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...s). Let α be some positive real parameter and β = α−1. The family of Jack polynomials Jλ(α), indexed by partitions, forms a basis of the algebra of symmetric functions with rational coefficients in α =-=[22, 33]-=-. We consider the transition matrix between this basis and the classical basis of power sums pµ, i.e. we write Jλ(α) = ∑ |µ|=|λ| θλµ(α) pµ. As a consequence of the Frobenius formula (see the argument ...

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...s). Let α be some positive real parameter and β = α−1. The family of Jack polynomials Jλ(α), indexed by partitions, forms a basis of the algebra of symmetric functions with rational coefficients in α =-=[22, 33]-=-. We consider the transition matrix between this basis and the classical basis of power sums pµ, i.e. we write Jλ(α) = ∑ |µ|=|λ| θλµ(α) pµ. As a consequence of the Frobenius formula (see the argument ...

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...form a basis of Zn. For i = 1, . . . , n the Jucys-Murphy elements Ji are defined by Ji = ∑ j<i(ji), where (ji) is a transposition. These commutative elements were introduced independently in [8] and =-=[25, 26]-=-. They do not belong to Zn. However Jucys and Murphy proved, by different means, that Zn is the abstract symmetric algebra S specialized at the Ji’s, i.e. that we have S[J1, . . . , Jn] = Zn. Given a ...

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...p 2 1(Aλ) + ( n+ 2 3 ) . (3.6) Since the moments may be written σk(λ) = fk(Aλ) with fk a symmetric function depending on n, we may also consider the central element M (k)n = fk(J1, . . . , Jn). Biane =-=[1, 2]-=- has shown that M (k) n = pi(Jkn+1), with pi the orthogonal projection of C[Sn+1] onto C[Sn]. By Jucys’ result, for any λ ` n we have M (k)n χ λ = σk(λ)χ λ. It is natural to study the equivalent expan...

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...classes form a basis of Zn. For i = 1, . . . , n the Jucys-Murphy elements Ji are defined by Ji = ∑ j<i(ji), where (ji) is a transposition. These commutative elements were introduced independently in =-=[8]-=- and [25, 26]. They do not belong to Zn. However Jucys and Murphy proved, by different means, that Zn is the abstract symmetric algebra S specialized at the Ji’s, i.e. that we have S[J1, . . . , Jn] =...

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...chur function associated with hooks. It should be emphasized that our method is not algebraic, but analytic. We do not work in the symmetric algebra C[Sn], but rather in the shifted symmetric algebra =-=[11, 29]-=-. Actually given a partition λ and χλ the character of the corresponding irreducible representation, by a celebrated result of Jucys and Murphy we have f(J1, . . . , Jn)χ λ = f(Aλ)χ λ, with Aλ the alp...

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... λ if it is symmetric in the r “shifted variables” λi − i/α. In analogy with symmetric functions, this defines S∗(α), the algebra of shifted symmetric functions with coefficients in Q[α]. We refer to =-=[29, 30, 31]-=-, or to [17, 19] for a short survey. It is known [19, Proposition 2] that the quantities θλµ(α) are shifted symmetric functions of λ, and form a basis of S∗(α). Moreover [17, Lemma 7.1], given a symme...

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...form a basis of Zn. For i = 1, . . . , n the Jucys-Murphy elements Ji are defined by Ji = ∑ j<i(ji), where (ji) is a transposition. These commutative elements were introduced independently in [8] and =-=[25, 26]-=-. They do not belong to Zn. However Jucys and Murphy proved, by different means, that Zn is the abstract symmetric algebra S specialized at the Ji’s, i.e. that we have S[J1, . . . , Jn] = Zn. Given a ...

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...ons. Thus we have fµ = z −1 µ gµ. Tables giving gµ for |µ| − l(µ) ≤ 14 are available on a web page [21]. 2.8 Dependence on n The notion of partial permutation of {1, . . . , n} has been introduced in =-=[7]-=-. It leads to define an abstract algebra B, a basis of which is formed by elements Bρ indexed by all partitions. There is an isomorphism ι between this algebra and the shifted symmetric algebra S∗, wh...

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...d over partitions ρ satisfying ρ = µ, or equivalently ρ̃ = µ. 7 3 The transition measure Given a partition λ ` n, the transition measure ωλ is a probability measure on the real line, studied by Kerov =-=[9, 10]-=- and others. For any i = 1, . . . , l(λ) + 1 we define the transition probabilities ci(λ) = Hλ Hλ(i) = 1 n+ 1 dimλ(i) dimλ , if the partition λ(i) exists, and 0 otherwise. We have easily ci(λ) = 1 λi ...

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...se of this paper is to present a general method to compute such an expansion. This problem had been solved before by Jucys [8] for f = ek, the elementary symmetric function, and by Lascoux and Thibon =-=[13]-=- for f = pk, the power-sum symmetric function. 1 Our method provides a new proof for these classical results, but also allows to handle many new cases. In this paper we consider f = hk, the complete s...

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... + (1− z) ∑ u≥2 umu ∑ r+s=u+1 C(r − 1)C(s− 1) C(u− 1) . But this is an obvious consequence of Proposition 8.3. Alternative expressions of C(r) may be obtained by using a result of Matsumoto and Novak =-=[23]-=-, which gives the leading coefficient of mλ(J1, . . . , Jn), with mλ a monomial symmetric function. Proposition 8.5. We have C(r) = 1 r + 1 Pr(1 r+1; z). Proof. Given an alphabet X and using λ-ring no...

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...p 2 1(Aλ) + ( n+ 2 3 ) . (3.6) Since the moments may be written σk(λ) = fk(Aλ) with fk a symmetric function depending on n, we may also consider the central element M (k)n = fk(J1, . . . , Jn). Biane =-=[1, 2]-=- has shown that M (k) n = pi(Jkn+1), with pi the orthogonal projection of C[Sn+1] onto C[Sn]. By Jucys’ result, for any λ ` n we have M (k)n χ λ = σk(λ)χ λ. It is natural to study the equivalent expan...

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...ition matrix between this basis and the classical basis of power sums pµ, i.e. we write Jλ(α) = ∑ |µ|=|λ| θλµ(α) pµ. As a consequence of the Frobenius formula (see the argument in the introduction of =-=[19]-=-), the quantities θλµ(α) generalize the central characters, i.e. we have θ λ µ(1) = θ λ µ = n! z−1µ χ̂ λ µ. Given a partition λ, the α-content of any node (i, j) ∈ λ is defined as j−1− (i−1)/α. We den...

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...s Jucys-Murphy specialization f(J1, . . . , Jn) ∈ Zn. By definition we have Cµ χ λ = θλµ χ λ. By taking eigenvalues we get f(Aλ) = ∑ |µ|=n aµ(n) θ λ µ. (2.2) for any partition λ ` n. It is well known =-=[11, 29, 3, 18]-=- that the central character θλµ is a shifted symmetric function of λ. Thus the previous equality holds in S∗. We may equivalently study this decomposition in S∗, rather than the original one in Zn. 2....

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...he symmetric algebra and the Jucys-Murphy elements might be generalized for α 6= 1. The only known exception is for α = 2 and α = 1/2, where a deep interpretation has been recently found by Matsumoto =-=[24]-=- in terms of odd Jucys-Murphy elements (J1, J3, . . . , J2n−1) of S2n. 47 12 Appendix Lemma. Let z be an indeterminate. The quantities a(k)µ (n) = ∑ ρ=µ c(k)ρ ( n− |µ| m1(ρ) ) (12.1) satisfy the recur...

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... side are non zero for respectively |σ| + 1 − l(σ) + 1 = k − 1 − 2i and |σ|+ 1− l(σ)− 1 = k − 1− 2i, with i ≥ 0. We conclude by induction on p. The following result was proved independently by Murray =-=[27]-=- and Novak [28]. Proposition 7.3. If |ρ| − l(ρ) = k and m1(ρ) = 0, we have c(k)ρ = l(ρ)∏ i=1 C(ρi − 1), with C(r) the Catalan number (2r)!/(r + 1)!r!. Proof. As shown above, for |σ| − l(σ) = k − 1 we ...

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...and Zn the center of C[Sn]. Given a partition µ with weight n, denote by Cµ the conjugacy class of permutations having cycle-type µ, viewed as the formal sum of its elements. Since Farahat and Higman =-=[5]-=- it is known that these classes form a basis of Zn. For i = 1, . . . , n the Jucys-Murphy elements Ji are defined by Ji = ∑ j<i(ji), where (ji) is a transposition. These commutative elements were intr...

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...s Jucys-Murphy specialization f(J1, . . . , Jn) ∈ Zn. By definition we have Cµ χ λ = θλµ χ λ. By taking eigenvalues we get f(Aλ) = ∑ |µ|=n aµ(n) θ λ µ. (2.2) for any partition λ ` n. It is well known =-=[11, 29, 3, 18]-=- that the central character θλµ is a shifted symmetric function of λ. Thus the previous equality holds in S∗. We may equivalently study this decomposition in S∗, rather than the original one in Zn. 2....

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...hod in the framework of Jack polynomials is briefly sketched at the end. 2 Generalities and notations We recall some notions about the symmetric group and its representations, referring the reader to =-=[4]-=- for an introduction. In this paper n is a fixed positive integer. 2 2.1 Permutations and partitions A partition λ = (λ1, ..., λr) is a finite weakly decreasing sequence of nonnegative integers, calle...

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...m of σk(λ) = ∑ |µ|=n s(k)µ (n) θ λ µ = ∑ ρ s(k)ρ ( n− |ρ| m1(ρ) ) θλρ̃ , which gives also the class expansion of the central element M (k) n . A very different proof was found independently by Féray =-=[6]-=-. Theorem 6.5. The class expansions pk(J1, . . . , Jn) = ∑ ρ c(k)ρ ( n− |ρ| m1(ρ) ) Cρ̃, M (k) n = ∑ ρ s(k)ρ ( n− |ρ| m1(ρ) ) Cρ̃ are connected by s(k)ρ = c (k) ρ∪(1). Proof. At the beginning of this ...

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...c in the r “shifted variables” λi − i/α. In analogy with symmetric functions, this defines S∗(α), the algebra of shifted symmetric functions with coefficients in Q[α]. We refer to [29, 30, 31], or to =-=[17, 19]-=- for a short survey. It is known [19, Proposition 2] that the quantities θλµ(α) are shifted symmetric functions of λ, and form a basis of S∗(α). Moreover [17, Lemma 7.1], given a symmetric function f ...

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... where fk = ∑ q,r≥0 q+2r≤k min(r,k−2r)∑ s=0 ( n+ r − 1 r − s ) ∑ |µ|=k−2r 〈 µ q 〉 s z−1µ pµ (3.4) is a symmetric function depending on n. Here 〈 µ q 〉 s is some positive integer explicitly known (see =-=[16]-=-, [17, p. 3459] or [18, p. 392]), in particular by a generating function. In view of Section 2.4, the moments σk(λ) are shifted symmetric functions. In this paper we shall mainly need the following el...

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...ero for respectively |σ| + 1 − l(σ) + 1 = k − 1 − 2i and |σ|+ 1− l(σ)− 1 = k − 1− 2i, with i ≥ 0. We conclude by induction on p. The following result was proved independently by Murray [27] and Novak =-=[28]-=-. Proposition 7.3. If |ρ| − l(ρ) = k and m1(ρ) = 0, we have c(k)ρ = l(ρ)∏ i=1 C(ρi − 1), with C(r) the Catalan number (2r)!/(r + 1)!r!. Proof. As shown above, for |σ| − l(σ) = k − 1 we have∑ r≥1 rmr(σ...

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... been recovered by φρ(t) = ∑ σ χσρψσ(t), which is a straightforward consequence of the classical Frobenius formula sρ = ∑ σ χρσ z −1 σ pσ. Tables giving φρ(t) for |ρ| ≤ 14 are available on a web page =-=[20]-=-. 9.2 Second method: expansion of φρ The difficulties encountered to solve the differential system (9.1) lead us to a very different approach. We start from the case z = 1 where, according to Theorem ...

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... symmetric function (depending on n), explicitly given in [18] in terms of auxiliary symmetric functions. Thus we have fµ = z −1 µ gµ. Tables giving gµ for |µ| − l(µ) ≤ 14 are available on a web page =-=[21]-=-. 2.8 Dependence on n The notion of partial permutation of {1, . . . , n} has been introduced in [7]. It leads to define an abstract algebra B, a basis of which is formed by elements Bρ indexed by all...