Polynomiality of shifted Plancherel averages and content evaluations

BLAISE PASCAL

Sho Matsumoto

Polynomiality of shifted Plancherel averages and content evaluations

Volume 24, n^o1 (2017), p. 55-82.

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Polynomiality of shifted Plancherel averages and content evaluations

Sho Matsumoto

Abstract

The shifted Plancherel measure is a natural probability measure on strict parti- tions. 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 SchurQ-functions.

Résumé

La mesure de Plancherel décalée est une mesure de probabilité naturelle sur les partitions strictes. Nous démontrons une propriété de polynomialité pour les moyennes de mesures de Plancherel décalées. Comme application, nous don- nons une nouvelle preuve de certaines formules d’évaluation des contenus obtenues par Han et Xiong très récemment. Nous utilisons, comme outil principal, les Q- fonctions de Schur factorielles.

1. Introduction

1.1. 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 = (λ₁, λ₂, . . . , λ_l) of positive integers. The integer`(λ) =l is thelength of λ and |λ|=^Pl_i=1λ_i is the size ofλ. If |λ|=n, we say thatλis a partition of n.

A partition λ is said to be strict if all λi 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 = ^S∞_n=0SP_n and OP = ^S∞_n=0OP_n. For convenience, we deal with

The author is supported by JSPS KAKENHI Grant Number 25800062, 24340003.

Keywords:strict partition, Plancherel measure, SchurQ-function, content.

Math. classification:05E05, 05A19, 60C05.

the empty partition∅ , which is the unique partition inSP₀ =OP₀ with length 0.

For each λ∈ SP, we consider the set

S(λ) ={(i, j)∈Z² |1≤i≤`(λ), i≤j≤λi+i−1}.

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.

Let λ, µ be strict partitions such thatS(λ) ⊃S(µ). Put k =|λ| − |µ|.

A standard tableau of shape S(λ/µ) is a sequence of strict partitions (λ⁽⁰⁾, λ⁽¹⁾, . . . , λ^(k)) such that λ⁽⁰⁾ = µ; λ^(k) = λ; and that for each i = 1,2, . . . , k, the diagram S(λ⁽ⁱ⁾) is obtained fromS(λ⁽ⁱ⁻¹⁾) by adding exactly one box. We denote by g^λ/µ the number of standard tableaux of shapeS(λ/µ). We set g^λ/µ = 0 unlessS(λ)⊃S(µ). Defineg^λ=g^λ/∅. 1.2. Shifted Plancherel measure

In this paper, we consider the following probability measure onSP_n, stud- ied in many papers, e.g., in [1, 10, 17].

Definition 1.1. Theshifted Plancherel measure Pn onSP_nis defined by Pn(λ) = 2^n−`(λ)(g^λ)²

n! .

This indeed defines a probability since the identity [9, Corollary 10.8]

X

λ∈SPn

2^n−`(λ)(g^λ)²=n!

holds. For example, if n = 5 then P5((5)) = ¹⁶_5!, P5((4,1)) = ⁷²_5!, and P5((3,2)) = ³²_5!.

For a function ϕonSP, we call En[ϕ] = ^X

λ∈SPn

Pn(λ)ϕ(λ) = ^X

λ∈SPn

2^n−`(λ)(g^λ)²

n! ϕ(λ)

the shifted Plancherel average of ϕ.

Let {x₁, x₂, . . .} be formal variables, then for each positive integer r, the r-th power-sum symmetric function is defined by

pr(x1, x2, . . .) =x^r₁+x^r₂+· · · .

It is well known that the algebra of symmetric functions is generated by the family {p_r}. We denote by Γ the subalgebra generated by {p_2m+1}m=0,1,2,.... Elements in Γ are sometimes called supersymmetric functions, adapted to strict partitions. The following theorem claims a polynomiality of En[f] for any supersymmetric functionf.

Theorem 1.2. Suppose that f is a supersymmetric function. Then

En[f] = ^X

λ∈SP_n

2^n−`(λ)(g^λ)²

n! fλ₁, λ₂, . . . , λ_`(λ) is a polynomial function in n.

We introduce a notationx^↓k as

x^↓k=x(x−1)(x−2)· · ·(x−k+ 1)

for a variable xand a positive integerk. Ifnis an integer with 0≤n < k then n^↓k= 0. We also setx^↓0 = 1.

In some supersymmetric functions f, we give the explicit expressions of En[f] as linear combinations of descending powersn^↓j. In fact, we will show

En[p₃] = ^X

λ∈SPn

2^n−`(λ)(g^λ)² n!

λ³₁+λ³₂+· · ·+λ³_`(λ)= 3n^↓2+n,

En[p5] = ^X

λ∈SPn

2^n−`(λ)(g^λ)² n!

λ⁵₁+λ⁵₂+· · ·+λ⁵_`(λ)= 40

3 n^↓3+ 15n^↓2+n, En[p²₃] = ^X

λ∈SPn

2^n−`(λ)(g^λ)² n!

λ³₁+λ³₂+· · ·+λ³_`(λ)²

= 9n^↓4+ 54n^↓3+ 31n^↓2+n.

1.3. A deformation

Fix a strict partition µofm. We define the measurePµ,n on SP_n+m by Pµ,n(λ) = m!

(n+m)!2n−`(λ)+`(µ)g^λ

g^µg^λ/µ (λ∈ SP_n+m). (1.1)

Note that Pµ,n(λ) = 0 unless S(λ) ⊃S(µ). We will prove that Pµ,n is a probability, i.e., ^P_λ∈SP

n+mPµ,n(λ) = 1. For a function ϕon SP, define Eµ,n[ϕ] = ^X

λ∈SP_n+m

Pµ,n(λ)ϕ(λ)

= ^X

λ∈SPn+m

m!

(n+m)!2n−`(λ)+`(µ)g^λ

g^µg^λ/µϕ(λ). The summation Eµ,n[ϕ] is considered in [8]. Note that E∅,n[ϕ] is nothing butEn[ϕ]. The following theorem is a slight extension of Theorem 1.2.

Theorem 1.3. Let µ be a strict partition. Suppose thatf is a supersym- metric function. Then Eµ,n[f] is a polynomial function inn.

1.4. Content evaluations

For each box= (i, j) inS(λ), we definec=j−iand call it thecontent of. We deal with symmetric functions evaluated by quantitiesc_b, where

bc = 1

2c(c+ 1).

Theorem 1.4. For any symmetric function F, there exists a unique su- persymmetric function Fb in Γ such that

Fb(λ₁, λ₂, . . . , λ_`(λ)) =F(_bc :∈S(λ))

for any λ∈ SP. Here F(_bc:∈S(λ)) is the specialization of the sym- metric function F(x₁, x₂, . . .) such that the first |λ| variables are substi- tuted by c_b for boxes in S(λ), and all other variables by 0.

From Theorems 1.3 and 1.4, we obtain the following corollary immedi- ately. This result was first obtained in [8] very recently.

Corollary 1.5. Letµbe a strict partition of m. For any symmetric func- tion F,

X

λ∈SP_n+m

m!

(n+m)!2n−`(λ)+`(µ)g^λ

g^µg^λ/µF(_bc :∈S(λ)) is a polynomial function in n.

In [8], the reason why they consider the quantity F(_bc : ∈ S(λ)) is not presented. We note that the multi-set {_bc | ∈ 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

En[_cp2] = ^X

λ∈SPn

2^n−`(λ)(g^λ)² n!

X

∈S(λ)

(_bc)²= 2

3n^↓3+1 2n^↓2,

En[p_b(1²₎] = ^X

λ∈SPn

2^n−`(λ)(g^λ)² n!





 X

∈S(λ)

cb







2

= 1

12(n^↓4+ 4n^↓3−8n^↓2−2n). Furthermore, we will give a new algebraic proof of the identity

Eµ,n[p_b1−p_b1(µ)] = 1

2n^↓2+|µ|n , which is given in [8, Theorem 1.3].

1.5. Related research and the aim

Let P_n be the set of all (not necessary strict) partitions of n. The (tradi- tional) Plancherel probability measure P^Plann on P_n is defined by

P^Plann (λ) = (f^λ)² n! ,

wheref^λ is the number of standard tableaux of shapeY(λ). HereY(λ) is the ordinary Young diagram ofλ:Y(λ) ={(i, j)∈Z² |1≤i≤`(λ), 1≤ j ≤ λi}. Let F be a symmetric function. In [25] (see also [7]), Stanley proves that the summation

X

λ∈Pn

P^Plann (λ)F(h² :∈Y(λ))

is a polynomial in n. Hereh denotes the hook length of the squarein the Young diagram Y(λ). Panova [22] shows an explicit identity for the symmetric function F =Fr(x1, x2, . . .) =^P_j≥1^Qr_i=1(xj −i²).

Moreover, Stanley [25] proves that the content evaluation X

λ∈Pn

P^Plan_n (λ)F(c :∈Y(λ)) (1.2) is also a polynomial in n. Olshanski [21] finds that the functions λ 7→

F(c : ∈ Y(λ)) are seen as shifted-symmetric functions in variables

λ1, λ2, . . . 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

X

λ∈Pn

P^Plan_n (λ) ^X

∈Y(λ) k−1

Y

i=0

c²−i²= (2k)!

((k+ 1)!)²n^↓(k+1) is obtained.

We emphasize the fact that the content evaluations are related to ma- trix 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

Z

U(N)

|u₁₁u22· · ·unn|²dU

=

∞

X

k=0

(−1)^kN^{−(n+k) X}

λ∈Pn

P^Plann (λ)h_k(c: ∈Y(λ)). Here hk are complete symmetric functions. More general identities (for other classical groups) can be seen in [18, 19].

The quantityF(c_b :∈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 _bc come from their projective version. Unfortunately, it is not known any direct connection between matrix integrals and the projective content evaluationF(_bc:∈ S(λ)).

Our results in this paper are seen as the counterparts of the con- tent evaluation [21] in the theory of the shifted Plancherel measure. As Olshanski does in [21], we employ factorial versions of symmetric func- tions. Specifically, we introduce a new family of supersymmetric functions (p_ρ)_ρ∈OP. The function p_ρ is also regarded as projective (or spin) irre- ducible 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_bρ, Ch_ρ, . . . in their articles.

We will provide explicit values of shifted Plancherel averages Eµ,n[p_ρ] for all strict partitions µand 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.

1.6. 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 iden- tities 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 onSPintroduced in [8]

and show that they are supersymmetric functions. We comment on some remaining questions in Section 7.

2. Supersymmetric functions

2.1. The algebra of supersymmetric functions

Asymmetric functionis a collection of polynomialsF = (F_N)_N=1,2,...with rational coefficients such that

• each F_N is symmetric inN commutative variables x₁, x₂, . . . , x_N;

• the stability relationFN+1(x1, . . . , xN,0) =FN(x1, . . . , xN) holds for all N ≥1.

We often write F asF(x1, x2, . . .) in infinitely-many variables x1, x2, . . .. For each r = 1,2, . . . , the r-th power-sum symmetric function p_r is given by

pr(x1, x2, . . .) =x^r₁+x^r₂+· · · .

It is well known that thep_rgenerate the algebra of all symmetric functions and are algebraically independent overQ.

Definition 2.1. Let Γ denote the subalgebra of symmetric functions gen- erated by p_2m+1,m = 0,1,2, . . .. We say elements in Γ to be supersym- metric functions.

Let us review supersymmetric functions along [16, Chapter III.8]. Define p_ρ=p_ρ1p_ρ2· · ·p_ρl

forρ= (ρ₁, ρ₂, . . . , ρ_l)∈ OP. Thep_ρform a linear basis of Γ by definition.

The scalar product on Γ is defined by

hp_ρ, pσi= 2^−`(ρ)zρδρσ (2.1) for ρ, σ∈ OP, where

z_ρ= ^Y

r≥1

r^mr(ρ)m_r(ρ)!,

andm_r(ρ) is the multiplicity ofrinρ:m_r(ρ) =|{i∈ {1,2, . . . , `(ρ)} |ρ_i = r}|.

Given an elementf in Γ and a strict partitionλ, we denote byf(λ) the value

f(λ1, λ2, . . . , λ_`(λ),0,0, . . .).

For example, p₃(λ) = λ³₁+λ³₂+· · ·+λ³_`(λ). Elements in Γ are uniquely defined by their values onSP, i.e. two elementsf, gin Γ coincide with each other if and only if it holds that f(λ) =g(λ) for every strict partitionλ.

2.2. 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.

Definition 2.2. Let λ = (λ1, λ2, . . . , λ_l) ∈ SP. Suppose that N ≥ l =

`(λ). We define a polynomialP_λ|N by

Pλ|N(x1, . . . , xN) = 1 (N −l)!

X

ω∈SN

ω









x^λ₁¹x^λ₂²· · ·x^λ_l^{l Y}

i:1≤i≤l, j:i<j≤N

xi+xj

xi−xj







 ,

where the symmetric groupSN acts by permuting the variablesx1, . . . , xN. If `(λ) > N, then we set P<sub>λ|N</sub>(x<sub>1</sub>, . . . , x<sub>N</sub>) = 0. Then the collection (P<sub>λ|N</sub>)<sub>N=1,2,...</sub> defines an elementP<sub>λ</sub> in Γ. DefineQ<sub>λ</sub> byQ<sub>λ</sub> = 2<sup>`(λ)</sup>P_λ. We callP_λ andQ_λ a SchurP-function and SchurQ-function, respectively.

Proposition 2.3.

(1) The family (P_λ)_λ∈SP forms a linear basis of Γ.

(2) hP_λ, Qµi=δλµ for λ, µ∈ SP.

(3) ^P_λ∈SP

n2^−`(λ)hf, Q_λihg, Q_λi=hf, gi for f, g∈Γ.

(4) Suppose |λ| ≥ |µ|. Then g^λ/µ = hp^|λ|−|µ|₁ Pµ, Qλi for λ, µ ∈ SP, where g^λ/µ is the number of standard tableaux of shapeS(λ/µ). In particular, g^λ =g^λ/∅ =hp^|λ|₁ , Q_λi.

Proof. (1), (2): See [16, Chapter III, (8.9) and (8.12)].

(3): By linearity, it is enough to show the identity for f = Pµ and g=Q_ν with|µ|=|ν|=n. Then both sides are equal toδ_µν by (2).

(4): We recall a special case of the Pieri-type formula for Schur P- functions ([16, Chapter III, (8.15)]): for µ∈ SP,

p1Pµ= ^X

µ⁺:µ⁺&µ

P_µ⁺,

where the sum runs over strict partitions µ⁺ obtained from µ by adding one box. Put k= |λ| − |µ|. The integer g^λ/µ is the number of sequences (λ⁽⁰⁾, λ⁽¹⁾, . . . , λ^(k)) of strict partitions such that λ⁽⁰⁾ =µ, λ^(k) =λ, and such that λ⁽ⁱ⁾ & λ⁽ⁱ⁻¹⁾ for each i = 1,2, . . . , k. Therefore we find the formula

p^k₁P_µ= ^X

λ∈SP_|µ|+k

g^λ/µP_λ.

(4) now follows from (2).

For a strict partition λand odd partitionρ of sizesk, we define

X_ρ^λ =hp_ρ, Qλi. (2.2)

Equivalently, the quantities X_ρ^λ are determined as transition matrices via p_ρ= ^X

λ∈SP_k

X_ρ^λP_λ or Q_λ = ^X

ρ∈OP_k

2^`(ρ)z_ρ⁻¹X_ρ^λp_ρ. (2.3)

The quantity X_ρ^λ is a character value for a projective representation of symmetric groups, see [9, Chapter 8]. One can compute values X_ρ^λ recur- sively if we use a Murnaghan–Nakayama rule ([16, Chapter III.8, Exam- ple 11]). Note that X₍₁^λk)=g^λ and Xρ^(k)= 1. Equivalently,

p^k₁ = ^X

λ∈SP_k

g^λP_λ and Q_(k)= ^X

ρ∈OP_k

2^`(ρ)z_ρ⁻¹pρ. Proposition 2.4. For ρ, σ∈ OP_k,

X

λ∈SP_k

2^{−`(λ)</sup>X<sub>ρ</sub><sup>λ</sup>X<sub>σ</sub><sup>λ</sup> =δρ,σ2<sup>−`(ρ)}zρ. Proof. It follows from (2.3) and Proposition 2.3 (2) that hp_ρ, p_σi=^{D X}

λ

X_ρ^λP_λ,^X

µ

X_σ^µP_µ^E=^X

λ,µ

X_ρ^λX_σ^µhP_λ, P_µi=^X

λ

X_ρ^λX_σ^λ2^{−`(λ)</sup>, the left hand side of which equals 2<sup>−`(ρ)}z_ρδ_ρσ by (2.1).

2.3. Factorial Schur P-functions

The next definition is due to A. Okounkov and given in [10].

Definition 2.5. Letλ= (λ₁, . . . , λ_l)∈ SP. Suppose that N ≥l=`(λ).

We introduce a polynomial P_λ|N^∗ by

P_λ|N^∗ (x1, . . . , xN) = 1 (N −l)!

X

ω∈SN

ω









x^↓λ₁ ¹x^↓λ₂ ²· · ·x^↓λ_l ^{l Y}

i:1≤i≤l, j:i<j≤N

xi+xj

xi−xj







 .

If `(λ) > N, then we set P<sub>λ|N</sub><sup>∗</sup> (x<sub>1</sub>, . . . , x<sub>N</sub>) = 0. The collection (P<sub>λ|N</sub><sup>∗</sup> )<sub>N=1,2,...</sub> defines an element P<sub>λ</sub><sup>∗</sup> in Γ. Define Q<sup>∗</sup><sub>λ</sub> by Q<sup>∗</sup><sub>λ</sub> = 2<sup>`(λ)</sup>P_λ^∗. We call P_λ^∗ and Q^∗_λ the factorial Schur P-function and Q-function, re- spectively.

Remark that P_λ is homogeneous, whereas P_λ^∗ is not. Let us review some properties for factorial SchurP-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.

(1) P_λ^∗ =Pλ+g, where g is a supersymmetric function of degree less than |λ|.

(2) The family (P_λ^∗)_λ∈SP forms a linear basis of Γ.

(3) P_µ^∗(λ) = 0 unless S(µ)⊂S(λ).

(4) It holds that

P_µ^∗(λ) =|λ|^↓|µ|g^λ/µ

g^λ =|λ|^↓|µ|hp^|λ|−|µ|₁ Pµ, Q_λi

g^λ . (2.4)

On the right hand side of (2.4), we can think|λ|−|µ|being nonnegative, because if |λ|<|µ|then|λ|^↓|µ|= 0.

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

x^k =

k

X

j=1

T(k, j)x^↓j (k= 1,2, . . .). (2.5) Proposition 2.7. Let λ be a strict partition of lengthl. Then

P_λ =

λ1

X

j1=1

· · ·

λl

X

jl=1

T(λ1, j1)· · ·T(λ_l, j_l)P_(j^∗_1,...,j

l). Here we set P_(j^∗

1,...,jl) = 0 if j₁, . . . , j_l are not pairwise distinct, and P_(j^∗_1,...,j

l)= (sgnπ)P_µ^∗

if (j1, . . . , jl) = (µ_π(1), . . . , µ_π(l)) for some strict partitionµ= (µ1, . . . , µl) of length l with a permutationπ.

Proof. The definition of P_λ|N^∗ (x₁, . . . , x_N) in Definition 2.5 makes sense even if (λ₁, . . . , λ_l) is replaced with any sequence of positive integers (j1, . . . , j_l). From the alternating property

π







 Y

i:1≤i≤l, j:i<j≤N

xi+xj

x_i−x_j









= (sgnπ) ^Y

i:1≤i≤l, j:i<j≤N

xi+xj

x_i−x_j

for a permutation π∈S_l acting on variables x1, . . . , x_l, we have P_(j^∗

π(1),...,jπ(l))|N(x₁, . . . , x_N) = (sgnπ)P_(j^∗

1,...,jl)|N(x₁, . . . , x_N). In particular, P_(j^∗

1,...,jl) = 0 if j_s = j_t for some s 6= t. Our proposition follows from the definitions of P_λ, P_µ^∗, and Stirling numbers.

3. New supersymmetric functions

Recall the fact that two families (P_λ)_λ∈SP and (P_λ^∗)_λ∈SP are liner bases of Γ. Define a linear isomorphism Ψ : Γ→Γ by

Ψ(P_λ) =P_λ^∗ (λ∈ SP).

Note that Ψ⁻¹(f) coincides with the top-degree term of f by Proposi- tion 2.6 (1).

Definition 3.1. For each ρ ∈ OP_k, we define the supersymmetric func- tion p_ρ by p_ρ= Ψ(p_ρ). From (2.3), we have

p_ρ= ^X

λ∈SP_k

X_ρ^λP_λ^∗.

For an odd partition ρ, we denote by ˜ρ the odd partition obtained from ρ by erasing parts equal to 1. For example, ifρ = (5,5,3,1,1), then ρ˜= (5,5,3). Note that |ρ|=|˜ρ|+m₁(ρ) and`(ρ) =`( ˜ρ) +m₁(ρ).

Proposition 3.2.

(1) p_ρ = p_ρ+g, where g is a supersymmetric function of degree less than |ρ|.

(2) The family (p_ρ)ρ∈OP forms a linear basis of Γ.

(3) For ρ∈ OP and λ∈ SP,

p_ρ(λ) =|λ|^↓|ρ|hp^|λ|−|˜₁ ^ρ|pρ˜, Q_λi

g^λ =|λ|^↓|ρ|X_ρ∪(1^λ_˜ |λ|−|ρ|˜)

g^λ . Here ρ˜∪(1^k) denotes the odd partition ( ˜ρ,1,1, . . . ,1

| {z }

k

).

(4) For ρ∈ OP and λ∈ SP,

p_ρ(λ) = (|λ| − |ρ|)˜ ^↓m1(ρ)p_ρ˜(λ).

(5) If |˜ρ|>|λ|, then p_ρ(λ) = 0.

Proof. (1): It follows immediately from (2.3) and Proposition 2.6 (1).

(2): It follows immediately from (1) and the fact that (pρ)_ρ∈OP form a linear basis of Γ.

(3): Set k=|ρ|. From Proposition 2.6 (iv), (2.3), and (2.2), we have p_ρ(λ) = ^X

µ∈SP_k

X_ρ^µP_µ^∗(λ) = ^X

µ∈SP_k

X_ρ^µ|λ|^↓khp^|λ|−k₁ P_µ, Q_λi g^λ

=|λ|^↓khp^|λ|−k₁ p_ρ, Q_λi

g^λ =|λ|^↓kX_ρ∪(1^λ |λ|−k)

g^λ . Note thatρ∪(1^|λ|−k) = ˜ρ∪(1^{|λ|−|˜ρ|}).

(4): Since |λ|^↓|ρ|=|λ|^↓|˜ρ|·(|λ| − |˜ρ|)^↓m1(ρ), (3) implies that p_ρ(λ) =|λ|^↓|˜ρ|·(|λ| − |˜ρ|)^↓m1(ρ)X_ρ∪(1^λ_˜ |λ|−|˜ρ|)

g^λ = (|λ| − |˜ρ|)^↓m1(ρ)p_ρ˜(λ). (5): If |λ| < |˜ρ|, then |λ|^↓|˜ρ| = |λ|(|λ| −1)· · ·(|λ| − |ρ|˜ + 1) = 0, and

therefore we obtain p_ρ(λ) = 0 from (3).

Substitutingρ= (1^k) in Proposition 3.2 (iv), we obtainp₍₁k)(λ) =|λ|^↓k. Using Stirling numbers defined in (2.5), we find

p₍₁k)=

k

X

j=1

T(k, j)p₍₁j).

More generally, a power-sum function pρ can be expanded as a linear combination of p_σ in the following way. First, we expand p_ρ in terms of Pλ by using (2.3). Second, eachPλ is expanded in terms of factorial Schur P-functionsP_µ^∗ by Proposition 2.7. Finally, eachP_µ^∗ is expanded in terms of p_σ by the formula

P_µ^∗ = ^X

σ∈OP|µ|

2^{−`(µ)+`(σ)}z_σ⁻¹X_σ^µp_σ, which is the image of the second equation on (2.3) under Ψ.

Example 3.3.

p₍₁₎=p₍₁₎,

p₍₁2)=p₍₁2)+p₍₁₎,

p₍₃₎=p₍₃₎+ 3p₍₁2)+p₍₁₎, p₍₁³₎=p₍₁3)+ 3p₍₁2)+p₍₁₎

p_(3,1) =p_(3,1)+ 3p₍₃₎+ 3p₍₁3)+ 7p₍₁2)+p₍₁₎, p₍₁⁴₎=p₍₁4)+ 6p₍₁3)+ 7p₍₁2)+p₍₁₎,

p₍₅₎=p₍₅₎+ 10p_(3,1)+35

3 p₍₃₎+ 40

3 p₍₁3)+ 15p₍₁2)+p₍₁₎,

p_(3,1,1) =p_(3,1,1)+ 7p_(3,1)+ 3p₍₁4)+ 9p₍₃₎+ 16p₍₁3)+ 15p₍₁2)+p₍₁₎, p₍₁⁵₎=p₍₁5)+ 10p₍₁4)+ 25p₍₁3)+ 15p₍₁2)+p₍₁₎.

Remark 3.4. For two ordinary partitions λ, µ, we consider Ch_µ(λ) =







|λ|^↓|µ|χ

λ

µ∪(1|λ|−|µ|)

f^λ if|λ| ≥ |µ|,

0 if|λ|<|µ|,

where χ^λ_µ∪(1|λ|−|µ|) is the value of the irreducible character χ^λ of the sym- metric 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 char- acters 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 func- tion p_ρ is a projective analog of Ch_µ since X_ρ^λ is a character value for a projective representation of symmetric groups.

4. Shifted Plancherel averages

4.1. 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

Eµ,n[p_ρ] = ^X

λ∈SPn+m

m!

(n+m)!2n−`(λ)+`(µ)g^λ

g^µg^λ/µp_ρ(λ).

Since Proposition 3.2 (iv) implies that

Eµ,n[p_ρ] = (n+m− |ρ|)˜ ^↓m1(ρ)Eµ,n[p_ρ˜], (4.1) it is sufficient to compute Eµ,n[p_ρ] for odd partitions ρwith no part equal to 1.

The following lemma is seen in [16, Chapter III.8, Example 11].

Lemma 4.1. For f, g∈Γ,

hp₁f, gi= 1 2

f, ∂

∂p₁g

.

Here the differential operator _∂p^∂

1 acts on functions inΓexpressed as poly- nomials in p1, p3, p5, . . ..

Theorem 4.2. Let ρ be an odd partition such thatm₁(ρ) = 0. Then Eµ,n[p_ρ] =p_ρ(µ). (4.2) Proof. First of all, ifn+m <|ρ|, then we havep_ρ(λ) = 0 for allλ∈ SP_n+m and p_ρ(µ) = 0 by virtue of Proposition 3.2 (5), so that Eµ,n[p_ρ] = 0 = p_ρ(µ). Consequently, we may assume|ρ| ≤n+m.

Using Proposition 2.3 (4) and Proposition 3.2 (3), we see that Eµ,n[p_ρ] = ^X

λ∈SPn+m

m!

(n+m)!2n−`(λ)+`(µ)g^λ

g^µhpⁿ₁Pµ, Qλi

×(n+m)^↓|ρ|hp^n+m−|ρ|₁ pρ, Qλi g^λ

= m!

g^µ

(n+m)^↓|ρ|

(n+m)!

X

λ∈SP_n+m

2^n−`(λ)hpⁿ₁Q_µ, Q_λihp^n+m−|ρ|₁ p_ρ, Q_λi

= m!

g^µ

(n+m)^↓|ρ|

(n+m)! 2ⁿhpⁿ₁Qµ, p^n+m−|ρ|₁ pρi, (4.3) where we have used Proposition 2.3 (3) for the last equality.

We next compute the scalar product hpⁿ₁Q_µ, p^n+m−|ρ|₁ p_ρi. Assume that

|ρ|> m. Expanding Qµ inpσ (see (2.3)), we have hpⁿ₁Q_µ, p^n+m−|ρ|₁ p_ρi= ^X

σ∈OPm

2^`(σ)z⁻¹_σ X_σ^µhp^n+m₁ ^1(σ)p_˜σ, p^{n−(|ρ|−m)}₁ p_ρi.

Here all the scalar products of the right hand side vanish by (2.1) because m₁(ρ) = 0 and n+m₁(σ) ≥ n > n −(|ρ| −m). Therefore it follows from (4.3) that

Eµ,n[p_ρ] = 0 if|ρ|> m .

On the other hand, p_ρ(µ) = 0 if |ρ| > m by Proposition 3.2 (5), hence Eµ,n[p_ρ] = 0 =p_ρ(µ) in that case.

We finally assume that |ρ| ≤m. Using Lemma 4.1 we have hpⁿ₁Q_µ, p^n+m−|ρ|₁ p_ρi= 1

2ⁿ

Q_µ, ∂

∂p₁ n

p^n+m−|ρ|₁ p_ρ

= (n+m− |ρ|)^↓n

2ⁿ hQ_µ, p^m−|ρ|₁ p_ρi

= (n+m− |ρ|)^↓n 2ⁿ

g^µ

m^↓|ρ|p_ρ(µ),

where in the last equality Proposition 3.2 (3) is applied. Combining this with (4.3) gives

Eµ,n[p_ρ] = (n+m)^↓|ρ|(n+m− |ρ|)^↓nm!

(n+m)!m^↓|ρ| p_ρ(µ).

A straightforward computation gives the desired expression.

Substituting ρ=∅ in Theorem 4.2, we obtain Eµ,n[1] = 1,

which shows thatPµ,n defined in (1.1) is a probability measure onSP_n+m and that Eµ,n is the average with respect toPµ,n.

Corollary 4.3. For ρ∈ OP_k,

En[p_ρ] =δ_ρ,(1k)n^↓k.

Proof. Notice that En[p_ρ] =n^↓m1(ρ)En[p_ρ˜] by (4.1). Substitutingµ=∅ in Theorem 4.2 implies that En[p_ρ˜] = δρ,∅_˜ . Therefore En[p_ρ] survives only if

ρ = (1^k), andEn[p₍₁k)] =n^↓k.

Now the polynomiality of Eµ,n[f] becomes trivial.

Proof of Theorems 1.2 and 1.3. Since thep_ρ,ρ∈ OP, form a linear basis of the algebra Γ of supersymmetric functions, we obtain Theorem 1.3 from Theorem 4.2. Theorem 1.2 is a special case of Theorem 1.3 withµ=∅.

4.2. Orthogonality for p_ρ

We can also easily compute the Pn-average of productsp_ρp_σ.

Theorem 4.4. Let ρ andσ be odd partitions such that m1(ρ) =m1(σ) = 0. Then

En[p_ρp_σ] =δ_ρ,σ2^|ρ|−`(ρ)z_ρn^↓|ρ|.

Proof. We may supposen≥max{|ρ|,|σ|}by virtue of Proposition 3.2 (5).

It follows from Proposition 3.2 (3) that En[p_ρp_σ] = ^X

λ∈SP_n

2^n−`(λ)(g^λ)² n! ·n^↓|ρ|

g^λ X_ρ∪(1^λ n−|ρ|)·n^↓|σ|

g^λ X_σ∪(1^λ n−|σ|)

= n^↓|ρ|n^↓|σ|

n!

X

λ∈SPn

2^n−`(λ)X_ρ∪(1^λ n−|ρ|)X_σ∪(1^λ n−|σ|).

Using Proposition 2.4 and the fact that m₁(ρ) =m₁(σ) = 0, it equals

= n^↓|ρ|n^↓|σ|

n! 2n−(`(ρ)+n−|ρ|)

z_ρ∪(1^n−|ρ|₎δρ,σ =n^↓|ρ|2^|ρ|−`(ρ)zρδρ,σ. Substituting σ =∅ in Theorem 4.4 recovers Corollary 4.3.

4.3. Bound for degrees

The following proposition is a direct consequence of Corollary 4.3.

Proposition 4.5. Letf be a supersymmetric function. If we expandf as a linear combination of p_ρ:

f = ^X

ρ∈OP (finite sum)

aρ(f)pρ,

then

En[f] =^X

r≥0

a₍₁^r₎(f)n^↓r.

In particular, if a₍₁^r₎(f) vanish for all r > k, then En[f] is of degree at most k.

Inspired by [12], we define a degree filtration of the vector space Γ by deg₁(p_ρ) =|ρ|+m1(ρ).

More generally, for f =^P_ρaρ(f)p_ρ∈Γ, define deg₁(f) = max

ρ:aρ(f)6=0(|ρ|+m₁(ρ)).

Proposition 4.6. The degree of En[f] as a polynomial in n is at most

1

2deg₁(f).

Proof. It is sufficient to check this for f =p_ρ. By virtue of Corollary 4.3, we have: if ρ 6= (1^k) then En[p_ρ] = 0; if ρ = (1^k) then En[p_ρ] = n^↓k and

deg₁(p_ρ) = 2k.

4.4. Examples

We show some explicit expressions of En[p_ρ], which are presented in Sub- section 1.2. In Example 3.3, we give expansions of some pρ in p_σ. By Corollary 4.3, we obtain the following identities immediately.

En[p₃] =En

hp₍₃₎+ 3p₍₁2)+p₍₁₎ⁱ= 3n^↓2+n , En[p5] =En

p₍₅₎+ 10p_(3,1)+35

3 p₍₃₎+ 40

3 p₍₁3)+ 15p₍₁2)+p₍₁₎

= 40

3 n^↓3+ 15n^↓2+n . Moreover, Theorem 4.4 gives

En

hp²₃ⁱ=En

h(p₍₃₎+ 3p₍₁2)+p₍₁₎)²ⁱ

=En

hp₍₃₎p₍₃₎+ 6p₍₃₎p₍₁2)+ 2p₍₃₎p₍₁₎+ 9p₍₁2)p₍₁2)

+ 6p₍₁2)p₍₁₎+p₍₁₎p₍₁₎ⁱ

= 12n^↓3+ 0 + 0 + 9n^↓2·n^↓2+ 6n^↓2·n+n²

= 9n^↓4+ 54n^↓3+ 31n^↓2+n . 5. Content evaluations

5.1. Supersymmetry

In this section, we give a proof of Theorem 1.4. Let λbe a strict partition and recall the shifted Young diagram

S(λ) =ⁿ(i, j)∈Z²

1≤i≤`(λ), i≤j ≤λ_i+i−1^o.