A Spin Analogue of Kerov Polynomials

Kerov polynomials describe normalized irreducible characters of the symmetric groups in terms of the free cumulants associated with Young diagrams. We suggest well-suited counterparts of the Kerov polynomials in spin (or projective) representation settings. We show that spin analogues of irreducible characters are polynomials in even free cumulants associated with double diagrams of strict partitions. Moreover, we present a conjecture for the positivity of their coefficients.


Characters of symmetric groups
Irreducible representations of symmetric groups S n are indexed by partitions λ of n, or equivalently by Young diagrams of size n. The corresponding character χ λ takes values at conjugacy classes C ν in S n , which are also indexed by partitions ν of n. The character values χ λ ν = χ λ (C ν ) have been studied for a long time by innumerable researchers, see, e.g., [20]. Recently, for several problems in the asymptotic representation theory, we often deal with the χ λ ν , fixing ν and letting λ vary. More precisely, for a partition ν of k and a partition λ of n with k ≤ n, we define Ch ν (λ) = n(n − 1) · · · (n − k + 1) where f λ := χ λ (1 |λ| ) denotes the dimension of the irreducible representation of the symmetric group S |λ| associated with λ. We set Ch ν (λ) = 0 whenever k > n. Then the Ch ν are functions on all Young diagrams. An approach from this point of view is sometimes referred to as dual approach or dual combinatorics to characters of symmetric groups [6,9].
Here we omit K 6 and K 8 because we will compare odd-numbered polynomials with Example 1.5 below. We can see a complete list of K k up to k = 11 in [2].
The polynomials K k in the theorem are called Kerov polynomials. We can observe that all coefficients in the above examples are nonnegative integers. This surprising phenomenon is very nontrivial and had been called the Kerov (positivity) conjecture. It was finally proved by Féray [7]. The authors in [5] obtained clearer combinatorial interpretations. Theorem 1.3 (Féray [7]). All coefficients of K k are nonnegative integers.
Recently, a generalization of Kerov polynomials involving Jack polynomials has been actively studied in [4,16,21]. In this paper, we work on research in another direction. Our aim here is to present an answer to the following question: What is the counterpart of the Kerov polynomials in the spin representation setting?

Spin representations
Spin (or projective) representation theory of symmetric groups was introduced by Schur. Currently, it can be understood via a double covering of the symmetric group or via the Hecke-Clifford algebra, see, e.g., [22]. In the spin case, the corresponding character values X λ ν are indexed by strict partitions λ and odd partitions ν. For a fixed odd partition ν, we can define a spin version p ν of (1.1), which is a function on the set of all strict partitions. The function p ν was first introduced by Ivanov [12, 13] (with notation p # ν ) and developed in author's recent work [18].

Results
To define a spin analogue of Kerov polynomials, we need to find a spin analogue of free cumulants R j (λ). Furthermore, it is natural to expect that the same statement of Theorem 1.3 holds in the spin setting. The aim in this paper is to establish a spin analogue of Kerov polynomials by using a spin analogue R 2j (λ) of R j , which is defined as the half of even free cumulants of the double diagram of a strict partition λ. Indeed, we obtain the following theorem for p 2k−1 with one-row odd partition ν = (2k − 1). Theorem 1.4. For each k = 1, 2, 3, . . . , there exists a polynomial K spin 2k−1 in k variables with rational coefficients, such that p 2k−1 (λ) = K spin 2k−1 (R 2 (λ), R 4 (λ), . . . , R 2k (λ)) for all strict partitions λ. Furthermore, K spin 2k−1 is of the form K spin 2k−1 = R 2k + (a polynomial in R 2 , R 4 , . . . , R 2k−2 of degree at most 2k − 2). Here we only show that the coefficients in K spin 2k−1 are rational numbers. The degree of R 2k is regarded as deg R 2k = 2k − 1. We call the polynomial K spin 2k−1 a spin Kerov polynomial. Example 1.5.
It seems that formulas in Example 1.5 resemble those in Example 1.2. The proof of Theorem 1.4 is accomplished by observing relations among some collections of generators in a symmetric function algebra Γ. We obtain formulas in Example 1.5 as the by-products. Not only Theorem 1.4 but also Example 1.5 is an important result in this paper because the latter one indicates a spin analogue of Theorem 1.3 to us. Conjecture 1.6 (spin Kerov conjecture). All coefficients in spin Kerov polynomials are nonnegative integers.

Discussion and outline
Comparing Example 1.5 with Example 1.2, we can find some interesting coincidences for coefficients. For example, the coefficient of linear terms R 2j in K spin 2k−1 likely coincides with that of R 2j in K 2k−1 . Furthermore, if 2s 2 + 4s 4 + · · · = 2k − 2, the coefficient of R s 2 2 R s 4 4 · · · in K spin 2k−1 likely coincide with 2 s 2 +s 4 +···−1 times the coefficient of R s 2 2 R s 4 4 · · · in K 2k−1 . We do not discuss their coincidence or combinatorial interpretations here. We leave them in future work.
We give the proof of Theorem 1.4 and how to derive Example 1.5 with precise definitions in Section 3 after the review of preliminary facts for free cumulants Section 2. As mentioned, the ordinary Kerov conjecture was first proved by Féray [7]. After that, the authors of [5] gave alternative proof by using Stanley-Féray polynomials. The Stanley-Féray polynomials were analyzed by using Young symmetrizers [8]. In order to attack Conjecture 1.6, it is a natural practice to study projective Young symmetrizers. The projective Young symmetrizer was studied in [15,19]. Unfortunately it was quite complicated, so it seems difficult to apply it to our problem.
In Section 4, we try another choice for free cumulants. The another free cumulants R 2k (λ) comes from a symmetrized double diagram D(λ) of λ. The symmetrized diagrams D(λ) are useful for the study of asymptotics of Plancherel measures in the spin setting (see [13] and [3,Chapter 4]). Since odd-numbered quantities R 2k−1 vanish by virtue of the symmetry of D(λ), they are simpler than free cumulants of D(λ) in a sense. Even if we replace R 2k with R 2k , we can show the existence of "spin Kerov polynomials" easily. However, the corresponding polynomial for p 2k−1 with k = 2 has a non-integer coefficient. In this sense, we view the R 2k as being unsuitable for spin Kerov polynomials. We thus believe that the R 2k are the most appropriate choice in the spin setting.

Preliminary
In the present section, we review the transition measures of a partition according to [11,14].
The number of nonzero λ i is called the length of λ and written as (λ).
We usually identity a partition with its Young diagram. We denote by λ = (λ 1 , λ 2 , . . . ) the conjugate partition of λ, i.e., the Young diagram of λ is the transpose of that of λ with respect to the diagonal line. Define d = d(λ) by the number of boxes on the diagonal line in the Young diagram of λ. The modified Frobenius notation For example, the modified Frobenius notation of λ = (5, 4, 4, 1, 1)

Free cumulants
Let µ be a probability measure on R with a compact support. Define the Cauchy transform of µ by where the M k [µ] are moments of µ: Free cumulants R k [µ] (k = 1, 2, . . . ) of µ are defined via the famous free cumulant-moment formula, see, e.g., [11,Chapter 1]. We do not need the explicit definition here. We only use the relation with some integer coefficients c ν .

Kerov's transition measures
We draw the Young diagram of a partition λ in Russian style, see, e.g., [11,Fig. 2.1]. Let be the corresponding local minimas and maximas, which are integers by definition. We call these numbers the Kerov interlacing coordinates of λ. It is known that they satisfy the relation For each partition λ, we define a probability measure m λ on R via the Cauchy transform .
This probability measure, called Kerov's transition measure of λ, is supported by the set for k ≥ 2.

Super power-sums
These are also determined via

Rayleigh measures
For each partition λ with Kerov's interlacing coordinate x 1 < y 1 < · · · < y r−1 < x r , we introduce an R-valued measure τ λ on R by where δ x is the Dirac measure at a point x. This is called the Rayleigh measure of λ. The moments are clearly given by

Relations between two measures
Two measures m λ and τ λ have the following relation, which is nothing but that the relation between complete symmetric functions and power-sum symmetric functions. Equivalently, Combining above formulas, we can express free cumulants R k (λ) in terms of functions p super j (λ) in principle. This fact will be applied in the next section.

Spin Kerov polynomials
Our goal of this section is to define the counterpart of the Kerov polynomial in the spin representation setting.
On the left, the Young diagram (in English style) is a collection of left-justified rows of boxes where the i-th row has λ i boxes reading from top to bottom. On the middle, the shifted diagram is obtained from the Young diagram by shifting the i-th row (i − 1) boxes to the right, for each i ≥ 2. Moreover, on the right, the Young diagram of D(λ), or the double diagram of λ, is obtained by combining the shifted diagram with its reflection (the gray area) on the diagonal.
A (not necessary strict) partition ρ = (ρ 1 , ρ 2 , . . . ) is said to be odd if all nonzero ρ i are odd. It is well known that the number of strict partitions of n coincides with that of odd partitions of n.
For each λ ∈ SP, denote by P λ Schur's P -function, see the definition in [17, Chapter III.8]. The family {P λ | λ ∈ SP} and {p ρ | ρ are odd partitions} form a linear basis of Γ, respectively. The quantity X λ ρ , where λ is a strict partition and ρ is an odd partition with |λ| = |ρ|, is defined via the relation Put g λ = X λ (1 |λ| ) . These quantities X λ ρ are integers and encode character values of irreducible spin representations of symmetric groups [10]. In particular, a positive integer g λ times a power of 2 coincides with the dimension of an irreducible spin representation of symmetric groups.
Each symmetric function f in Γ is regarded as a Q-valued function on SP, by For two functions f, g ∈ Γ, it holds f (λ) = g(λ) for all λ ∈ SP if and only if f = g [12, Proposition 6.2].
Remark 3.3. The formulas in Example 3.2 are also obtained in the following way. First, we expand p ρ in terms of factorial Schur P -functions P * λ . Second, we expand each P * λ in terms of (ordinary) Schur P -functions P ν . Finally, we expand each P ν in terms of odd power-sum symmetric functions p σ . See [18,Example 3.3] for details.
Proof . Using (2.4) and (3.1), we have Taking the logarithm of the right hand side and expanding it at z = ∞, Changing variable j → n = 2k + j + 1, we have On the other hand, from (2.4) we have log φ(z; D(λ)) = ∞ n=1 1 nz n p super n (D(λ)).
Comparing the coefficient of z −n in the above equations, we obtain the desired identity.

Moments of Rayleigh measures
Recall the Rayleigh measure τ λ defined in Section 2.5.

Proof of main theorem
For each k = 1, 2, 3, . . . and λ ∈ SP, we set The degree of R 2k in Γ is 2k − 1 as we proved in Corollary 3.7.

Computations for spin Kerov polynomials
In this subsection, we explain how to obtain formulas in Example 1.5. We use Okounkov's idea which is employed in [2] for ordinary Kerov polynomials. Recall the function G m λ (z) defined in Section 2.3, which is the Cauchy transform of the transition measure associated with λ.
Proposition 3.8. For each strict partition λ, we have Proof . Immediate from Lemma 2.1 and the identity (3.5).
The expression (3.7) can be rewritten by (3.4) as Using this with computer, we can obtain expansions of R 2k in terms of power-sums.

Another double diagrams
In this section, we deal with another double diagram of a strict partition λ. It is considered in [13] and De Stavola's thesis [3]. We denote it by D(λ) and call it the symmetrized double diagram. We do not give its explicit definition here, see [3, Fig. 11]. The diagram D(λ) is not a Young diagram associated to any partition but we can extend notions in Section 2 to such diagrams, see [11,Chapter 2.2]. The diagram D(λ) is drawn in the Russian style and has Kerov's interlacing coordinate where x −i = −x i , y −i = −y i , and x 0 = 0. Note that these numbers except x 0 belong to Z + 1 2 . The corresponding Kerov's transition measure m D(λ) is characterized by its Cauchy transform Compare this lemma with Proposition 3.8. The remaining discussion is the same with that in the previous section, so we write only results.
For each k = 1, 2, 3, . . . and λ ∈ SP, we set Then it has the expression It is easy to see that R k = 0 for odd k. We remark that R k (D(λ)) does not vanish even if k is odd.
Example 4.4. p 1 = R 2 , p 3 = R 4 + 7 4 R 2 . From this example, we find that coefficients of polynomials in Proposition 4.3 are not integers. This indicates that these polynomials are ineligible for "spin Kerov polynomials". We thus believe that the R 2k defined in Section 3 are the most appropriate choice for spin Kerov polynomials.