Normalized characters of symmetric groups and Boolean cumulants via Khovanov's Heisenberg category

In this paper, we study relationships between the normalized characters of symmetric groups and the Boolean cumulants of Young diagrams. Specifically, we show that each normalized character is a polynomial of twisted Boolean cumulants with coefficients being non-negative integers, and conversely, that, when we expand a Boolean cumulant in terms of normalized characters, the coefficients are again non-negative integers. The main tool is Khovanov's Heisenberg category and the recently established connection of its center to the ring of functions on Young diagrams, which enables one to apply graphical manipulations to the computation of functions on Young diagrams. Therefore, this paper is an attempt to deepen the connection between the asymptotic representation theory and graphical categorification.

1. Introduction 1.1. Backgrounds. Study of the asymptotic behavior of representations of symmetric groups as the group size goes to infinity was initiated by Vershik and Kerov [VK81], and has been developing as a central subject of the so-called asymptotic representation theory. The asymptotic representation theory has not only become a field that gives general ideas of studying towers of algebraic structures such as compact groups [Voi76, VK82, Bia95, CŚ09, BG15, CNŚ18], Hecke algebras [FM12], quantum groups [Sat19, Sat21c, Sat21b, Sat21a], among others apart from symmetric groups [Bia98,Ker03,BO17], but also, due to its amenability to probability theory, given rise to a new stream of probability theory, integrable probability (for survey, see e.g. [BG12,BP14,Cor14]), which has now been widening its perspective even independently of representation theory.
In the case of symmetric groups, the irreducible representations are labelled by Young diagrams (see e.g. [Ful96]). One approach to the asymptotic representation of symmetric groups is to study asymptotic behavior of characters viewed as functions of Young diagrams. As a result, several observables defined on Young diagrams show up to describe the asymptotic behavior. Most prominently, Biane [Bia98] pointed out that free cumulants play significant roles in the asymptotic representation of symmetric groups, and opened up a new approach from free probability and random matrix theory [HP00,NS06,MS17]. In this paper, however, we are concerned with another series of observables, Boolean cumulants [SW93].
Another pillar of this paper is Khovanov's Heisenberg category introduced in [Kho14] for the purpose of categorifying the infinite dimensional Heisenberg algebra. It is a category whose morphisms are given by planar diagrams modulo local relations that reflect induction and restriction of representations of symmetric groups. It was already anticipated in [Kho14] that Khovanov's Heisenberg category could be related to free probability theory in the spirit of [GJS10], which emphasized that the planar structure is shared by subfactors from operator algebras, free probability and random matrices. Remarkably in this direction, [KLM19] established an isomorphism between the center of Khovanov's Heisenberg category and the ring of shifted symmetric functions, and in particular clarified that elements of the center of Khovanov's Heisenberg category give rise to observables on Young diagrams.
We view the results of [KLM19] as what allows for graphical calculus of observables on Young diagrams. In the rest of the paper, we investigate relation between normalized characters of symmetric groups and Boolean cumulants standing on this viewpoint.
1.2. Symmetric groups. For n ∈ Z >0 , let S n be the n-th symmetric group. We write Y n for the set of Young diagrams with n cells. Then the isomorphism classes of irreducible representations of S n are labelled by Y n ; for λ ∈ Y n , let (ρ λ , V λ ) be an irreducible representation corresponding to λ. See e.g. [Ful96] for accounts of representations of symmetric groups (or [OV96,VO05] for an alternative approach.) We define the character of this representation by where e ∈ S n is the unit element. Note that in this convention, we have χ λ (e) = 1. Obviously the definition is independent of the choice of an irreducible representation of type λ. The character χ λ only depends on conjugacy classes of S n , which are in one-to-one correspondence with the set P n of partitions of n. Although the set P n of partitions is naturally identified with the set of Young diagrams Y n , we distinguish them because of their different roles. We write χ λ π for the value of χ λ on the conjugacy class corresponding to a partition π ∈ P n .
Given a partition λ ∈ Y, we consider the following function of z: where (x 1 , . . . , x n ) and (y 1 , . . . , y n−1 ) are the local maxima and local minima of the profile of λ, respectively. Due to the interlacing property (1.1), there exists a unique probability measure ν λ on R supported on {x 1 , . . . , x n } so that the function G λ is its Cauchy transform: This probability measure ν λ is called the transition measure of λ. Obviously, when we expand the Cauchy transform around infinity, the coefficients give the moments of the probability measure in such a way that Note that the property (1.2) ensures that M 1 (λ) = 0, i.e., the transition measure ν λ is centered. We introduce another function; the multiplicative inverse of G λ (z): The Boolean cumulants B k (λ), k ∈ Z >0 of λ are defined [SW93] as the coefficients of the expansion of H λ (z) around z = ∞ so that It is immediate that B 1 (λ) = M 1 (λ) = 0. For our purpose, the twisted Boolean cumulants are also important. They are just the sign change of the Boolean cumulants [RŚ08]:B k (λ) = −B k (λ), k ∈ Z >0 , or in other words, the function H λ is expanded so that 1.4. Main results. In the previous two subsections, we introduced several families of functions on Y. One is that of normalized characters Σ π of the symmetric groups labelled by partitions π ∈ P. We also saw the moments, the Boolean cumulants, and the twisted Boolean cumulants depending on Young diagrams, hence we can regard them as functions on Y. In the following, borrowing terminologies from integrable probability, we often call functions on Y observables.
Let us, in particular, take the twisted Boolean cumulants {B k : k ∈ Z >0 }. Our first main result states that any normalized character up to sign is a polynomial of twisted Boolean cumulants with non-negative integer coefficients. Note that the first twisted Boolean cumulantB 1 ≡ 0 is a constant function, and can be eliminated in the following discussion.
To state the main result, the following notations are convenient: for π ∈ P, we write ℓ(π) for the length of π, i.e., the number of parts in π. We also set |π| = n if π ∈ P n . The following result was presented in [RŚ08] as a conjecture.
Conversely, when we expand Boolean cumulants in terms of normalized characters, we again observe non-negative integer coefficients. Theorem 1.3. For any k ∈ Z ≥2 , the Boolean cumulant B k is expanded in terms of normalized characters so that where m k π ∈ Z ≥0 , π ∈ P. Furthermore, m k π = 0 unless |π| − ℓ(π) ≡ k mod 2.
Organization. In the next Section 2, we introduce Khovanov's Heisenberg category and see some of its properties that are used in subsequent sections. In Section 3, we study the center of Khovanov's Heisenberg category. In particular, we review the construction of observables on the Young diagrams out of the center of Khovanov's Heisenberg category given by [KLM19]. We prove Theorems 1.1 and 1.2 in Section 4, and Theorem 1.3 in Section 5. Finally in Section 6, we make a few comments on related topics and future directions.

Khovanov's Heisenberg category
2.1. Definition. We recall the category H that was introduced in [Kho14]. It is a C-linear monoidal category generated by objects Q + and Q − . For a sequence ε = (ε 1 , . . . , ε n ) with ε i ∈ {+, −}, i = 1, . . . , n, we write Q ε = Q ε 1 ⊗ · · · ⊗ Q εn . Then objects of H are finite direct sums of Q ε with various sequences ε. The unit object 1 = Q ∅ corresponds to the empty sequence. We will express the objects Q + and Q − by upward and downward arrows as Then for a sequence ε = (ε 1 , . . . , ε n ), the object Q ε can be depicted by the corresponding sequence of upward and downward arrows. For instance, the sequence ε = (+ + − + −−) gives For ε = (ε 1 , . . . , ε m ) and ε ′ = (ε ′ 1 , . . . , ε ′ n ), the morphism space Hom H (Q ε , Q ε ′ ) is the C-vector space spanned by oriented chord diagrams that connect the graphical expressions of Q ε and Q ε ′ with matched orientations, modulo isotopy. For instance, the following diagram is a morphism from Q ++−+−− to Q −+ : The morphisms are subject to the following local relations: Composition of morphisms is given by the concatenation of diagrams.
Remark 2.1. In [Kho14] studied was the Karoubi envelop of H, which is usually called Khovanov's Heisenberg category. In fact, it gives a categorification of the infinite dimensional Heisenberg algebra. In the present paper, we do not need the Karoubi envelop since, in the end, we will focus on the center of the category that does not differentiate between H and its Karoubi envelop. Because of this reason, we call H Khovanov's Heisenberg category by abuse of terminology.

Some properties.
Here we collect some properties of H that we will use in the subsequent parts. First we state a trivial fact easily deduced from the definition of H.
For each σ ∈ S n , we draw its image under ϕ n as Next we introduce a shorthand symbol := for the right turn.
Lemma 2.3. We have the following local relations: Proof. As a warm-up for manipulation of diagrams, we show a proof of the first relation. The second relation in (2.2) gives We can use (2.1) to both terms in the right hand side. For the first term, we can see that where we used the second relation in (2.1) in the former equality to "pull the loop rightward", and the first relation in the latter equality to resolve the consecutive two crossings. The latter term is obviously = .
Hence we obtain the desired identity.
We also introduce a notation for multiple insertion of dots: Lemma 2.4. For any k ∈ Z ≥0 , we have the following relation: where the last sum is understood as empty unless k ≥ 2.
Proof. For each i = 0, 1, . . . , k − 1, let us consider the following diagram: Moving one dot from the right of the upward arrow to the left using Lemma 2.3, it becomes The same Lemma 2.3 allows one to transform the second diagram in the right hand side in the following way: Consequently, we obtain the relation where the last sum in the right hand side is understood as empty if i = k − 1. When we sum up (2.4) from i = 0 to i = k − 1, we can see that as is desired.
Proposition 2.5. For any k ∈ Z ≥0 , we have the relation where m ij , n l ∈ Z ≥0 for all i, j, l.
Proof. When k = 0, the assertion is obvious from Lemma 2.4. Assume that the assertion is true up to some k, then from Lemma 2.4, we have Due to the induction hypothesis, each term in the last sum is expressed as . Therefore, the assertion is also ture for k + 1.

Center of Khovanov's Heisenberg category
3.1. Basic description. By definition, the center of the category H is the commutative algebra Z(H) = End H (1).
Let us recall the generators of Z(H) studied in [Kho14]. We set Obviouslyc 0 = 1 andc 1 = 0, hence as for the seriesc k , only those for k ≥ 2 are nontrivial algebra elements. For n ∈ Z ≥0 and k 1 , k 2 ≤ n, we write k 1 (n) k 2 for C[S n ] considered as a (C[S k 1 ], C[S k 2 ])bimodule. Also, for a C[S k 2 ]-module M , we write M for the C[S k 1 ]-module. When either k 1 = n or k 2 = n, we will suppress the notation by simply writing (n) k 2 = n (n) k 2 or k 1 (n) = k 1 (n) n unless there is confusion. Using these notions, compositions of induction and restriction functors are understood as bimodules and natural transformations are as bimodule homomorphisms. For instance, the functor The category S also admits a graphical expression. We can depict induction and restriction functors by arrows that separate numbered regions as Then the composition of functors is expressed by aligning these arrows. In particular, for any object of S k , the rightmost region is numbered k. If a region negatively numbered appears, we regard the object as the zero object.
3.3. Functor F k : H → S k . Due to the obvious similarity of the graphical expression of H and S, we can relate them by certain functors. To be precise, since regions in the graphical expression of S are numbered, we have to specify the number of the rightmost region to consider a functor from H. Let us fix k ∈ Z ≥0 and consider the assignments of objects F k : H → S k that sends Q + and Q − to the appropriate induction and restriction functors, and the tensor product ⊗ to the composition • of functors. For instance, Q ++−+ is sent to Proposition 3.2. Defining the assignment of morphisms by identifying diagrams naturally, F k : H → S k defines a functor.
In fact, the relations imposed on the morphism space of H illustrate the commutation relations of induction and restriction functors.
Note that the endomorphism ring End S k (Id C[S k ]-Mod ) consists of bimodule automorphisms on C[S k ] over (C[S k ], C[S k ]). In other words, the endomorphism ring is simply the center of C[S k ]: Therefore the functor F k induces the ring homomorphism (denoted by the same symbol)

Construction by Kvinge-Licata-Mitchell.
For λ ∈ Y k , we can consider the character χ λ as a function on Z(C[S k ]) by linearly extending it to C[S k ] and restricting it to Z(C[S k ]). Furthermore, when we recall the isomorpshisms (see e.g. [Lam01]) of algebras, we can see that the irreducible character χ λ restricted on Z(C[S k ]) is an algebra homomorphism. We write C[Y] for the ring of complex-valued functions on Y.
Then it is an injective algebraic homomorphism.
Proof. Since the functor F k induces an algebraic homomorphism Z(H) → Z(C[S k ]), and the character χ λ is also an algebraic homomorphism on Z(C[S k ]), it is obvious that their composition, Φ is an algebraic homomorphism (the algebraic structure of C[Y] is defined by the pointwise multiplication.) In [KLM19], the authors showed that algebraically independent generators of Z(H) are sent to algebraically independent elements of C[Y], which implies the injectivity.
Remark 3.4. In [KLM19] established was an isomorphism between Z(H) and the ring of shifted symmetric functions. For our purpose, however, the embedding of Z(H) into C[Y] is sufficient.
The following result from [KLM19] is what we rely on in the proof of our main result.
Remark 3.6. In [KLM19], it was also shown that Φ(c k ) = M k , k ∈ Z ≥0 . In fact the relations between c k , k ∈ Z ≥0 andc k , k ∈ Z ≥0 seen in Proposition 3.1 are equivalent to the functional relation In this section, we prove Theorems 1.1 and 1.2 at the same time.
4.1. Alternative statement. Due to Theorem 3.5, the first half of our main result Theorem 1.1 is equivalent to that for any π ∈ P, (−1) ℓ(π) α π is expressed by a polynomial of −c k , k = 0, 1, . . . with non-negative integer coefficients.
Here we show a slightly stronger result than Theorem 1.1. For a partition π ∈ P n and a sequence i = (i 1 , i 2 , . . . , i n ) ∈ (Z ≥0 ) n , we set We prove the following theorem in this section.

4.2.
Proof of Theorem 4.1. We prove it by induction in terms of |π|. In the base case |π| = 1, the only possibility of π is π = (1). Hence for any i = (i) with i ∈ Z ≥0 , Therefore, we can take P (1),(i) = y i for each i ∈ Z ≥0 , which is certainly of degree i. Let us assume that the assertion is true for all π up to |π| ≤ n, and prove that the assertion is again true for π ∈ P n+1 .
We consider two cases separately. The first case is that π = (n + 1) is of length 1. In this case, we observe for any sequence i = (i 1 , . . . , i n , i n+1 ) ∈ (Z ≥0 ) n+1 that Applying Lemma 2.3 repeatedly to move the dots located on the most internal curl over the crossing, we have By the induction hypothesis, there exists a non-negative integer coefficient polynomial P π,i (y 0 , y 1 , . . . ) such that the first term in the right hand side is expressed as Furthermore each monomial appearing in P π,i is of degree at most n + |i|.
Let us consider each summand in the second term. We shall apply Proposition 2.5 repeatedly to move the internal hoop outside . For each b = 0, 1, . . . , i n+1 − 1, we have . By the induction hypothesis, there exists a non-negative integer coefficient polynomial P b π,i (y 0 , y 1 , . . . ) such that the above element is expressed as Furthermore, all monomials appearing in P b π,i are of degrees at most n + |i| − 2.

5.
Proof of Theorem 1.3 5.1. Alternative statement. The strategy of proving Theorem 1.3 is simiar to the proofs of Theorems 1.1 and 1.2. That is, we employ the graphical calculus of the center of Khovanov's Heisenberg category.
Here we present a stronger result. For σ ∈ S n , we write ℓ(σ) for the Coxeter length of σ, i.e., the minimal number of simple transpositions s i , i = 1, . . . , n − 1 needed to express σ. Also recall the notational convention introduced after Proposition 2.2.
Note that the expansion in Theorem 5.1 is not unique. In fact, for any σ ∈ S n and ρ ∈ S n−1 , we have the identity Nevertheless, the fact that the coefficients are non-negative integers is independent of the way of expansion. Note also that the Coxeter length is invariant under conjugation. It is easy to deduce Theorem 1.3 from Theorem 5.1.
Under the correspondence of Theorem 3.5, we obtain the desired expansion to complete the proof. 5.2. Proof of Theorem 5.1. We prove Theorem 5.1 by induction in k. When k = 0, the assertion is obviously true.
Assume that the assertion is true for all k up to n. Applying Lemma 2.3 repeatedly, we have From the induction hypothesis, the first term in the right hand side is expanded so that where m ′ σ ∈ Z ≥0 and m ′ σ = 0 unless ℓ(σ) ≡ n mod 2. Notice that this is already the desired form of expansion and nothing is to be done for it.
Summing up all the contributions, we obtain the desired expansion.
6. Discussions 6.1. Free cumulants. In asymptotic representation theory, instead of Boolean cumulants, free cumulants [Spe94] are more often discussed. In contrast to that the generating function of Boolean cumulants is given by the multiplicative inverse of the Cauchy transform, the generating function of free cumulants is defined by the inverse mapping of the Cauchy transform; since the transition measure of a Young diagram λ is compactly supported, its Cauchy transform G λ is a conformal map from a neighborhood of infinity to a neighborhood of the origin. Its inverse map K λ , often expressed as K λ = G −1 λ to distinguish it from the multiplicative inverse H λ = G −1 λ , is Laurent expanded around the origin as Here appearing R k (λ), k = 2, 3 . . . are the free cumulants of λ.
One importance of free cumulants in asymptotic representation theory lies in the fact that they describe asymptotic behavior of characters of symmetric groups [Bia98]. Furthermore it is known that normalized characters corresponding to single-row partitions are expressed as polynomials (Kerov polynomials) of free cumulants R k , k = 2, 3, . . . with coefficients being non-negative integers [Bia03,Fér09,DFŚ10].
Despite such importance of free cumulants, at the moment, graphically concise expressions of them are lacking. It could be interesting to search for graphical expressions of free cumulants 6.2. Combinatorial interpretation. Since we have shown Theorem 1.1 by induction in the size of partitions, it is not clear if the coefficients of the polynomials admit combinatorial interpretation. It would be interesting to try to understand Theorem 1.1 combinatorially. Note that the coefficients of Kerov polynomials have a clear combinatorial interpretation [Fér09,DFŚ10].
6.3. Other variants, deformations. There are quantum deformation [LS13] and higher level extensions [MS18,Bru18] of Khovanov's Heisenberg category. These generalizations have been incorporated by [BSW20] into the category denoted by Heis k (z, t), where z and t are invertible elements of the ground field and k ∈ Z. It is interesting to study if these quantum Heisenberg categories can give useful tools to study asymptotic representation of some quantum deformed algebraic structures.
One candidate could be a family of cyclotomic Hecke algebras. Given a polynomial f (x) = f 0 + f 1 x + · · · + f l x l of degree l such that f 0 f 1 = 1, the cyclotomic Hecke algebras H f n , n ∈ Z ≥0 are defined as quotients of affine Hecke algebras. As an analogue of the functors F k : H → S k that we explained in Subsection 3.3, there is a functor [BSW20] Heis −l (z, f −1 under the matching z = q − q −1 with q being the standard quantization parameter of Hecke algebras. Then it is an interesting problem to ask if the construction of [KLM19] can be extended to this setting. Another direction of deformation is the Jack deformation. Although graphical realization of Jack polynomials are still absent, the Jack inner product has been graphically realized by [LRS18] in the context of Frobenius Heisenberg categorification [CL12,RS17] (see also [Sav19] for the Frobenius Heisenberg categorification). One can ask if the results therein are useful in the context of asymptotic representation theory. Toward this direction, it is also important to look for the graphical realization of free cumulants. In fact, there are several attempts to understand the Jack deformed analogue of Kerov polynomials [Las09,DFŚ14].
The analogue of [KLM19] associated with projective (spin) representations of symmetric groups has been established in [KOR20] using the twisted Heisenberg category [CS15,OR17]. It is plausible that our main results have analogues in the case of projective representations. We will report some results along this line in the forthcoming paper. Note that spin normalized characters and the spin analogue of Kerov polynomials have been also studied in [Mat18,MŚ20].