A Free Cumulant? Jonathan Novak and Piotr Sniady´

WHAT IS... ? a Free Cumulant? Jonathan Novak and Piotr Sniady´

Dedicated to Roland Speicher on the occasion of his ﬁftieth birthday.

Cumulants are quantities coupled to combinato- Formula (1) is also well known to probabilists. n rial notions of “connectivity” and probabilistic In stochastic applications, mn = mn(X) = EX is notions of “independence”. There are two princi- the moment sequence of a random variable, and pal species of cumulants: classical and free. Our the quantities cn(X) implicitly defined by (1) are discussion begins with the more widely known called the cumulants of X. This term was coined classical cumulants. by R. Fisher and J. Wishart in 1932. Cumulants Suppose one wishes to determine the num- were, however, first investigated by the Danish ber cn of connected graphs on the vertex set astronomer T. Thiele as early as 1889. He called [n] = {1,...,n}. The total number of (possibly them half-invariants because (n) disconnected) graphs on [n] is mn = 2 2 , and X,Y independent ⇒ cn(X + Y ) = cn(X) + cn(Y ). because any graph is the disjoint union of its connected components, we have This linear behavior is what gives cumulants their advantage: in most situations cumulants are the (1) mn = X Y c|B|, “right” quantities to work with. For example, the π∈P(n) B∈π universality of the standard Gaussian distribu- where the sum runs over all partitions π = B1 ⊔ tion is reflected in the simplicity of its cumulant B2 ⊔... of [n] into disjoint nonempty subsets. The sequence 0, 1, 0, 0, 0,... . sequence (mn) thus recursively determines (cn) Let us now consider a geometric variation of our via (1). initial graph-counting problem. Given a graph G on More generally, if mn is the number of “struc- [n], we may represent the vertices of G as points tures” that can be placed on [n], and cn is the on a circle and the edges of G as line segments number of “connected structures” on [n] of the joining these points. The resulting picture may be same type, then the sequences (mn) and (cn) are re- connected even if G is not (Figure 1). Let κn denote lated as in (1). This fundamental enumerative link, the number of “geometrically connected” graphs which is often expressed in terms of generating on [n]. As before, we consider a partition π of [n] functions, is ubiquitous in mathematics. Promi- which tells which vertices of G belong to the same nent examples come from enumerative algebraic connected component of its geometric realization. geometry, where connected covers of Riemann surfaces are counted in terms of all covers, and quantum field theory, where sums over connected 3 2 Feynman diagrams are computed in terms of sums over all diagrams. 4 1

Piotr Sniady´ is a professor in the Polish Academy of Sci- ences and the University of Wroclaw. His email address is 5 6 [email protected]. Jonathan Novak is a postdoctoral fellow at MSRI and the Figure 1. Example of a geometrically University of Waterloo. His email address is j2novak@ connected graph. math.uwaterloo.ca.

300 Notices of the AMS Volume 58, Number 2 It turns out that π is always noncrossing: if we Free probability theory was initiated by represent its blocks as convex polygons, they will Voiculescu in the 1980s, who used it to solve be disjoint (Figure 2). It follows that several previously intractable problems in operator algebras. Thus it is a very new field of = (2) mn X Y κ|B|, mathematics, albeit an exceptionally successful ∈N C ∈ π (n) B π one. It therefore came as a surprise when P. Biane where now the summation runs over noncrossing and S. Kerov discovered that free cumulants partitions of [n]. play an important role in a very classical the- Remarkably, just like the cn’s of formula (1), ory, namely the representation theory of the the κn’s of formula (2) may be realized as symmetric groups. half-invariants. Recall that a noncommutative Given a finite group G and a conjugacy class C probability space is a complex algebra A to- of G (viewed as an element of the group algebra gether with a linear functional ϕ : A → C sending C[G]), one knows that C acts as a scalar operator 1A to 1. One regards the elements of A as random in any irreducible representation of C[G]. The variables, with ϕ playing the role of expectation. value of this scalar defines a function on conju- The decision to view noncommutative algebras as gacy classes, called the central character of the probability spaces is justified a posteriori by the irreducible representation. Understanding an irre- fact that this framework supports a new notion of ducible representation amounts to computing its independence, called free independence, modeled central character which, typically, is very difficult. on the free product of algebras. Free independence In the case of the symmetric group G = SN , free is the key to a rich noncommutative probability cumulants shed substantial light on this question. theory—D. Voiculescu’s free probability theory— One begins by considering the Jucys-Murphy ele- which is now widely studied and well known to ment XN = (1 N + 1) ++ (N N + 1) ∈ C[SN+1], appear in a wide variety of contexts, for example, which is the sum of all transpositions interchang- in the large dimension limit of random matrix ing N + 1 with a smaller number. The spectrum theory. of XN can be completely understood in any ir- The realization that formula (2) holds the key reducible representation; thus XN and simple to free half-invariants is due to R. Speicher. functions of XN may be considered “known”. Given a random variable X in a noncommu- One then defines the nth moment mn = mn(XN ) n tative probability space, we associate with its of XN to be that part of the expansion of XN which n belongs to C[SN ]. These moments are not scalars moment sequence mn(X) = ϕ(X ) the free cumu- but central elements in C[SN ], and they give rise lant sequence κn(X) via (2). This is completely analogous to the construction of classical cumu- to free cumulants κn = κn(XN ) via (2). Remark- lants, the only difference being that the lattice of ably, the conjugacy class Ck of k-cycles can be all partitions has been replaced by the lattice of expressed as a polynomial in the free cumulants noncrossing partitions. Fundamentally, this means of Xn, and furthermore one has the first-order = + that set-theoretic connectivity has been replaced approximation Ck κk+1 lower order terms. with geometric connectivity. Speicher showed that This is not the only surprising appearanceof the free cumulant concept. For example, R. Stanley has X,Y freely indep. ⇒ κn(X + Y ) = κn(X) + κn(Y ). shown that the free cumulants of M. Haiman’s park- Indeed, the relationship between free indepen- ing function symmetric functions are precisely dence and free cumulants mirrors the relationship the complete symmetric functions. M. Lassalle has between classical independence and classical cu- formulated conjectures linking free cumulants mulants in every conceivable way. For example,the to Jack polynomials. Where will free cumulants analogue of the Gaussian distribution in free prob- appear next? ability theory is the Wigner semicircle distribution, and its universality is reflected in the simplicity of References Philippe Biane its free cumulant sequence 0, 1, 0, 0, 0,... . [1] , Free probability and combinatorics. In Proceedings of the International Congress of Math- ematicians, Vol. II (Beijing, 2002), pages 765–774, Beijing, 2002, Higher Ed. Press. [2] , Free probability for probabilists. In Quantum probability communications, Vol. XI (Grenoble, 1998), QP-PQ, XI, pages 55–71, World Sci. Publ., River Edge, NJ, 2003. [3] Alexandru Nica and Roland Speicher, Lectures on the combinatorics of free probability, volume 335 of London Mathematical Society Lecture Note Series, Figure 2. Noncrossing (left) versus a crossing Cambridge University Press, Cambridge, 2006. (right) partition.

February 2011 Notices of the AMS 301