Polynomiality of shifted Plancherel averages and content evaluations

The shifted Plancherel measure is a natural probability measure on strict partitions. We prove a polynomiality property for the averages of the shifted Plancherel measure. As an application, we give alternative proofs of some content evaluation formulas, obtained by Han and Xiong very recently. Our main tool is factorial Schur $Q$-functions.


Partitions
Following to Macdonald's book [16], let us recall the basic knowledge on strict partitions. A partition is a finite weakly-decreasing sequence λ = (λ 1 , λ 2 , . . . , λ l ) of positive integers. The integer (λ) = l is the length of λ and |λ| = l i=1 λ i is the size of λ. If |λ| = n, we say that λ is a partition of n.
A partition λ is said to be strict if all λ i are pairwise distinct, and λ is said to be odd if all λ i are odd integers. Let SP n be the set of all strict partitions of n and OP n the set of all odd partitions of n. The fact that their cardinalities coincide is well known: |SP n | = |OP n |. Set SP = ∞ n=0 SP n and OP = ∞ n=0 OP n . For convenience, we deal with the empty partition ∅ , which is the unique partition in SP 0 = OP 0 with length 0. For each λ ∈ SP, we consider the set The set S(λ) is usually drawn in a graphical way, and called the shifted Young diagram of λ. Each element = (i, j) ∈ S(λ) is often called a box of λ.

Shifted Plancherel measure
In this paper, we consider the following probability measure on SP n , studied in many papers, e.g., in [1,10,17]. Definition 1.1. The shifted Plancherel measure P n on SP n is defined by P n (λ) = 2 n− (λ) (g λ ) 2 n! .

Theorem 1.2. Suppose that f is a supersymmetric function. Then
is a polynomial function in n.
We introduce a notation x ↓k as for a variable x and a positive integer k. If n is an integer with 0 ≤ n < k then n ↓k = 0. We also set x ↓0 = 1.
In some supersymmetric functions f , we give the explicit expressions of E n [f ] as linear combinations of descending powers n ↓j . In fact, we will show

A deformation
Fix a strict partition µ of m. We define the measure P µ,n on SP n+m by Note that P µ,n (λ) = 0 unless S(λ) ⊃ S(µ). We will prove that P µ,n is a probability, i.e., λ∈SP n+m P µ,n (λ) = 1. For a function ϕ on SP, define The summation E µ,n [ϕ] is considered in [8].

Content evaluations
For each box = (i, j) in S(λ), we define c = j −i and call it the content of . We deal with symmetric functions evaluated by quantities c , where is a polynomial function in n.
In [8], the reason why they consider the quantity F ( c : ∈ S(λ)) is not presented. We note that the multi-set { c | ∈ S(λ)} forms the collection of squares of eigenvalues with respect to projective analogs of Jucys-Murphy elements, see [20,23,26,27] and also [1,Theorem 3.2]. We give the explicit expressions of some content evaluations. In fact, we will show Furthermore, we will give a new algebraic proof of the identity which is given in [8, Theorem 1.3].

Related research and the aim
Let P n be the set of all (not necessary strict) partitions of n. The (traditional) Plancherel probability measure P Plan n on P n is defined by where f λ is the number of standard tableaux of shape Y (λ). Here Y (λ) is the ordinary Young diagram of λ: Let F be a symmetric function. In [25] (see also [7]), Stanley proves that the summation is a polynomial in n. Here h denotes the hook length of the square in the Young diagram Y (λ). Panova [22] shows an explicit identity for the symmetric function . Moreover, Stanley [25] proves that the content evaluation is also a polynomial in n. Olshanski [21] finds that the functions λ → F (c : ∈ Y (λ)) are seen as shifted-symmetric functions in variables 59 λ 1 , λ 2 , . . . and obtains an alternative algebraic proof for the polynomiality of (1.2). Some explicit formulas for particular F are obtained in [5,6,14,15,18,19]. Just as an example, in [6] the identity We emphasize the fact that the content evaluations are related to matrix integrals ( [18,19]). For example, let us consider the unitary group U (N ) with the normalized Haar measure dU and suppose n ≤ N . Then Weingarten calculus gives the following identity U (N ) Here h k are complete symmetric functions. More general identities (for other classical groups) can be seen in [18,19]. The quantity F ( c : ∈ S(λ)) in Subsection 1.4 is a natural projective analog of F (c : ∈ Y (λ)), because the c are eigenvalues of Jucys-Murphy elements of the symmetric groups, while the c come from their projective version. Unfortunately, it is not known any direct connection between matrix integrals and the projective content evaluation F ( c : ∈ S(λ)).
Our results in this paper are seen as the counterparts of the content evaluation [21] in the theory of the shifted Plancherel measure. As Olshanski does in [21], we employ factorial versions of symmetric functions. Specifically, we introduce a new family of supersymmetric functions (p ρ ) ρ∈OP . The function p ρ is also regarded as projective (or spin) irreducible character values of the symmetric groups. For ordinary partitions, the counterpart is the normalized linear character, which has been studied in e.g. [2,3,4,12,24], and written as p # ρ , χ ρ , Ch ρ , . . . in their articles. We will provide explicit values of shifted Plancherel averages E µ,n [p ρ ] for all strict partitions µ and odd partitions ρ.
As mentioned above, Corollary 1.5 is obtained by Han and Xiong [8]. Our purpose in this paper is to provide more insight for their result, based on the theory of factorial Schur Q-functions. As a result, we can obtain some new identities given in Subsections 1.2 and 1.4 in a simple way.

Outline of the paper
The paper is organized as follows. Section 2 gives definitions and basis properties of Schur P -and factorial Schur P -functions. A more detailed description can be seen in [10,11,16]. In Section 3 we introduce new supersymmetric functions p ρ and provide some necessary properties. In Section 4 we give the proofs of Theorem 1.2 and Theorem 1.3. New identities presented in Subsection 1.2 are also proved. In Section 5 we give a proof of Theorem 1.4 and present some examples of content evaluations. In Section 6 we deal with some family of functions on SP introduced in [8] and show that they are supersymmetric functions. We comment on some remaining questions in Section 7.

The algebra of supersymmetric functions
A symmetric function is a collection of polynomials F = (F N ) N =1,2,... with rational coefficients such that We often write F as F (x 1 , x 2 , . . . ) in infinitely-many variables x 1 , x 2 , . . . . For each r = 1, 2, . . . , the r-th power-sum symmetric function p r is given by It is well known that the p r generate the algebra of all symmetric functions and are algebraically independent over Q.
The p ρ form a linear basis of Γ by definition. The scalar product on Γ is defined by Given an element f in Γ and a strict partition λ, we denote by f (λ) the value . Elements in Γ are uniquely defined by their values on SP, i.e. two elements f, g in Γ coincide with each other if and only if it holds that f (λ) = g(λ) for every strict partition λ.

Schur P -functions
Let us review the Schur P -function, which is the particular t = −1 case of the Hall-Littlewood function with parameter t. We use the definition in [10]. See also [16,Chapter III.8] and [9] for details.
(3): By linearity, it is enough to show the identity for f = P µ and g = Q ν with |µ| = |ν| = n. Then both sides are equal to δ µν by (2).
For a strict partition λ and odd partition ρ of sizes k, we define Equivalently, the quantities X λ ρ are determined as transition matrices via The quantity X λ ρ is a character value for a projective representation of symmetric groups, see [9,Chapter 8]. One can compute values X λ ρ recursively if we use a Murnaghan-Nakayama rule ([16, Chapter III.8, Exam- Proof. It follows from (2.3) and Proposition 2.3 (2) that the left hand side of which equals 2 − (ρ) z ρ δ ρσ by (2.1).

Factorial Schur P -functions
The next definition is due to A. Okounkov and given in [10].
Remark that P λ is homogeneous, whereas P * λ is not. Let us review some properties for factorial Schur P -functions. See [10,11] for detail. We note that (1)-(3) in the next proposition are immediately comfirmed from definitions, while the proof of (4) requires a more careful work. Proposition 2.6.
Next we give a formula for an expansion of P λ in terms of P * µ . Recall the Stirling numbers T (k, j) of the second kind defined by Proposition 2.7. Let λ be a strict partition of length l. Then T (λ 1 , j 1 ) · · · T (λ l , j l )P * (j 1 ,...,j l ) .
Note that Ψ −1 (f ) coincides with the top-degree term of f by Proposition 2.6 (1).
Substituting ρ = (1 k ) in Proposition 3.2 (iv), we obtain p (1 k ) (λ) = |λ| ↓k . Using Stirling numbers defined in (2.5), we find More generally, a power-sum function p ρ can be expanded as a linear combination of p σ in the following way. First, we expand p ρ in terms of P λ by using (2.3). Second, each P λ is expanded in terms of factorial Schur P -functions P * µ by Proposition 2.7. Finally, each P * µ is expanded in terms of p σ by the formula which is the image of the second equation on (2.3) under Ψ.

Remark 3.4. For two ordinary partitions λ, µ, we consider
where χ λ µ∪(1 |λ|−|µ| ) is the value of the irreducible character χ λ of the symmetric group S |λ| at conjugacy class associated with µ ∪ (1 |λ|−|µ| ). The functions Ch µ on the set of all partitions are called the normalized characters of symmetric groups, and have rich properties and applications. See [4,13,24]. Note that the function is written as p # µ in [13]. Our function p ρ is a projective analog of Ch µ since X λ ρ is a character value for a projective representation of symmetric groups.

Proof of Polynomiality
In the present section we give a proof of Theorems 1.2 and 1.3. Let m be a nonnegative integer. Fix µ ∈ SP m . For each ρ ∈ OP, we consider the summation Since Proposition 3.2 (iv) implies that Here the differential operator ∂ ∂p 1 acts on functions in Γ expressed as polynomials in p 1 , p 3 , p 5 , . . . .
Substituting ρ = ∅ in Theorem 4.2, we obtain E µ,n [1] = 1, which shows that P µ,n defined in (1.1) is a probability measure on SP n+m and that E µ,n is the average with respect to P µ,n .

Orthogonality for p ρ
We can also easily compute the P n -average of products p ρ p σ .

Bound for degrees
The following proposition is a direct consequence of Corollary 4.3.

Proposition 4.5.
Let f be a supersymmetric function. If we expand f as a linear combination of p ρ : In particular, if a (1 r ) (f ) vanish for all r > k, then E n [f ] is of degree at most k.

Examples
We show some explicit expressions of E n [p ρ ], which are presented in Subsection 1.2. In Example 3.3, we give expansions of some p ρ in p σ . By Corollary 4.3, we obtain the following identities immediately.

Supersymmetry
In this section, we give a proof of Theorem 1.4. Let λ be a strict partition and recall the shifted Young diagram For each = (i, j) ∈ S(λ), its content c is defined by c = j − i. We find as multi-sets.
for any strict partition λ.
Proof. Consider the function The Taylor expansion of log Φ(u; λ) at u = 0 is On the other hand, since we see that In this expression, the Taylor expansion of log Φ(u; λ) at u = 0 is Comparing coefficients in two expressions of the Taylor expansion, we obtain the desired formula. c (c + 1) k is supersymmetric. Moreover, if we set p 0 (λ) = |λ|, then p 0 is also supersymmetric.

Remarks on functions introduced by Han and Xiong
We identify a strict partition λ with its shifted Young diagram S(λ) as usual. A box = (i, j) in S(λ) is said to be an outer corner of λ if we obtain a new strict partition by removing the box from S(λ). A box ∈ Z 2 is said to be an inner corner of λ if we obtain a new strict partition by adding the box to S(λ). Denote by O λ and by I λ the set of all outer and inner corners of λ, respectively. For example, if λ = (5, 4, 2), then we have O λ = {(2, 5), (3, 4)} and I λ = {(1, 6), (3,5), (4, 4)}.
For each integer k ≥ 1, we define a function ψ k on SP by In their paper [8], Han and Xiong first introduced those functions. Remark that these are denoted by q k (or by Φ k ) in their articles with slight change ψ k = 2 k q k . Our purpose in this short section is to give an alternative simple expression of ψ k and to show that they are supersymmetric.
In particular, ψ k is a supersymmetric function.
Proof. Taking the logarithm of (6.1), we find Expanding the logarithm functions and comparing the coefficient of u k on both sides, we obtain the first equality in the theorem. The remaining equality is obtained by applying the binomial theorem for the first equality.

Hook evaluations
Recall the ordinary Plancherel measure P Plan n on partitions. As we mention in Subsection 1.5, Stanley [25] (see also [7]) proves that the summation λ∈Pn P Plan n (λ)F h 2 : ∈ Y (λ) is a polynomial in n for any symmetric function F , where h denotes the hook length of the square in the Young diagram Y (λ). What is the analog of this result for the shifted Plancherel measure on strict partitions?