terms of the lattice structure of non-crossing partitions. We show the usefulness of that result by giving direct and conceptually simple proofs for a lot of results about R-diagonal elements. Our investigations do not assume the trace property for the considered linear functionals. Introduction Free probability theory, due to Voiculescu [17, 18], is a non-commutative probability theory where the classical concept of “independence” is replaced by a non-commutative analogue, called “freeness”. Originally this theory was introduced in an operator-algebraic context for dealing with questions on special von Neumann algebras. However, since these beginnings free probability theory has evolved into a theory with a lot of links to quite different fields. In particular, there exists a combinatorial facet: main aspects of free probability theory can be considered as the combinatorics of non-crossing partitions. There are two main approaches to freeness: • the original approach, due to Voiculescu, is analytical in nature and relies on special Fock space constructions for the considered distributions. • the approach of Speicher [14, 15, 16] is combinatorial in nature and describes freeness in terms of so-called free cumulants – these objects are defined via a precise combinatorial description involving the lattice of non-crossing partitions; a lot of questions on freeness reduce in this approach finally to combinatorial problems on non-crossing partitions. The relation between these two approaches is given by the fact that the free cumulants appear as coefficients in the operators constructed in the Fock space approach. This connection was worked out by Nica [6]. The second author was supported by a Heisenberg fellowship of the DFG. 1 2 BERNADETTE KRAWCZYK AND ROLAND SPEICHER Here, we will investigate one fundamental problem in the combinatorial approach and show that there is a beautiful combinatorial structure behind this. In the combinatorial approach to freeness one defines, for a given linear functional φ on a unital algebra A, so-called free cumulants kn (n ∈ N), where each kn is a multi-linear functional on A in n arguments. The connection between φ and the kn is given by a combinatorial formula involving the lattice of non-crossing partitions. (The name “cumulants” comes from classical probability theory; there exist analogous objects with that name, the only difference is that there all partitions instead of non-crossing partitions appear.) It seems that many problems on freeness are easier to handle in terms of these free cumulants than in terms of moments of φ. In particular, the definition of freeness itself becomes much handier for cumulants than for moments. Since cumulants are multi-linear objects this implies that for problems involving the linear structure of the algebra A cumulants are quite easily and effectively to use. For problems involving the multiplicative structure of A, however, it is not so clear from the beginning that cumulants are a useful tool for such investigations. Nevertheless in a lot of examples it has turned out that this is indeed the case. In a sense, we will here present the unifying reason for these positive results. Namely, dealing with multiplicative problems reduces on the level of cumulants essentially to the problem of understanding the structure of cumulants whose arguments are products of variables. Here, in Section 2, we will show that this can be understood quite well and that there exists a nice and simple combinatorial description for such cumulants. That this formula is also useful will be demonstrated in Section 3. We will reprove and generalize a lot of results around the multiplication of free random variables. In particular, we will consider an important special class of distributions, so-called R-diagonal elements. These were introduced by Nica and Speicher in [8]. However, the investigations and characterizations in [8, 9] were not always straightforward and used a lot of ad hoc combinatorics. Our approach here will be much more direct and conceptually clearer. Furthermore, we will get in the same spirit direct proofs of results of Haagerup and Larsen [2, 5] on powers of R-diagonal elements. An important point to make is that all earlier investigations on Rdiagonal elements were always restricted to a tracial frame – i.e., φ was assumed to satisfy the trace condition φ(ab) = φ(ba) for all a, b ∈ A. In contrast, our approach does not rely on this assumption, so all our results are also valid for non-tracial φ. Thus we do not only get simple proofs for known results but also generalizations of all these results to COMBINATORICS OF FREE CUMULANTS 3 the general, non-tracial case. (That non-tracial R-diagonal elements appear quite naturally can, e.g., be seen in [13], where such elements arise in the polar decomposition of generalized circular elements). Our Propositions 3.5 and 3.9 were inspired by and prove some conjectures of the recent work [10]. There the notion of R-diagonality is also treated in the non-tracial case and some of our results of Section 3 are proved there for the general case, too. However, the approach in [10] is quite different from the present one and relies on Fock space representations and freeness with amalgamation. The paper is organized as follows. In Section 1, we give a short and self-contained summary of the relevant basic definitions and facts about free probability theory and non-crossing partitions. In Section 2, we state and prove our main combinatorial result on the structure of free cumulants whose arguments are products and, in Section 3, we apply this result to derive various statements about R-diagonal elements. 1. Preliminaries In this section we provide a short and self-contained summary of the basic definitions and facts needed for our later investigations. 1.1. Non-commutative probability theory. 1)We will always work in the frame of a non-commutative probability space (A, φ). This is, by definition, a pair consisting of a unital ∗-algebra A and a unital linear functional φ : A → C. (φ unital means that φ(1) = 1.) The elements a ∈ A are called non-commutative random variables, or just random variables in (A, φ). Let a1, . . . , an be random variables in a non-commutative probability space (A, φ). Let C〈X1, . . . , Xn〉 denote the algebra of polynomials in n non-commuting indeterminants – i.e., the algebra generated by n free generators. Then the linear functional μa1,...,an : C〈X1, . . . , Xn〉 → C given by linear extension of Xi(1) . . . Xi(m) 7→ φ(ai(1) · · ·ai(m)) (m ∈ N, 1 ≤ i(1), . . . , i(m) ≤ n) is called the joint distribution of a1, . . . , an. The joint distribution of a and a is also called the ∗-distribution of a. Consider random variables ai and bi (1 ≤ i ≤ n) in (A, φ). Then a1, . . . , an and b1, . . . , bn have the same joint distribution, if the following equation holds for all m ∈ N, 1 ≤ i(1), . . . , i(m) ≤ n: φ(ai(1) · · ·ai(m)) = φ(bi(1) · · · bi(m)) . 4 BERNADETTE KRAWCZYK AND ROLAND SPEICHER 2) Note that all our considerations will be on the algebraic (or combinatorial) level, thus we will not require that φ is a positive functional. However, it is well known that freeness – the crucial structure in our investigations is compatible with positivity properties. The requirement that our probability space should be a ∗-algebra and not just an algebra is only for convenience, since, in Section 3, we will need the ∗ for dealing with Haar unitaries and R-diagonal elements. In all statements where no ∗ appears we could also replace the requirement “∗-algebra” by “algebra”. 3) Most of the questions which we will investigate in Section 3 were up to now only considered for tracial linear functionals. We stress that all our considerations do not use the trace property, i.e. we will not use the equation φ(ab) = φ(ba). 1.2. Partitions. 1) Fix n ∈ N. We call π = {V1, . . . , Vr} a partition of S = (1, . . . , n) if and only if the Vi (1 ≤ i ≤ r) are pairwisely disjoint, non-void tuples such that V1 ∪ · · · ∪ Vr = S. We call the tuples V1, . . . , Vr the blocks of π. The number of components of a block V is denoted by |V |. Given two elements p und q with 1 ≤ p, q ≤ n, we write p ∼π q, if p and q belong to the same block of π. We get a linear representation of a partition π by writing all elements 1, . . . , n in a line, supplying each with a vertical line under it and joining the vertical lines of the elements in the same block with a horizontal line. Example: A partition of the tuple S = (1, 2, 3, 4, 5, 6, 7) is π1 = {(1, 4, 5, 7), (2, 3), (6)} =̂ 1 2 3 4 5 6 7 . If we write a block V of a partition in the form V = (v1, . . . , vp) then this shall always imply that v1 < v2 < · · · < vp. 2) A partition π is called non-crossing, if the following situation does not occur: There exist 1 ≤ p1 < q1 < p2 < q2 ≤ n such that p1 ∼π p2 6∼π q1 ∼π q2: 1 · · · p1 · · · q1 · · · p2 · · · q2 · · · n The set of all non-crossing partitions of (1, . . . , n) is denoted by NC(n). In the same way as for (1, . . . , n) one can introduce non-crossing partitions NC(S) for each finite linearly ordered set S. Of course, NC(S) COMBINATORICS OF FREE CUMULANTS 5 depends only on the number of elements in S. In our investigations, non-crossing partitions will appear as partitions of the index set of products of random variables a1 · · ·an. In such a case, we will also sometimes use the notation NC(a1, . . . , an). (If some of the ai are equal, this might make no rigorous sense, but there should arise no problems by this.) If S is the union of two disjoint sets S1 and S2 then, for π1 ∈ NC(S1) and π2 ∈ NC(S2), we let π1 ∪ π2 be that partition of S which has as blocks the blocks of π1 and the blocks of π2. Note that π1 ∪ π2 is not automatically non-crossing. 3) Let π, σ ∈ NC(n) be two non-crossing partitions. We write σ ≤ π, if every block of σ is completely included in a block of π. Hence, we obtain σ out of π by refining the block-structure. For example, we have {(1, 3), (2), (4, 5), (6, 8), (7)} ≤ {(1, 3, 7), (2), (4, 5, 6, 8)}. The partial order≤ induces a lattice structure onNC(n). In particular, given two non-crossing partitions π, σ ∈ NC(n), we have their join π ∨ σ, which is the unique smallest τ ∈ NC(n) such that τ ≥ π and τ ≥ σ. The maximum of NC(n) – the partition which consists of one block with n components – is denoted by 1n. The partition consisting of n blocks, each of which has one component, is the minimum of NC(n) and denoted by 0n. 4) The lattice NC(n) is self-dual and there exists an important antiisomorphism K : NC(n) → NC(n) implementing this self-duality. This complementation map K is defined as follows: Let π be a noncrossing partition of the numbers 1, . . . , n. Furthermore, we consider numbers 1̄, . . . , n̄ with all numbers ordered like 1 1̄ 2 2̄ . . . n n̄ . The complement K(π) of π ∈ NC(n) is defined to be the biggest σ ∈ NC(1̄, . . . , n̄)=̂NC(n) with π ∪ σ ∈ NC(1, 1̄, . . . , n, n̄) . Example: Consider the partition π := {(1, 2, 7), (3), (4, 6), (5), (8)} ∈ NC(8). For the complement K(π) we get K(π) = {(1), (2, 3, 6), (4, 5), (7, 8)} , as can be seen from the graphical representation: 6 BERNADETTE KRAWCZYK AND ROLAND SPEICHER 1 1̄ 2 2̄ 3 3̄ 4 4̄ 5 5̄ 6 6̄ 7 7̄ 8 8̄ . 5) Non-crossing partitions and the complementation map were introduced by Kreweras [4]; for further combinatorial investigations on that lattice, see, e.g., [1, 12]. 6) The main combinatorial ingredient of Theorem 2.2 will be joins with special partitions σ whose blocks consist of neighbouring elements, like π ∨ {(1), (2), . . . , (l, . . . , l + k), . . . , (n)}. This is given by uniting the blocks of π containing the elements l, . . . , l + k, and we say that we obtain π∨{(1), (2), . . . , (l, . . . , l + k), . . . , (n)} by connecting the elements l, . . . , l + k. Example: Considering the partition π = {(1, 8), (2, 3), (4, 5, 7), (6)} =̂ 1 2 3 4 5 6 7 8 we have π ∨ {(1, 2, 3, 4), (5), (6), (7), (8)} = {(1, 2, 3, 4, 5, 7, 8), (6)} =̂ 1 2 3 4 5 6 7 8 . 1.3. Free cumulants. Given a unital linear functional φ : A → C we define corresponding (free) cumulants (kn)n∈N kn : A n → C, (a1, . . . , an) 7→ kn(a1, . . . , an) indirectly by the following system of equations: