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Polynomiality of Plancherel Averages of Hook-Content Summations for Strict, Doubled Distinct and Self-Conjugate Partitions

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Polynomiality of Plancherel Averages of Hook-Content Summations for Strict, Doubled Distinct and Self-Conjugate Partitions Polynomiality of Plancherel averages of hook-content summations for strict, doubled distinct and self-conjugate partitions Guo-Niu Han and Huan Xiong∗ Abstract. The polynomiality of shifted Plancherel averages for summations of contents of strict partitions were established by the authors and Matsumoto independently in 2015, which is the key to determining the limit shape of random shifted Young diagram, as explained by Matsumoto. In this paper, we prove the polynomiality of shifted Plancherel averages for summations of hook lengths of strict partitions, which is an analog of the authors and Matsumoto’s results on contents of strict partitions. In 2009, the first author proved the Nekrasov-Okounkov formula on hook lengths for integer partitions by using an identity of Macdonald in the frame- e work of type A affine root systems, and conjectured that the Plancherel av- erages of some summations over the set of all partitions of size n are always polynomials in n. This conjecture was generalized and proved by Stanley. e Recently, P´etr´eolle derived two Nekrasov-Okounkov type formulas for C and e Cˇ which involve doubled distinct and self-conjugate partitions. Inspired by all those previous works, we establish the polynomiality of t-Plancherel aver- ages of some hook-content summations for doubled distinct and self-conjugate partitions. 1. Introduction A strict partition is a finite strict decreasing sequence of positive integers λ¯ = ¯ ¯ ¯ ¯ ¯ (λ1, λ2,..., λℓ) (see [17, 31]). The integer |λ| = 1≤i≤ℓ λi is called the size and ℓ(λ¯)= ℓ is called the length of λ.¯ For convenience, let λ¯ = 0 for i>ℓ(λ¯). A strict P i partition λ¯ is identified with its shifted Young diagram, which means that the i-th row of the usual Young diagram is shifted to the right by i boxes. A standard shifted Young tableau of the shape λ is obtained by filling in the boxes of the shifted Young diagram of λ with numbers from 1 to |λ| such that the numbers strictly increase along every row and every column. The shifted Plancherel measure Pn for strict partitions with size n is defined by (see [3, 13, 18, 19, 20]) n−ℓ(λ¯) 2 n−ℓ(λ¯) 2 (fλ¯ ) 2 fλ¯ Pn(λ¯) := = , n! H(λ¯) ¯ ¯ where fλ¯ is the number of standard shifted Young tableaux of shape λ, and H(λ) is the product of all hook lengths of boxes in the shifted Young diagram of λ¯. For a function ϕ on the set of strict partitions, its shifted Plancherel average is defined 2010 Mathematics Subject Classification. 05A15, 05A17, 05A19, 05E05, 05E10, 11P81. Key words and phrases. random partition, Plancherel measure, strict partition, doubled dis- tinct partition, self-conjugate partition, hook length, content, difference operator. ∗ Huan Xiong is the corresponding author. 1 ∗ 2 GUO-NIUHANANDHUANXIONG by (see [18]) n−ℓ(λ¯) 2 2 (f¯ ) E [ϕ] := P (λ¯)ϕ(λ¯)= λ ϕ(λ¯). n n n! ¯ ¯ |λX|=n |λX|=n The polynomiality of shifted Plancherel averages for functions related to contents of strict partitions (the definition for contents and hook lengths of strict partitions are given in the beginning of Section 2) were established by the authors [12] and Matsumoto [18] recently. Theorem 1.1 ([12, 18]). Suppose that Q is a given symmetric function. Then n−ℓ(λ¯) c 2 f¯ c E [Q : ∈ λ¯ ]= λ Q : ∈ λ¯ n 2 ¯ 2 ¯ H(λ) |λX|=n is a polynomial in n, where c is the content of the box in the shifted Young diagram of λ¯. Theorem 1.1 is important in the study of the limit shape problems of random shifted Young diagram, as explained by Matsumoto [19]. In Section 2 we derive the following hook length analog for Theorem 1.1. Theorem 1.2. Suppose that k is a given nonnegative integer and µ¯ is a given strict partition. Then, ¯ |λ¯|−ℓ(λ¯) ℓ(λ) 2 f¯ (1.1) λ/µ¯ 2 h2k + (4k − 1) λ¯2k H(λ¯) i ¯ ¯ i=1 |λ/Xµ¯|=n h∈HX(λ) X is a polynomial in n of degree at most k +1, where fλ/¯ µ¯ is the number of standard skew shifted Young tableaux of shape λ/¯ µ¯, and H(λ¯) is the multiset of hook lengths of boxes in the shifted Young diagram of λ¯. k ℓ(λ¯) ¯2k We mention that the term (4 − 1) i=1 λi is necessary in the formula (1.1), 2k since E [ ¯ h ] is not a polynomial of n when k ≥ 1. n h∈H(λ) P The followingP so-called Nekrasov-Okounkov formula z q|λ| 1 − = (1 − qk)z−1, h2 λ∈P h∈H(λ) k≥1 X Y Y where P is the set of all integer partitions λ with |λ| denoting the size of λ and H(λ) the multiset of hook lengths associated with λ (see [7]), was discovered in- dependently several times: first, by Nekrasov and Okounkov in their study of the theory of Seiberg-Witten on supersymmetric gauges in particle physics [21]; then, proved by Westbury using D’Arcais polynomials [33]; finally, by the first author us- ing an identity of Macdonald [17] in the framework of type A affine root systems [7]. Moreover, he asked to find Nekrasov-Okounkov type formulas associated with other root systems [8, Problem 6.4], and conjectured that the Plancheree l average 1 n! h2k H(λ)2 |λX|=n h∈HX(λ) Polynomiality of Plancherel averages of hook-content summations 3 Figure 1. From strict partitions to doubled distinct partitions. is always a polynomial in n for any k ∈ N, where H(λ) = h∈H(λ) h. This con- jecture was proved by Stanley in a more general form. In particular, he showed Q that 1 n! Q (h2 : h ∈ H(λ)) Q (c : c ∈C(λ)) H(λ)2 1 2 |λX|=n is a polynomial in n for any symmetric functions Q1 and Q2, where C(λ) is the multiset of contents associated with λ (see [29]). For some special functions Q1 and Q2 the latter polynomial has explicit expression, as shown by Fujii-Kanno- Moriyama-Okada [5] and Panova [24]. The doubled distinct partition of a strict partition λ¯, denoted by λ¯λ¯, is defined to be the usual partition whose Young diagram is obtained by adding λ¯i boxes to the i-th column of the shifted Young diagram of λ¯ for 1 ≤ i ≤ ℓ(λ¯) (see [6, 25, 26]). For example, (6, 4, 4, 1, 1) is the doubled distinct partition of (5, 2, 1) (see Figure 1). For each usual partition λ, let λ′ denote the conjugate partition of λ (see [6, 17, 25, 26]). A usual partition λ is called self-conjugate if λ = λ′. The set of all doubled distinct partitions and the set of all self-conjugate partitions are denoted by DD and SC respectively. For each positive integer t, let Ht(λ)= {h ∈ H(λ): h ≡ 0 (mod t)} h. be the multiset of hook lengths of multiples of t. Write Ht(λ)= h∈Ht(λ) Recently, P´etr´eolle derived two Nekrasov-Okounkov type formQulas for C and Cˇ which involve doubled distinct and self-conjugate partitions. In particular, he obtained the following two formulas in [25, 26]. e e Theorem 1.3 (P´etr´eolle [25, 26]). For positive integers n and t we have 1 1 (1.2) = n , if t is odd; Ht(λ) (2t) n! λ∈DD,|λ|=2nt #HtX(λ)=2n 1 1 (1.3) = n , if t is even. Ht(λ) (2t) n! λ∈SC,|λ|=2nt #HtX(λ)=2n We call n DD (2t) n! Pt,n (λ) := (t odd) Ht(λ) and n SC (2t) n! Pt,n(λ) := (t even) Ht(λ) ∗ 4 GUO-NIUHANANDHUANXIONG the t-Plancherel measure for doubled distinct and self-conjugate partitions with size n, respectively. For a function ϕ on the sets of doubled distinct or self-conjugate partitions, its t-Plancherel average is defined by n DD DD (2t) n! Et,n [ϕ] := Pt,n ϕ(λ)= ϕ(λ) (t odd) Ht(λ) λ∈DD,|λ|=2nt λ∈DD,|λ|=2nt #HtX(λ)=2n #HtX(λ)=2n and n SC SC (2t) n! Et,n[ϕ] := Pt,nϕ(λ)= ϕ(λ) (t even) Ht(λ) λ∈SC,|λ|=2nt λ∈SC,|λ|=2nt #HtX(λ)=2n #HtX(λ)=2n respectively. Actually, the t-Plancherel measure in this paper can be generalized to be defined on the set of general partitions (see [4]). As we know, the linear and projec- tive representations of symmetric groups can be labelled bijectively by partitions and strict partitions, respectively (see [17]). Therefore, the ordinary and shifted Plancherel measure gives natural probability measure on such representations. Sim- ilarly, when t is a prime number, the t-blocks in the modular representations of sym- metric groups can be labelled bijectively by t-core partitions. In fact, the famous Nakayama conjecture [14, Theorem 6.2.21] states that two irreducible characters of the symmetric groups are in the same t-block if and only if their corresponding partitions have the same t-core. The t-Plancherel measure thus gives natural prob- ability measure on t-blocks in the modular representations of symmetric groups. Also, as the ordinary and shifted Plancherel measure are closely related to certain numbers of paths in Young’s lattice, our t-Plancherel measure can be seen as a gen- eralized probability measure arising from counting the paths (see (4.3) and (5.2)) in Stanley’s t-differential poset theory [30]. Inspired by previous works, we establish the polynomiality of t-Plancherel av- erages of some hook-content summations for doubled distinct and self-conjugate partitions.
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