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Extensions of character formulas by the Littlewood decomposition

In 2015, the author proved combinatorially character formulas expressing sums of the (formal) dimensions of irreducible representations of symplectic groups, refining some works of Nekrasov and Okounkov, Han, King, and Westbury. In this article, we obtain generalizations of these character formulas, by using a bijection on integer partitions, namely the Littlewood decomposition, for which we prove new properties. As applications, we derive signed generating functions for subsets of integer partitions, and new hook length formulas.


Introduction
Studying irreducible characters of classical Lie groups is classical, as they link many domains of mathematics, such as combinatorics, number theory, or representation theory (see for instance [7] and the references cited there). An example of such a connection is implicitly given by results of King [6] and King-El-Samra [1]. Indeed, in [6], a modification rule for the characters of the Lie algebra of type A is proved. In [1] the dimensions of formal irreducible characters of SU are computed. Combining these results, one can equate an arbitrary power of the Dedekind η function with a combinatorial sum over integer partitions, involving hook lengths. Here, recall that the Dedekind η function is a weight 1/2 modular form defined as η(x) = x 1/24 k≥1 (1 − x k ). This gives for instance the now well-known Nekrasov-Okounkov formula [9,Equation (6.12)], that Nekrasov and Okounkov obtained through the Seiberg-Witten theory in 2006 and which was also proved independently by Westbury in [12].
Another proof of this formula, having a more combinatorial flavour, is due to Han and uses both Macdonald formula for the affine root systems of type A [4, Equation (1.3)] and a famous bijection due to Garvan-Kim-Stanton [2]. Following Han's approach and using Macdonald formula for affine root systems of types C, the author proved in [10, Theorem 1.1] the following Nekrasov-Okounkov type formula in type C. For any complex number z, with the notations and definitions of Section 2, we have: to −1 if h is the hook length of a box strictly above the diagonal in the Ferrers diagram of λ and to 1 otherwise. The formula expressed in the following theorem, is a new Nekrasov-Okounkov type formula, and corresponds to type Cˇ. Actually, these two results generalize most of Macdonald formulas in the classical affine types B, Bˇ, C, Cˇ, BC, D.
Theorem 1.1. For any complex number z, the following expansion holds: where the sum ranges self-conjugate partitions λ, δ λ is equal to 1 (respectively −1) if the Durfee square of λ is of even (respectively odd) size, and ε h is equal to −1 if h is the hook length of a box strictly above the diagonal in the Ferrers diagram of λ and to 1 otherwise.
We prove this theorem in a combinatorial way, following Han's approach, by using a Macdonald formula in type Cˇand by introducing a notion of generalized hook length. Here again, it appears that when z is a positive integer, Theorem 1.1 can also be proved algebraically, by using results of [6] and [1]. More precisely, Equations (1) and (2) become in this case character formulas which express sums of the (formal) dimensions of irreducible representations of symplectic groups.
In another direction, Han generalized the Nekrasov-Okounkov formula by adding two extra parameters (one being a positive integer and the other one a complex number) through the Littlewood decomposition, which is a classical bijection between partitions and some vectors of partitions [4,5]. In the same way, the author generalizes (1) in [10,Theorem 1.2] by adding two extra parameter, however only in the case where t = 2t + 1 is a odd positive integer and z a complex number: It is natural to ask whether one can also generalize Theorem 1.1 and (1) in the t even case through this bijection. The answer is positive, as we will see in the present paper.
To do this, we will need new properties of this decomposition. Indeed, as the partitions involved in Theorem 1.1 and (1) have special forms, we have to study their images under the Littlewood decomposition. Some specificities of the images (according to the parity of the aforementioned positive integer) will lead us to two different generalizations for each formula (1) and (2). We state below the one corresponding to (1) in the t even case. Theorem 1.2. Let t = 2t be an even positive integer. With the notations and definitions of Section 2, we have: where the sum ranges over doubled distinct partitions, and H t (λ) is the multi-set of hook lengths of λ which are integral multiples of t.
We now give the two generalizations of Theorem 1.1.
Let t be a positive integer. For any complex numbers y and z, we have: To prove these theorems, we will need to understand precisely the behaviour of the two statistics δ λ and ε h under the Littlewood decomposition. This will require some methods coming from combinatorics on words.
From Theorem 1.2 and Theorem 1.3, we are able to derive many applications, among which new signed generating functions for self-conjugate and doubled distinct partitions. However we will only highlight here consequences regarding hooklength formulas. From (3) is extracted in [10,Equation (9)] the following symplectic hook-length formula, for any positive integer n and odd positive integers t: Here we can state an analog in type Cˇfor self-conjugate partitions.
Theorem 1.4. We have for any positive integer n and any even positive integer t: We would also like to emphasize we are also able to state a new companion formula for (7) (respectively (8)) when t is even (respectively odd), but the right-hand sides are less explicit and are expressed as the coefficients of the Taylor expansion of some exponential functions (see Corollary 3.3).
The paper is organized as follows. In Section 2, we recall the definitions and notations regarding partitions, and we introduce the Littlewood decomposition. Next we prove new properties of this decomposition, and in particular we examine the behaviour of the two statistics δ λ and ε h . In Section 3, we prove our generalizations (Theorem 1.2 and 1.3), from which we derive the aforementioned consequences. We postpone in Section 4 the proof of Theorem 1.1, where we introduce the notion of generalized hook length.

Littlewood decomposition for integer partitions
In all this section, t is a fixed positive integer.
2.1. Definitions. We recall the following definitions, which can be found in [11]. A partition λ = (λ 1 , λ 2 , . . . , λ ) of the integer n ≥ 0 is a finite non-increasing sequence of positive integers whose sum is n. The λ i 's are the parts of λ, := (λ) is its length, and n its weight, denoted by |λ|. Each partition can be represented by its Ferrers diagram as shown in Figure 1, left. (Here we represent the Ferrers diagram in French convention.)  For each box v = (j, k) in the Ferrers diagram of λ (with j ∈ {1, . . . , } and k ∈ {1, . . . , λ j }), we define the hook of v as the set of boxes u such that either u lies on the same row and on the right of v, or u lies on the same column and above v. The hook length h := h v of v is the cardinality of its hook (see Figure 1, center). The principal diagonal is the diagonal including the box (1, 1), and a box (and its hook) is principal if it belongs to the principal diagonal. The Durfee square of λ is the greatest square included in its Ferrers diagram, the length of its side is the Durfee length, denoted by D(λ). We define two statistics associated with the partition λ.
Definition 2.1. Let λ be a partition. The first statistic is denoted by δ λ and defined as (−1) D(λ) . The second statistic, ε h , is defined on each box of λ as −1 if h := h v is a hook length of a box v strictly above the principal diagonal, and 1 otherwise.
The hook length multi-set of λ, denoted by H(λ), is the multi-set of all hook lengths of λ. We say that a partition λ is a t-core if and only if no hook length of λ is a multiple of t. Recall [11, p. 468] that λ is a t-core if and only if H(λ) does not contain the integer t. We denote by P the set of partitions and by P (t) the subset of t-cores. The conjugate partition λ * of a partition λ is the one whose Ferrers diagram is the reflection of that of λ with respect to the principal diagonal.
Recall that a partition λ can be represented by its Frobenius coordinates where a i = λ i − i and b i = λ * i − i, for i ∈ {1, . . . , D(λ)}. We define the set of self-conjugate partitions SC as the set of partitions λ such that a i = b i for all i ∈ {1, . . . , D(λ)}, and similarly we define the set of doubled distinct partitions DD as the set of partitions λ such that a i = b i + 1 for all i ∈ {1, . . . , D(λ)} (see an example in Figure 2 below). We define the set of self-conjugate t-cores SC (t) (respectively doubled distinct t-cores DD (t) ) as the subset of P (t) made of self-conjugate (respectively doubled distinct) partitions.
2.2. The Littlewood decomposition. We follow Han [4] here. The Littlewood decomposition is a classical bijection which maps each partition to its t-core and t-quotient (see for example [11, p. 468]). Let W be the set of bi-infinite binary sequences beginning with infinitely many 0's and ending with infinitely many 1's. Each element w of W can be represented by a sequence but the representation is not unique. Indeed, for any fixed integer k the sequence (b i+k ) i represents w. The canonical representation of w is the unique sequence (c i ) i = · · · c −2 c −1 c 0 c 1 c 2 · · · such that: We put a dot symbol "." between the letters c −1 and c 0 in the bi-infinite sequence (c i ) i when it is the canonical representation. There is a natural one-to-one correspondence between the set of partitions P and W, which we recall now. Let λ be a partition. We encode each horizontal edge of the Ferrers diagram of λ by 1 and each vertical edge by 0. Reading these (0, 1)-encodings from top to bottom and left to right yields a binary word u. By adding infinitely many 0's to the left and infinitely many 1's to the right of u, we get an element w = · · · 000u111 · · · ∈ W. The map ψ : λ → w (10) is a one-to-one correspondence between P and W. The canonical representation of ψ(λ) will be denoted by C λ .  For example (see Figure 2), for λ = (4, 2, 1, 1), we have u = 10010110, so that the image ψ(λ) = w = · · · 00010010110111 · · · and C λ = · · · 0001001.0110111 · · · .
Notice that the symbol "." in the canonical representation of λ corresponds in the Ferrers diagram to the north east corner of its Durfee square. The size of the latter is the number of 1's before the symbol "." in the canonical representation.
Notice that this canonical representation can be used to describe self-conjugate and doubled-distinct partitions. Indeed, if v is a finite binary word, we define f (v) as the reverse word of v in which we exchange the letters 0 and 1. Then, λ is selfconjugate if and only if its canonical representation is of the form · · · 00v.f (v)11 · · · ; and λ is doubled-distinct if and only if its canonical representation is of the form · · · 00v.1f (v)11 · · · . Now we fix a partition λ, and we show how to associate a vector of partitions with λ (involving the fixed integer t).
We split the canonical representation C λ = (c i ) i into t sections, i.e. we form the subsequence v k = (c it+k ) i for each k ∈ {0, . . . , t−1}. The partitions λ 0 , λ 1 , . . . , λ t−1 are defined as ψ −1 (v 0 ), ψ −1 (v 1 ), . . . , ψ −1 (v t−1 ), respectively. Notice that the subsequence v k is not necessarily the canonical representation of λ k . For each subsequence v k , we continually replace the subword 10 by 01. The final resulting sequence is of the form · · · 000111 · · · and is denoted by w k . The t-core of λ is the partitionλ such that the t sections of the canonical representation Cλ are exactly w 0 , w 1 , . . . , w t−1 (an equivalent definition using ribbons can be found in [11, p. 468]).

Remark 2.3.
In the proof of this theorem, properties (ii) and (iii) are consequences of the following fact, which will be useful in the sequel: the boxes of λ are in oneto-one correspondence with ordered pairs of integers (i, j) such that i < j, c i = 1 and c j = 0 in C λ . Moreover the hook length of the box associated with the pair (i, j) is equal to j − i.

2.3.
Littlewood decomposition of self-conjugate partitions. Let t be a positive integer. We know from [2, Section 7] that the restriction of the Littlewood decomposition Ω to the set SC is a bijection with the set of vectors of the form (λ, λ 0 , λ 1 , . . . , λ t−1 ) ∈ SC (t) × P t , such that λ t−i−1 = λ i * for 0 ≤ i ≤ t − 1 (where we recall that λ i * is the conjugate partition of λ i ). Notice that this implies that the partition λ t is self-conjugate when t = 2t + 1 is odd (see Figure 4 for an example). To prove our two last generalizations of Theorem 1.1 stated in Theorem 1.3 below, we will need new properties of the Littlewood decomposition Ω of selfconjugate partitions, which precisely describe the behavior of the two statistics δ λ and ε h when we apply Ω to a self-conjugate partition.
Lemma 2.4. Let t be a positive integer and let λ be a self-conjugate partition. Set (λ, λ 0 , λ 1 , . . . , λ t−1 ) its image under Ω. The following properties hold: Let v = (j, k) be a box in λ i and denote by v * the box (k, j) in λ i * = λ t−1−i , and by V and V * the boxes in λ canonically associated with v and v * respectively. If V is strictly above (respectively strictly below) the principal diagonal of λ, then V * is strictly below (respectively strictly above) the principal diagonal of λ. In particular, we have Proof. Let λ be a self-conjugate partition, and let Ω(λ) = (λ, λ 0 , λ 1 , . . . , λ t−1 ) be its image under Ω. Setṽ := ψ(λ), and v i := ψ(λ i ) for 0 ≤ i ≤ t − 1 where we recall that ψ is defined in (10). We will focus here on the case where t is odd, which is the most complicated one. Indeed, in this case, the partition λ t in the t-quotient is self-conjugate and requires a specific attention. When t is even, all partitions in the t-quotient play the same role, which simplifies the proof.
Assume from now on that t is odd. We prove (i) by induction on the number of boxes in the principal diagonal of λ. It is true if λ is empty, as the t-core and the t quotient contain empty partitions. If λ is non-empty, we denote by λ the doubled distinct partition obtained by deleting the largest principal hook length of λ. Set Ω(λ ) := (λ , λ 0 , λ 1 , . . . , λ t−1 ),ṽ := ψ(λ ), and v i := ψ(λ i ) for 0 ≤ i ≤ t − 1. By the induction hypothesis, we have δ λ = δλ δ λ t . Deleting the largest principal hook length of λ corresponds, in terms of words, to turn the first 1 of ψ(λ) into a 0, and the last 0 of ψ(λ) into a 1. Two cases can occur.
Case 1: the first 1 in ψ(λ) belongs to v t (see the example in Figure 5 below). This implies that there are exactly kt + t + 1 letters between this 1 and the symbol "." (including this 1), where k is an integer. As λ is self-conjugate, there are also kt + t + 1 letters between the symbol "." and the last letter 0 (including this 0). So the last 0 belongs also to v t . Turning the first 1 into a 0 and the last 0 into a 1 actually changes only v t and deletes the largest principal hook of λ t . The t-coreλ does not change. So λ t is equal to the partition λ t in which we delete the largest principal hook and Figure 5. Illustration of the proof of (i) Case 1, for λ = (5, 5, 5, 4, 3) as in Figure 4, and λ = (4, 4, 3, 2).
Case 2: the first 1 in ψ(λ) belongs to a word v i (with 0 ≤ i ≤ t − 1 and i = t ) and the last 0 in ψ(λ) belongs to the word v t−1−i . Turning the first 1 into a 0 deletes the first column in λ i and turning the last 0 into a 1 deletes the first row in λ t−1−i . The partition λ t does not change, so δ λ t = δ λ t . We want actually to prove that the parity of the Durfee square ofλ is different from the parity of the one ofλ . By continually replacing the subword 10 by 01 in v i (respectively v t−1−i ), we obtain a subword of the form · · · 000111 · · · , where the last 0 is in position k 1 (respectively k 2 ), where k 1 and k 2 are integers. By continually replacing the subword 10 by 01 in v i (respectively v t−i ), we also obtain a subword of the form · · · 000111 · · · , but here the last 0 is in position k 1 +1 (respectively k 2 −1). According to the respective distances of k 1 and k 2 to the "." symbol inṽ, the Durfee square ofλ increases or decreases by 1 the Durfee square ofλ. So δ λ = −δλ δ λ 0 = δλδ λ 0 . This ends the proof of (i).
We now prove (ii). Let v = (j, k) be a box of λ 0 and set V its canonically associated box in λ. If λ t is empty, the property is trivial. Otherwise, as λ t is self-conjugate, we can write v 0 = · · · 00w.f (w)11 · · · , where w is a word beginning by 1, and as before, f (w) is the reverse word of w in which we exchange the letters 0 and 1. Assume that v is strictly above the principal diagonal of λ t and in its Durfee square (the three other cases are similar and left to the reader). Using the one-toone correspondence between boxes and ordered pairs of integers, we can decompose w in the following way: w = w 1 1w 2 where there are exactly k − 1 occurrences of 1 in w 1 . We can also decompose f (w) as f (w) = w 3 0w 4 , where there are exactly j − 1 occurrences of 0 in w 4 . As λ t is self-conjugate and as j > k, f (w 1 ) is a suffix of w 4 . This situation also holds in the canonical representation of ψ(λ). We can write ψ(λ) = · · · 00u 1 1u 2 .u 3 0u 4 11 · · · , where the pair made of the first 1 after u 1 and the first 0 after u 3 corresponds to the box V . The word f (u 1 ) is a suffix of u 4 , and u 1 contains strictly less 1's than the number of 0's in u 4 , so the box V is strictly above the principal diagonal of λ.
We finally prove (iii). Notice first that for a parity reason, the boxes V and V * can not belong to the principal diagonal of λ. Assume without loss of generality that k < j (and so V is strictly above the principal diagonal in λ) . We can write v i = · · · 00w 1 1w 2 0w 3 11 · · · , where w 1 begins by 1, w 3 ends by 0, and there are k − 1 occurrences of 1 in w 1 and j − 1 occurrences of 0 in w 3 (where here the 1 and the 0 before and after w 2 correspond to the box v).
where u 1 begins by 1, u 3 ends by 0, and there are j − 1 occurrences of 1 in u 1 and k − 1 occurrences of 0 in u 3 . Here, the 1 and the 0 before and after u 2 correspond to the box v * . These informations allow us to describe ψ(λ) in the following way: ψ(λ) = · · · 00x 1 1x 2 1x 3 0x 4 0x 5 11 · · · , where the 1 after x 1 comes from the k th occurrence of 1 in v i , the 1 after x 2 comes from the i th occurrence of 1 in v t−1−i , the 0 after x 3 comes from the i th occurrence of 0 reading v i from right to left and the 0 after x 4 comes from the k th occurrence of 0 reading v t−1−i from right to left. The box V corresponds to the ordered pair given by the 1 after x 1 and the 0 after x 3 . Therefore there is at least one more occurrence of 0 in x 4 0x 5 than the number of occurrences of 1 in x 1 , so the box V is strictly above the principal diagonal in λ. The box V * corresponds to the ordered pair given by the 1 after x 2 and the 0 after x 4 , so V * is strictly below the principal diagonal in λ.
The properties in Lemma 2.4 can be checked in the example of Figure 4. The first property of the above lemma allows us to compute the following signed generating function of self-conjugate t-cores. Lemma 2.5. Let t be a positive integer. We have Proof. We only prove here the case t = 2t + 1, the other case t = 2t being similar. By the Littlewood decomposition of self-conjugate partitions and Lemma 2.4 (i) we obtain: By enumerating self-conjugate partitions according to their principal hook length, we also derive: As the generating function of integer partitions is well-known, formula (12) follows.

2.4.
Littlewood decomposition of doubled distinct partitions. Let t be a positive integer. It is already known (see [2,Bijection 3]) that the restriction of Ω to the set of doubled distinct partitions gives a bijection with the set of vectors Note that if t = 2t is even, then the partition λ t is self-conjugate. This property can be checked on Figure 3, for λ = (8, 7, 6, 5, 3, 2, 1) ∈ DD and t = 3.
To prove our generalization of (1), namely Theorem 1.2, we will need new properties of the Littlewood decomposition for doubled distinct partitions. The following Theorem express these properties, the easier t odd case is already known (see [10,Lemma 4.2]). The t even case is similar to Lemma 2.4, therefore we do not detail the proof. Lemma 2.6. Let t be a positive integer and write t = 2t or t = 2t + 1 according to the parity of t. Let λ be a doubled distinct partition, and let (λ, λ 0 , λ 1 , . . . , λ t−1 ) be its image under Ω. The following properties hold: (i) δ λ = δλδ λ 0 if t is odd and δ λ = δλδ λ 0 δ λ t if t is even.
(ii) Let v be a box of λ 0 and let V be its canonically associated box in λ (in the sense of Remark 2.3). The box v is strictly above the principal diagonal of λ 0 if and only if V is strictly above the principal diagonal of λ. (iii) Let v = (j, k) be a box of λ i , with 1 ≤ i ≤ t . Denote by v * = (k, j) the box of λ 2t +1−i = λ i * and by V and V * the boxes of λ canonically associated with v and v * . If V is strictly above (respectively below) the principal diagonal in λ i , then V * is strictly below (respectively above) the principal diagonal in λ. Property (i) of the previous lemma allows us to compute the following signed generating function of doubled distinct t-cores, which is surprisingly simpler than the unsigned one given in [2]. As the proof relies on the same arguments as the one of Lemma 2.5, it is omitted.
Lemma 2.7. Let t = 2t be an even positive integer. The following equality holds: Notice that the t odd case is already known, but not useful here.

Generalizations of character formulas
3.1. Proof of Theorem 1.2. Here we prove Theorem 1.2, which will be seen to generalize (1), and we derive a first application.
Proof of Theorem 1.2. Let t = 2t be a positive integer and let λ be a doubled distinct partition. We first transform the expression by using the Littlewood decomposition of λ. Set Ω(λ) = (λ, λ 0 , λ 1 , . . . , λ t−1 ). According to Lemma 2.6 (i), we have δ λ = δλδ λ 0 δ λ t . Let B i be the multi-set of hook lengths in H t (λ) coming from the boxes of λ which correspond to the ones of λ i , for 0 ≤ i ≤ t − 1. According to Theorem 2.2 (iii) and Lemma 2.6 (ii), we have: Similarly, according to Theorem 2.2 (iii) and Lemma 2.6 (iii) applied for i = t , we have: Let v = (j, k) be a box of λ i , with 1 ≤ i ≤ t − 1. Denote by v * = (k, j) the box of λ 2t +1−i = λ i * and by V and V * the boxes of λ canonically associated with v and v * . By Lemma 2.6 (iii), one of the two boxes V and V * is strictly below the principal diagonal of λ, and the other is strictly above the principal diagonal of λ. So we have: where the last equality follows from h V = th v according to Theorem 2.2 (iii). Multiplying this over all boxes V in B i gives: Using (16)-(18), and Theorem 2.2 (ii), we can rewrite (15) as follows: We sum this over all doubled distinct partitions λ. The left-hand side becomes the one of (4), while the right-hand side can be written as a product of four terms. The first one is through Lemma 2.7. The second is by (1) applied with x replaced by y 2 x t and 2z + 2 replaced by t. The third is according to the classical Nekrasov-Okounkov formula [4, Equation (1. 3)] applied with x replaced by y 2 x t and 2z + 2 replaced by t. The last is by Theorem 1.3 applied with x replaced by y 2 x t and z replaced by z/2.
Notice that when y = 0 in Theorem 1.2, we recover Lemma 2.7.
Corollary 3.1 (z = −b/y, y → 0 in Theorem 1.2). We have for any even positive integer t = 2t : Proof. By setting z = −b/y, and letting y → 0 in Theorem 1.2, the left-hand side becomes exactly the one of (23). The limit on the right-hand side can be computed through the classical however we omit the details here.

3.2.
Proof of Theorem 1.3. By using the properties of the Littlewood decomposition of self-conjugate partitions given in Lemma 2.4, we can prove two generalizations of the Nekrasov-Okounkov formula in type Cˇ(namely (2)).
Proof of Theorem 1.3. The proof is similar to the one of Theorem 1.2, and we do not give all the details here. We only emphasize that, at the final step, when we sum over self-conjugate partitions, the right-hand side can be written as a product of a smaller number of terms. In the t odd case, we obtain three terms, one being similar to the left-hand side of (19) (here the sum ranges over self-conjugate partitions), the second being similar to the left-hand side of (21) and the third being similar to the leftt-hand side of (22). The first term can be computed through Lemma 2.5, the others in the same way as in the proof of Theorem 1.2. In the t even case, only the two first terms appear.
Notice that when we set y = 0 in Theorem 1.3, we recover Lemma 2.5, and when we set y = t = 1 in Theorem 1.3, we recover (2). So Theorem 1.3 unifies the Macdonald identities generalized by Theorem 1.1 and the signed generating function of self-conjugate t-cores. Similarly to Corollary 3.1, we have the following consequence.
3.3. Hook formulas. In this section, we prove our generalized hook formula in Theorem 1.4.
Proof of the hook formula (8). To prove (8), we extract the coefficients of b 2n x tn in (25). We then have to simplify signs. Indeed, extracting the coefficient of b 2n x 2tn on both sides of (25) gives: Note that when λ is self-conjugate, all its principal hook lengths are odd. As the sum of all the principal hook lengths is equal to |λ| = 2n, it is even. Therefore there is an even number of principal hook lengths, and δ λ = 1. As t is even, none of the principal hook lengths of λ is an integral multiple of t. So, if a box (j, k) of λ has a hook length which is an integral multiple of t, the box (k, j) is distinct from the box (j, k). Moreover, as λ is self-conjugate, the hook length of (k, j) is also a multiple of t. Summarizing, we have ε (j,k) ε (k,j) = −1, and besides Unlike the case of (25), trying to extract coefficients in equations (23) and (26) to obtain hook formulas as explicit as (7) and (8) and λ∈SC,|λ|=tn

A Nekrasov-Okounkov formula in type CW
e prove in this section a Nekrasov-Okounkov type formula, for the affine type . Recall that in this type, one of the Macdonald formulas can be written in the following form [8, p. 137 (6)(b)], for all integers t ≥ 2: where c 2 is an explicit constant, and where the sum ranges over the t-tuples v := (v 1 , . . . , v t ) ∈ Z t such that v i = 2i − 1 mod 4t. We show here the generalization of the previous formula, namely Theorem 1.1. The idea of the proof is the following: starting from (30), we apply a bijection to rewrite the right-hand side of this identity in terms of self-conjugate t-cores and next simply in terms of self-conjugate partitions. Finally, an argument of polynomiality will allow us to conclude.
Let us precise the organization of this section. In Section 4.1, we introduce the notion of generalized hook length and its properties. In Section 4.2, we study a bijection due to Garvan-Kim-Stanton that we use to complete the proof of Theorem 1.1 in Section 4.3. Finally, we study consequences of this Theorem in Section 4.4. 4.1. Self-conjugate t-cores and generalized hook lengths. In all this section, t is a fixed integer.
Recall that the definitions of t-cores and self-conjugates partitions have been given in Section 2.1. We study more precisely in this section the properties of such partitions.
We can first observe the following facts. The set of principal hook lengths of a self-conjugate partition contains only odd positive integers. Moreover, if ∆ is a finite set of odd positive integers, then there is a unique self-conjugate partition λ such that ∆ is the set of principal hook lengths of λ.
In order to study the self-conjugate partitions (and t-cores), we need to extend the notion of principal hook length. Definition 4.1. Let λ be a self-conjugate partition and let (i, j) be a pair of integers such that 1 ≤ i, j ≤ λ 1 . We define the generalized hook length h i,j of the box (i, j) as: Notice that, as λ is self-conjugate, if a box (i, j) belongs to the Ferrers diagram of λ, then the notions of hook length and generalized hook length of (i, j) coincide. If the box (i, j) does not belong to the Ferrers diagram of λ, a direct calculation shows that h i,j is the opposite of the number of boxes u such that u does not belong to the Ferrers diagram of λ and such that either u lies on the same line and on the left of (i, j), or u lies on the same column and below (i, j). Notice also that, as λ is self-conjugate, all of its principal generalized hook lengths (the hook lengths of the boxes (i, i) with 1 ≤ i ≤ λ 1 ) are odd. Moreover, we can prove the following lemma.
Lemma 4.2. Let λ be a self-conjugate partition and consider ∆ g its set of principal generalized hook length. Let i be an odd integer such that 1 ≤ i ≤ 2λ 1 − 1 = h 1,1 . The two following properties are satisfied: (i) either i or −i belongs to ∆ g , (ii) if i belongs to ∆ g then −i does not belong to ∆ g .
Proof. We first prove (ii). Let i be an odd integer such that 1 ≤ i ≤ λ 1 . Assume that i and −i both belong to ∆ g , and set (j 1 , j 1 ) (respectively (j 2 , j 2 )) the principal box with generalized hook length equal to i (respectively −i). Then the generalized hook length of the box (j 1 , j 2 ) is equal to: This is a contradiction with the fact that h j1,j2 is the cardinal (or the opposite of the cardinal) of a non-empty set. Thus (ii) is satisfied. There are in ∆ g exactly λ 1 odd integers (by definition of a generalized hook length), which are all different as the generalized hook length strictly decrease along the principal diagonal, which are all in absolute value smaller or equal to h 1,1 , and these integers satisfy the property (ii). So, according to the pigeonhole principle, either i or −i belongs to ∆ g . For example, for the self-conjugate partition λ = (4, 2, 1, 1) of Figure 7, the set {7, 1, −3, −5} of its principal generalized hook length satisfies the properties of Lemma 4.2.

A bijection between self-conjugate 2t-cores and vectors of integers.
In this section, we introduce a bijection involving self-conjugate 2t-cores. Definition 4.3. Let λ be a self-conjugate 2t-core. For all i ∈ {1, . . . , t}, we set: It is not clear at this step that the set t i=1 {∆ 2i−1 } characterizes the 2t-core λ.
Let λ be a t-core, we define the vector φ(λ) := (n 0 , n 1 , . . . , n t−1 ) as follows. We label the box (i, j) of λ by (j − i) modulo t. We also label the boxes in the (infinite) column 0 in the same way, and we call the resulting diagram the extended t-residue diagram (see Figure 9 below). A box is called exposed if it is at the end of a row of the extended t-residue diagram. The set of boxes (i, j) of the extended t-residue diagram satisfying t(r − 1) ≤ j − i < tr is called a region and labelled r. We define n i as the greatest integer r such that the region labelled r contains an exposed box with label i.  . The extended 3-residue diagram of the 3-core λ = (7, 5, 3, 1, 1). The exposed boxes are circled.
Proof. The fact that φ 1 (λ) is a bijection is (almost) already known, as it is a direct consequence of [2,Section 7], and the equality (32) comes from [2, (7.4)]. Equation (33) comes from [10, Theorem 3.2] (applied with t + 1 replaced by 2t and a empty doubled distinct partition). Recall that the proof of this theorem comes from a precise study of the action of the bijection φ on self-conjugate partitions.
In Figure 8, we see that the 6-core has for image under φ 1 the vector (−1, 1, 0), and the relations (33) are satisfied.
Remark 4.6. There are three consequences of the previous theorem. Let λ be a self-conjugate 2t-core and set ∆ its set of principal hook lengths.
(i) There can not be in ∆ both an integer equal to 2i − 1 − 2t mod 4t and an integer equal to −(2i − 1) − 2t mod 4t. (ii) If h > 4t belongs to ∆, then h − 4t also belongs to ∆. (iii) If a finite subset of N satisfies the two former properties (i) and (ii) and does not contain any even integer, then it is the set of principal hook lengths of a self-conjugate and doubled distinct 2t-core.
By writing v i = 4tn i +2i−1 for all integers i ∈ {1, . . . , t}, we can replace the sum in Macdonald formula (30) by a sum ranging over self-conjugate 2t-cores, thanks to our bijection φ 1 and to (33). Using the equality v 2 8t = |λ| + (2t − 1)(2t + 1) 24 , coming from (32), we eliminate the factor of modularity x 1/24 into Macdonald formula. This leads to  where the product on the right-hand side ranges over the pairs of integers i < j such that i and j are odd and between 1 and 2t − 1. In all the next section, all products denoted by i<j should be interpreted in this way.

4.3.
Last steps of the proof of Theorem 1.1. In order to complete the proof of Theorem 1.1, we want to rewrite the product on the right-hand side of (34). The strategy to perform this rewriting consist into making a recurrence on the number of principal hooks of the self-conjugate partition involved in (34). Let λ be a non-empty self-conjugate 2t-core, and set ∆ its set of principal hook lengths. We denoted by h 1,1 the length of its largest principal hook. By deleting this hook in the Ferrers diagram of λ, we obtain again a (eventually empty) self-conjugate 2t-core, which we denote by λ . We also denote by ∆ the set of principal hook lengths of λ , and we consider the integers ∆ i which are associated with λ through Definition 4.3.
Lemma 4.7. If i 0 is the unique integer such that ∆ i0 = h 11 , then we have: Proof. First, notice that To prove (35), we examine the consequences on the numbers ∆ i of the deletion of the principal hook of length h 1,1 . As we have ∆ = ∆ \ {h 1,1 }, for all j = i 0 , we have ∆ j = ∆ j . Indeed, the only maximum of the principal hook length congruency classes modulo 2t + 2 which can be changed by this deletion is ∆ i0 . We can deduce that: Now we have to consider three different cases depending on the value of h 1,1 = ∆ i0 .
In the rest of the proof, we need the following notion defined in [4]. (ii) for all a ∈ A such that a = −1, −2, . . . , −2t + 1, we have a ≥ 1 and a ≡ 0 mod 2t; (iii) let b > a ≥ 1 be two integers such that a ≡ b mod 2t. If b ∈ A, then a ∈ A.
Let A be a 2t-compact set. An element a ∈ A is 2t-maximal if for any integer b > a such that a ≡ b mod 2t, b / ∈ A (i.e. a is maximal in its congruency class modulo 2t). The set of 2t-maximal elements is denoted by max 2t (A). It is clear by definition of compact sets that A is uniquely determined by max 2t (A). We here recall the following lemma [10, Lemma 3.5].
Lemma 4.9. For any 2t-compact set A, we have: Lemma 4.10. With the same notations as above, we define the set E as: If ∆ j > 0, then we have ∆ j < h 1,1 . So, ∆ j belongs to ∆, and h 1,1 + ∆ j 2 is an element of H . This element is maximal in its congruency class modulo 2t in H .
Indeed if we assume that h 1,1 + ∆ j 2 + 2t is an element of H , then we can deduce that ∆ j + 4t belongs to ∆, which contradict the maximality of ∆ j .
The fact that all these elements are maximal in their congruency class in H can be proved in the same way as formerly.
If ∆ j < 0, the same arguments (taking care of the case where ∆ j does not belong to ∆ g , as h 1,1 is smaller than ∆ j in absolute value) allow us to show that h 1,1 + ∆ j 2 and h 1,1 − ∆ j − 4t 2 both belong to H . Their maximality can be showed by one of the two previous manners used above.
Lemma 4.11. Let λ be a self-conjugate 2t-core and (n 1 , . . . , n t ) ∈ Z t such that λ = φ 1 (n). The following equality holds: where we recall that the product on the left-hand side ranges over the pairs of integers i < j such that i and j are odd and between 1 and 2t − 1.
Proof. This is at this step that we make an induction on the number of principal hooks of the partition λ, by deleting in the partition at each step of the induction the principal hook of greatest length (that is the hook coming from the box (1, 1) of the partition). Here, we denote by P the left-hand side of (38), and we use the bijection φ 1 to transform P on a product involving the integers ∆ i , in the same way as when we establish (34) Using the same notations as the ones before Lemma 4.7, we define Lemma 4.7 ensures us that: Now, Lemma 4.10 show that the set E defined in (37) is the max 2t (H) of the set H, and the positive elements of H are the hook lengths of the first column of the Ferrers diagram of λ, with the exception of h 1,1 . We can apply Lemma 4.9 to show that where the product on the right-hand side ranges over the hook lengths of the first column of λ, without h 1,1 . This right-hand side can be immediately re-write as: where the product is over the hook length of all boxes belonging to the principal hook coming from the box (1, 1). So, we showed that with the same condition as above for the product. The expected result then follows by making the announced recurrence, after remarking that there is exactly D(λ) steps in the recurrence, that each step gives rise to a minus sign, which explains the term δ λ . The base case of the recurrence corresponds to an empty partition λ . In this case, we have by definition ∆ i = i−2t for any odd integer i such that 1 ≤ i ≤ 2t − 1, and so i<j (2t + ∆ i ) 2 − (2t + ∆ j ) 2 = i<j (i 2 − j 2 ) = 1 c 2 .
We can now prove Theorem 1.1.
To obtain (41), we set −2t := 2v − 1 in (2), and we consider the self-conjugate partitions which can have 2v − 1 as hook length only in their principal diagonal. These partitions are in bijection with the vectors involved in Macdonald formula (41) that we obtain after the substitution x → x 1/2 .