On self-Mullineux and self-conjugate partitions

The Mullineux involution is a relevant map that appears in the study of the modular representations of the symmetric group and the alternating group. The fixed points of this map are certain partitions of particular interest. It is known that the cardinality of the set of these self-Mullineux partitions is equal to the cardinality of a distinguished subset of self-conjugate partitions. In this work, we give an explicit bijection between the two families of partitions in terms of the Mullineux symbol.


Introduction
Let n be a non negative integer. It is well known that the isomorphism classes of complex irreducible representations of the symmetric group S n is indexed by the set of partitions of n. Let λ be a partition of n (written λ ⊢ n) and S λ the associated irreducible CS n -module. Tensoring S λ with the sign representation ε of S n results in the irreducible representation S λ ′ of S n , where λ ′ is the conjugate partiton of λ ([JK81, 2.1.8]). This procedure allows to understand, by Clifford theory, all complex irreducible representations of the alternating group A n . Indeed, let λ be a partition of n, • If λ = λ ′ then S λ ↓ An ≃ S λ ′ ↓ An is irreducible.
A natural question is then what happens when we change the characteristic of the field. Let p = 2 be a prime and F an algebraically closed field of characteristic p. It is well known that the number of isomorphism classes of irreducible representations of the symmetric group S n over F , is equal to the number of conjugacy classes of p-regular elements of S n ([Isa06, 15.11]), which in turn is in bijection with the p-regular partitions of n ([JK81, 6.1.2]). In this setting also, by Clifford theory, understanding the tensor product with the sign representation allows to obtain a classification of irreducible F A nmodules. However, the conjugate of a p-regular partition is not necessarily p-regular, so tensoring with the sign representation in this case does not amount to conjugating the corresponding partition.
In [Mul79a], G. Mullineux defined a bijection m on the set of p-regular partitions of n, which is an involution, and conjectured that for a p-regular partition λ with associated irreducible F S n -module D λ we have D λ ⊗ ε = D m(λ) . Later, in [Kle96], A. Kleshchev described a different algorithm to compute m(λ) and in [FK97], B. Ford and A. Kleshchev proved Mullineux conjecture to be true. Mullineux conjecture was also proven to be true in [BO94] by C. Bessenrodt and J.B. Olsson by using yet another description of the Mullineux bijection m. Other properties of this map have been studied in [Mul79b], [BO98], [BOX99]. Hence, tensoring with the sign representation in the modular case amounts to applying m on partitions, which makes the Mullineux map a p-analogue of conjugation of partitions. This way Laboratoire de Mathématiques de Reims FRE 2011, Université de Reims Champagne Ardenne, Moulin de la Housse BP 1039, 51687 REIMS cedex 2, France E-mail address: ana.bernal@univ-reims.fr.
we have a classification of irreducible representations of A n in characteristic p as follows. Let λ be a p-regular partition of n, • If λ = m(λ) then D λ ↓ An ≃ D m(λ) ↓ An is irreducible.
• If λ = m(λ) then D λ ↓ An splits into two irreducible, non-isomorphic F A n -modules D λ + and (1) {D λ ↓ An | λ ⊢ n, λ p-regular and λ = m(λ)} ⊔ {D λ + , D λ − | λ ⊢ n, λ p-regular and λ = m(λ)} is a complete set of non-isomorphic irreducible F A n -modules, considering only one partition λ for each couple {λ, m(λ)} with λ = m(λ) ([For97, 2.1]). Following such an indexing of irreducible modular representations of A n , it is natural to inquire about the set of p-regular partitions such that λ = m(λ). The definition of the Mullineux map m is quite complicated as are the different descriptions mentioned above, even if they are explicit. Therefore describing its fixed points is not easy. So that, in characteristic p, it is not straightforward to obtain a reasonably simple indexing set for the irreducible F A n -modules.
In fact, the number of fixed points of the Mullineux map, or self-Mullineux partitions, is equal to the number of partitions of n with different odd parts, none of them divisible by p ([AO91, Prop. 2]). This number is, in turn, equal to the number of self-conjugate partitions with diagonal hook-length not divisible by p ([JK81, 2.5.11]). We refer to the latter as BG-partitions (see Definition 3.1 for details). There is an elementary algebraic argument to see this (Appendix A). Thus, it is natural to ask for an explicit bijection between the self-Mullineux partitions and the BG-partitions.
The Mullineux map can be defined in terms of a symbol called the Mullineux symbol, defined on p-regular partitions. In this work we introduce a new symbol, defined on self-conjugate partitions. From such a symbol, associated to a BG-partition, we describe how to reconstruct a BG-partition and a self-Mullineux partition, and this algorithm provides our bijection.
A further motivation for finding an explicit bijection can also be given in the context of the representation theory of the symmetric group and of the alternating group. In [BG10], O. Brunat and J.-B. Gramain have shown the existence of a p-basic set for the symmetric group, which, by restriction, gives a p-basic set for the alternating group. However, this set, which provides a natural indexing set for the modular irreducible representations is not explicit and it would be ideal to give a complete description of it. One thing we know about such a set is that it always contains the set of BG-partitions. Hence, it is convenient to have a better understanding of them. More generally, this work can be seen as a first step to give a new natural way to label the modular irreducible representations of the symmetric group, for which tensoring with the sign representation is easier to describe combinatorially. We hope to come back to this problem later.
A bijection between the set of self-Mullineux partitions of n and partitions of n with different odd parts, none of them divisible by p can alternatively be derived from a bijection between two more general sets defined by C. Bessenrodt in [Bes91]. However, the two approaches are quite different because our bijection is defined directly between the sets of our interest. Moreover, we obtain a different bijection (see Remark 3.18).
The paper is organized as follows. In Section 2 we recall some definitions about partitions and the definition of the Mullineux map. Section 3 contains the main result of this paper; we define a symbol on self-conjugate partitions and we show how through this symbol we obtain the mentioned explicit bijection. Finally, in Section 4 we prove that the inverse procedure of reconstructing a unique BG-partition from a self-Mullineux partition is well defined, which confirms that this is a one to one correspondence without the need of knowing beforehand that there exists a bijection between both sets of partition.

Preliminaries: the Mullineux map
In this section we recall some general definitions about partitions and the definition of the Mullineux map, as defined by G. Mullineux in [Mul79a]. We follow closely definitions in [FK97].
A partition λ is a sequence λ = (λ 1 , λ 2 , . . . , λ k , . . .) of non-negative integers such that λ 1 ≥ λ 2 ≥ · · · ≥ λ k ≥ · · · , containing only finitely many non-zero terms. Let n ∈ N be such that |λ| = λ i = n. We say that λ is a partition of n, which we write λ ⊢ n. We denote by Par(n) the set of partitions of n. We call |λ| the size of λ. The integers λ i are called the parts of the partition λ. The number of non-zero parts is the length of λ and is denoted l(λ). The Young diagram of a partition λ is the set whose elements are called nodes. We represent the diagram as an array of boxes in the plane with the convention that i increases downwards and j increases from left to right. A partition is often identified with its Young diagram. For example, the diagram of λ = (5, 2, 2, 1) is The conjugate (or transpose) partition of λ = (λ 1 , . . . , λ k ) is the partition λ ′ of n defined as λ ′ j = #{i | λ i ≤ j}, which amounts to transposing the Young diagram [λ] with respect to its main diagonal. If λ = (5, 2, 2, 1), as above, then λ ′ = (4, 3, 1, 1, 1) and its Young diagram is If λ = λ ′ we say that λ is self-conjugate. For a positive integer e, λ is said to be e-regular if it does not contain e parts λ i = 0 which are equal. The partition λ = (5, 2, 2, 1) above is not 2-regular but it is 3-regular. We denote by Reg e (n) the set of e-regular partitions of n.
Let (i, j) be a node of a partition λ. We define the (i, j)-th hook of λ, or the (i, j) λ -th hook, as the set of nodes in [λ] to the right or below the node (i, j), that is, the nodes (k, l) such that k = i and j ≤ l ≤ λ i , or l = j and i ≤ k ≤ λ ′ j . The hook-length of λ at (i, j) ∈ [λ], denoted here h λ ij is the number of nodes in the (i, j) λ -th hook, that is We omit λ from the notation when there is no ambiguity. A partition which is equal to its (1, 1)-th hook is called a hook.
For any positive integer m, a m-hook (respectively (m)-hook ) is a hook of length m (respectively divisible by m). We call a node (i, i), for 1 ≤ i ≤ l(λ), a diagonal node and the set of diagonal nodes is the diagonal of λ. A (i, i)-th hook is referred to as a diagonal hook.
The rim of λ is the set of nodes (i, j) ∈ [λ] such that (i + 1, j + 1) / ∈ [λ], in other words, it is the south-east border of the Young diagram [λ]. For example, the rim of λ = (9, 6, 3, 1) is represented in the following diagram by colored boxes Let us label the nodes of the rim with positive integers from the top right to the bottom left, as shown in the following figure 4 3 2 1 7 6 5 10 9 8 11 Let p be a prime. The first p-segment of the rim consists of the nodes corresponding to integers less or equal than p. If the last node (i, j) of the first p-segment is in the last row of [λ], then [λ] only has one p-segment. If not, let l be the smallest label on row i + 1. The second p-segment of the rim consists of the nodes labelled by l ≤ m ≤ l + p − 1. Repeating this procedure we will eventually reach the bottom row of the diagram and it is clear that all p-segments have p nodes, except possibly the last one. The p-rim of λ is defined as the union of all the p-segments.
We denote a λ the number of nodes in the p-rim of λ. Define diagrams λ (0) , λ (1) , . . . , λ (l) as follows. Put λ (0) = λ and for i ≥ 1 put where we choose l maximal with respect to λ (l) = ∅; so λ (l+1) = ∅. We call the p-rim of λ (i) the i-th p-rim of λ. Let a i = a λ (i) be the number of nodes of the i-th p-rim of λ and r i the number of rows of λ (i) , that is, r i = l(λ (i) ). The Mullineux symbol of λ (introduced in [BO94]) is (2) G p (λ) = a 0 a 1 · · · a l r 0 r 1 · · · r l .
Proposition 2.4. Let p = 2 be a prime and λ a p-regular partition of a non-negative integer n. Set The entries of G p (λ) satisfy l i=0 a i = n. Moreover, if a 0 , . . . , a l , r 0 , . . . , r l are positive integers such that these inequalities are satisfied then there exists exactly one p-regular partition λ of n such that G p (λ) = a 0 a 1 · · · a l r 0 r 1 · · · r l .
Remark 2.5. It is easy to recover the p-regular partition λ from its Mullineux symbol G p (λ); start with the hook λ (l) of size a l and length r l , and for i = l −1, l −2, . . . , 0, add the i-th p-rim (consisting of a i nodes) to λ (i+1) from the bottom to the top, starting by placing a node on the first free placement in row r i . Then, adding nodes either on top (whenever it is possible) or to the right of the last added node until having added the last node of the p-segment and add the following p-segment starting on the first free placement of the row on top of the last added node. This procedure finishes at the first row. This algorithm is more precisely described in [FK97,§1].
Let λ be a p-regular partition of n, with Mullineux symbol (2) and let ε i be as in Proposition 2.4. Define s i = a i + ε i − r i . In [Mul79a,4.1] it is shown that the arrangement a 0 a 1 · · · a l s 0 s 1 · · · s l , corresponds to the Mullineux symbol of a p-regular partition. We are now able to define the Mullineux map m.
Definition 2.6. With the above notations, we define m(λ) as the unique p-regular partition such that G p (m(λ)) = a 0 a 1 · · · a l s 0 s 1 · · · s l .
Because of Proposition 2.4, m(λ) is well defined, and from the definition we can see that m is an involution.
Remark 2.7. If p > n, then Reg p (n) = Par(n) and irreducible F S n -modules are therefore indexed by all partitions of n. In this case, the Mullineux map coincides with conjugation: m(λ) = λ ′ .

From BG-partitions to self-Mullineux partitions
Let p be a prime. Let us define the set of BG-partitions.
Definition 3.1. We call BG-partition any self-conjugate partition with no diagonal (p)-hooks, that is, any partition λ such that p ∤ h λ ii for every 1 ≤ i ≤ l(λ) with (i, i) ∈ [λ]. We denote BG p the set of BG-partitions and BG n p the set of BG-partitions of a non-negative integer n. As said in the introduction, the aim of this work is to give an explicit bijection between BGpartitions and self-Mullineux partitions. In this section we describe such bijection: we define the BG-symbol associated to a BG-partition, which is a two-row arrangement of positive integers. We prove that BG-symbols of BG-partitions are Mullineux symbols of self-Mullineux partitions and that this association is injective, resulting in a bijection.
3.1. BG-symbol. We introduce a symbol, defined in general for self-conjugate partitions. This symbol is in some way inspired by the Mullineux symbol. In a similar way as the Mullineux symbol, which is defined by counting nodes on the p-rims of a sequence of partitions, the BG-symbol is defined by counting elements in a set of nodes called the p-rim* which is a symmetric analogue of the p-rim.
Let λ be a self-conjugate partition and denote by Rim p (λ) ⊆ [λ] the set of nodes in the p-rim of [λ]. Set that is, U λ consists of the nodes of the p-rim which are above (or on) the diagonal of [λ]. We denote This can be seen as the nodes in U λ reflected across the diagonal of λ.
Definition 3.4. Let λ be a self-conjugate partition. We define ε * λ := a * λ mod 2. It is the parity of the number of nodes on the p-rim* of λ.
Remark 3.5. For a self-conjugate partition λ, from the definition of p-rim* we have ε * λ = 0 ⇔ a * λ is even. ⇔ Rim * p (λ) has no diagonal nodes. This way, the number of nodes in U λ , that is, the number of nodes of the p-rim* of λ over (or on) the diagonal is Let λ be a self-conjugate partition. We define diagrams λ (0) * , λ (1) * , . . . , λ (l) * in an analogue way as for the Mullineux symbol, by considering the p-rim* instead of the p-rim. Put λ (0) * = λ and for i ≥ 1 put where we chose l maximal with respect to λ (l) * = ∅; so λ (l+1) * = ∅. We call the p-rim* of λ (i) * the i-th p-rim* of λ.
Remark 3.6. Notice that the p-rim* is only defined for self-conjugate partitions, but we claim that the diagrams λ (i) * are well defined, given the fact that Rim * p (λ) is symmetric in the sense that (u, v) ∈ Rim * p (λ) if and only if (v, u) ∈ Rim * p (λ). Therefore, removing these nodes from [λ] to obtain λ (1) * results again in a self-conjugate partition and then so it is for every λ (i) * . In other words, if λ (i) * is self-conjugate, then λ (i+1) * is self-conjugate.
For the partition λ (i) * , obtained by succesively removing nodes on the p-rim*, starting with λ = All these values associated to self-conjugate partitions may seem technical, and they are better understood by means of the Young diagram, see the following example.
In this diagram, each i-th 3-rim* is shown in a different shade.
The following lemmas will allow us to prove that two different self-conjugate partitions correspond to different BG-symbols. Lemma 3.12 is an analogue of [Mul79a, 2.1]. Its proof is quite technical and the arguments are easier to understand with an example, see Example 3.13.
Lemma 3.11. Let λ be a self-conjugate partition. If a * λ is an even number, then p | a * λ . Proof. From the definition (or see Remark 3.5), a * λ is even if and only if U λ ∩ L λ = ∅. Then the p-rim* of λ does not contain diagonal nodes. From the definition of Rim * p (λ), it means that the set U λ only contains p-segments of length p. And then the same is true for L λ . Therefore The converse is not true in general, for example, if p = 3 and λ = (5, 3, 2, 1, 1), we have that a * λ = 9. Lemma 3.12. Letλ be a self-conjugate partition, ε ∈ {0, 1} and m, a residue modulo p, such that m = 0 if ε = 0. Then, there exists a unique self-conjugate partition λ such that (i) a * λ ≡ ε (mod 2); Moreover, ifλ ∈ BG p , and p ∤ 2m + 1 when ε = 1, then λ ∈ BG p .
Proof. Given ε ∈ {0, 1} and m, a residue modulo p, let us see that there is a unique way to add nodes toλ to obtain a self-conjugate partition λ such that the added nodes are the p-rim* of λ. Let us study how nodes (i, j) over the diagonal (i ≤ j) must be added. This will determine all nodes that must be added (if (i, j) is added toλ, then (j, i) is added as well).
Now, let (i, j) be the first node that we add (with i fixed as before byλ and ε) and j ∈ {λ d +1, d+1} depending onλ and ε. Starting from this node, it is clear that there is a unique way to add nodes such that (i), (ii), and (iii) hold: If the position (i + 1, j) just above (i, j) is empty inλ, we add a node in that position, otherwise we add a node in (i, j + 1). We repeat this procedure for adding nodes until we have added m nodes (including (i, j) if ε = 0, not including (i, j) if ε = 1). If the last added node is in row 1 we stop here. If it is added in row k > 1, we add a node in row k − 1 in position (k − 1,λ k−1 + 1) and we restart the procedure to keep adding nodes until having added p nodes. We iterate this procedure, of adding groups of p nodes, until reaching the first row. This way we added nodes over the diagonal. Finally for each node (a, b) added, we add its reflection through the diagonal (b, a). And we obtain a self-conjugate partition λ.
It remains to verify that λ (1) * =λ. If ε = 1, it is straightforward that λ (1) * =λ. Since when removing the nodes of the p-rim* of λ over the diagonal we eventually reach a diagonal node, and then just remove the reflection of the removed nodes. It is clear that in this case we obtainλ. If ε = 0, the condition m = 0 says that a p-segment of λ will eventually reach the row d and this p-segment has exactly p nodes, so that there is no ambiguity when removing p-segments and λ (1) * =λ.
For the last part of the theorem, suppose thatλ ∈ BG p , and let us see that λ obtained as above is also in BG p . In other words, we are assuming thatλ does not contain any diagonal (p)-hooks and we want to show that the same is true for λ.
Suppose that λ has a diagonal (p)-hook, say the (i, i) λ -th hook, that is h λ ii = pk for some integer k > 0. For a partition µ, we set the convention h µ ij = 0 if (i, j) / ∈ [µ]. Sinceλ ∈ BG p , then the (i, i) λ -th hook is different from the (i, i)λ-th hook since if they were equal, λ would have a (p)-hook, which is not possible. Therefore (i,λ i + 1) ∈ [λ]. Since this node is not in [λ], by definition, it is on the p-rim* of λ, in particular, it belongs to a p-segment of Rim * p (λ) above the diagonal. Consider the two cases: this p-segment starts at row i, or this p-segment starts before row i, that is, this p-segment starts at a row j for 1 ≤ j < i.
• If the p-segment containing node (i,λ i + 1) starts at row i. Let (i, j) be the first node of this p-segment and (a, b) its last node (i ≤ a). Then a ≤ b because this segment is above the diagonal. Let N be the number of nodes on this p-segment. Then we have: where the last identity holds if b > a (since this implies that (a, b − 1) ∈λ). Then we have If a = b, this p-segment is the last segment in U λ and ε * λ = 1. So that N = m + 1 and we get h λ ii = 2m + 1. This contradicts p ∤ 2m + 1. If a < b, then the last node of this p-segment, (a, b) is not a diagonal node so that N = p and we get h λ ii = 2p + hλ aa , which implies p | hλ aa , a contradiction.
Proposition 3.14. Let p = 2 be a prime. Two different self-conjugate partitions have different BGsymbols. In other words, the BG-symbol gives rise to an injective map from self-conjugate partitions to the set of two-row positive integer symbols.
As it turns out, the BG-symbol of a BG-partition is a Mullineux symbol of some self-Mullineux partition. Denote by M p the set of Mullineux symbols of the self-Mullineux partitions M p .
Proposition 3.15. Let p be a prime and λ a BG-partition. The BG-symbol of λ, bg p (λ) is the Mullineux symbol of some self-Mullineux partition. That is We postpone the proof of this proposition to Section 3.2, since for it we need some technical lemmas. Recall, from Proposition 2.4 and Remark 2.5, that to a Mullineux symbol corresponds a unique p-regular partition. So that the Mullineux symbol determines a bijection between p-regular partitions and their Mullineux symbols. In particular, to a symbol in M p corresponds a unique self-Mullineux partition in M p . As a corollary from Proposition 3.14, Proposition 3.15 and from the fact that the sets BG n p and M n p have the same number of elements, we obtain the following result.
Theorem 3.16. We have that bg p (BG p ) = M p , and the BG-symbol provides an explicit bijection between BG-partitions and self-Mullineux partitions. This bijection is given by associating to a BG-partition λ its BG-symbol bg p (λ), which corresponds to a unique self-Mullineux partition. This bijection restricts to bijections between BG n p and M n p for every n ∈ N.
Remark 3.17. If we consider the Mullineux symbol G p as a bijection from the set of p-regular partitions into its image in the set of two-row arrangements of integers. Then, the bijection in Theorem 3.16, from BG p to M p is given precisely by G −1 p • bg p .
Remark 3.18. In [AO91], G. Andrews and J. Olsson prove a general partition identity, which depends on some integer parameters. A special case of this identity is the fact that the number of self-Mullineux partitions of a non-negative integer n equals the number of partitions of n with different odd parts, none of them divisible by p, which is in turn equal to the number of BG-partitions. Now, in [Bes91], C. Bessenrodt shows a combinatorial proof of the Andrews-Olsson identity, which provides, by choosing the right parameters, an explicit bijection between BG n p and M n p . The bijection from Theorem 3.16 is obtained in a more direct way and it is different from Bessenrodt's bijection. In particular, for p = 5 and n = 20, the partition (7, 6, 3, 2 2 ) ∈ M 20 5 is mapped to partition (9, 3, 2, 1 6 ) ∈ BG 20 5 under Bessenrodt's bijection, and it is mapped to (7, 5, 2 3 , 1 2 ) ∈ BG 20 5 under bijection from Theorem 3.16.

3.2.
Proof of the main result. In this subsection we prove Proposition 3.15. For this proof we need some technical lemmas.
In the set of BG-partitions, the implication in Lemma 3.11 becomes an equivalence: Lemma 3.21. Let λ ∈ BG p . Then a * λ is even if and only if p | a * λ . Proof. As already noted, the fact that a * λ implies p | a * λ is proved in Lemma 3.11. Suppose that p | a * λ . If a * λ is odd, then Rim * p (λ) contains a diagonal node. Then U λ is formed by p-segments of length p and one last p-segment of length possibly less than p, which, in this case contains the diagonal node. Let us name B the set of nodes in this last p-segment, and let A be the set

Figure 2
The set Rim * p (λ) is formed by length p segments and also B. Therefore, since p | a * λ , we have that |A| = p. Now, let a = (i, j) be the first node of the segment B, that is i = min{r | (r, s) ∈ B} and j = max{s | (r, s) ∈ B}. We have that the (i, i)-th hook contains exactly |A| = p nodes. See Figure 2b.
This means that λ has a diagonal (p)-hook, which is a contradiction because λ ∈ BG p .
We obtain the following corollary from Remark 3.5 and Lemma 3.21.
(3) Rim * p (λ) has no diagonal nodes. (4) p | a * λ . Consider a partition λ ∈ M p , that is, a fixed point of the Mullineux map. This is a condition that depends only on the columns of the Mullineux symbol of λ. Therefore, the partition λ (1) obtained by removing the p-rim of λ is also a fixed point of the Mullineux map, since its Mullineux symbol is obtained by removing the first column of the Mullineux symbol of λ. The following lemma is an analogue property for partitions in BG p .
Lemma 3.23. If λ ∈ BG p then λ (1) * ∈ BG p . In other words, if λ is a BG-partition, then, removing its p-rim* results in a BG-partition.
Proof. In Remark 3.6, we said that if λ is self-conjugate, then so it is for λ (1) * . In particular, if λ ∈ BG p , then λ (1) * is self-conjugate. It remains to prove that λ (1) * does not have any diagonal (p)-hooks.

Figure 3
There are now two possible cases: either (i, µ i + 1) is the last node of a p-segment of U λ (the nodes on the p-rim* of λ over the diagonal), or it is not the last node of the p-segment to which it belongs. Let us examine these two cases.
Suppose (i, µ i + 1) is the last node of a p-segment of U λ , and this p-segment starts on a node (a, b). See Figure 3b.
Then, the (a, a) λ -th hook has length equal to the length of the (i, i) µ -th hook plus twice the length of the p-segment of Rim * p (λ) containing the node (i, µ i + 1), that is h λ a,a = p + h µ i,i + p = p + pk + p = p(k + 2), so that λ contains diagonal (p)-hook, which is impossible.
Suppose now that (i, µ i + 1) is not the last node of a p-segment of U λ . First, notice that the node (i + 1, . This is true because (i, µ i + 1) is in the p-rim* of λ. We claim that (i + 1, µ i + 1) ∈ Rim * p (λ) ⊆ [λ]. In Figure 4, node (i + 1, µ i + 2) is illustrated as a red cross (meaning it is not in [λ]) and node (i + 1, µ i + 1) as a blue colored box (as are their opposites with respect to the diagonal). Indeed, (i + 1, µ i + 1) ∈ Rim * p (λ) because, since (i, µ i + 1) is not the last node of a p-segment, then the next node of its p-segment is either to the left or down. But the node to the left of (i, µ i + 1), that is, (i, µ i ) is not in the p-rim* of λ since it is in µ, so that the next node of this p-segment is (i + 1, µ i + 1), which is then in Rim * p (λ). The fact that (i + 1, µ i + 1) ∈ Rim * p (λ) ⊆ [λ] and (i + 1, µ i + 2) / ∈ [λ] implies that λ i+1 = µ i + 1 and therefore the (i + 1, i + 1) λ -th hook has length h λ (i+1,i+1) = h µ (i,i) = pk, that is, λ has a diagonal (p)-hook, a contradiction.
Proof of Proposition 3.15. Let us first state which properties characterize elements in M p . That is, if λ ∈ M n p which conditions characterize its Mullineux symbol G p (λ) = a 0 a 1 · · · a l r 0 r 1 · · · r l .
We also know that λ is the only p-regular partition whose Mullineux symbol satisfies properties (1)-(5) from Proposition 2.4. This way, the properties that characterize Mullineux symbols of partitions in M p are equivalent to the following properties l i=0 a i = n, and (4) a i = 2r i − ε i .
On the other hand, from the definition of ε * i and Corollary 3.22, we have that Let λ ∈ BG n p . Let us see that its BG-symbol bg p (λ) = a * 0 a * 1 · · · a * l r * 0 r * 1 · · · r * l is in M p by verifying properties (1)-(3) for a * i , ε * i and r * i : From the definition of the sequence a * 0 , . . . , a * l , it is clear that (3) holds. We have that (4) is satisfied from Remark 3.5. Let us first now that (2) holds. Since λ (l) * is not the empty partition, r * l ≥ 1. On the other hand, the partition λ (l) * is a hook and is self-conjugate; more precisely λ (l) * = (r * l , 1 a * l −r * l ). Then a * l = |λ (l) * | is odd, so that ε * l = a * l mod 2 = 1. Suppose that r * l ≥ p + ε * l = p + 1. This means that the first p-segment over the diagonal of λ (l) * consists of p nodes. But then, there are more nodes remaining in the first row of [λ (l) * ] that are not in the p-rim*, but this contradicts the maximality of l.
It remains to prove (1). A key element for this task is Lemma 3.19, which roughly says that truncating a BG-partition to some particular row results in a p-regular partition. The idea is to use the fact that this superior part, being a p-regular partition, satisfies properties from Proposition 2.4, which uses numbers from the Mullineux symbol, and these will give us information about r * i and ε * i , which are numbers appearing in the BG-symbol.
We claim that Rim p (λ) and Rim p (λ (1) ) end at the same row; row l(λ). This is not obvious since it could be possible that the p-rim of a partition µ, which always contains nodes in the last row of µ, row l(µ), contains every node in this last row, and then µ (1) does not have any nodes in row l(µ). But in our setting, this is not the case. Indeed, by definition, every node of a partition is in some i-th p-rim of the partition. In particular, the diagonal node (l(λ), l(λ)) is in the j-th p-rim of λ for some j > 1 since Rim p (λ) and Rim p (λ (1) ) do not contain diagonal nodes. On the other hand the p-rim of any partition contains nodes in the last row of the partition and since l(λ) is the last row of both λ and λ (j) , then it is also the last row of λ (1) . So that both λ and λ (1) contain nodes in row l(λ). Now, the fact that Rim p (λ) and Rim p (λ (1) ) end at the same row means that l(λ) = l(λ ′ ), that is r − r ′ = 0.
So that ε ≤ r − r ′ < p + ε, as we wanted to show, since in this case ε = 0.

Figure 6
As in the previous case, we consider the partition λ to which we have removed the first l(λ) − 1 columns and we still denote it by λ. We use the same notations as before for λ and λ (1) . We illustrate (7, 5, 2 3 , 1 2 ) by thicker lines in Figure 6b.

Figure 7
As before, we consider the partition λ to which we have removed the first l(λ) − 1 columns and we still denote it by λ. We use the same notations as before for λ and λ (1) . We illustrate (6, 5 2 , 3 2 , 1) by thicker lines in Figure 7b.
On the other hand, since we have that ε ′ = 0, by the same argument as in case (i), we have that ε ′ = 0.

Figure 8
Notice that in this case it is not necessarily true that a = r and a ′ = r ′ . Since λ contains the node (l(λ), l(λ) − 1) which is under the diagonal of λ, where the p-rim* does not behave as the p-rim. For the partition (7, 4, 3, 2, 1 3 ), this node is the node (3, 2), which in this case is in the p-rim* of (7, 4, 3, 2, 1 3 ). But it could be the case that the node (l(λ), l(λ) − 1) is not in the p-rim* of λ but in the p-rim* of λ (1) * . This depends on the divisibility of r by p.
Recall that r = # U λ is the number of nodes in the p-rim* of λ that are above (or on) the diagonal of λ. Let us consider the two cases: p | r and p ∤ r.

Figure 9
4. From self-Mullineux partitions to BG-partitions From Theorem 3.16, we know that the notion of BG-symbol induces an algorithm for the correspondence between BG-partitions and self-Mullineux partition, since it defines an injective mapping between sets of the same cardinality. Then we know that to each Mullineux symbol of a self-Mullineux partition, corresponds a unique BG-partition. Moreover, from the definition of the BG-symbol, and Lemma 3.12, we know how to find the BG-partition associated to such a Mullineux symbol under this correspondence. In this section we prove that this inverse algorithm is well defined, that is, we prove that applying it to a Mullineux symbol of a self-Mullineux partition results in a BG-partition. This confirms Theorem 3.16 without using the information about cardinalities.
Proposition 4.1. Let p be a prime and λ a self-Mullineux partition. The Mullineux symbol of λ, G p (λ) is the BG-symbol of some BG-partition. That is Proof. We proceed by induction on l, the length of the Mullineux symbol.
Let l = 0. And let S = a l r l ∈ M p , that is, S = G p (λ) for some λ ∈ M p . Let ε l = 0 if p | a l and ε l = 1 otherwise. Since S has exactly one column, then λ = λ (l) is a hook. On the other hand, since λ is fixed by the Mullineux map, we know that We claim that ε l = 1. If ε l = 0, that is, if p | a l , then the p-segments that form Rim p (λ) = [λ] are all of length exactly p. We know that λ is a p-regular hook, this means that λ is formed by exactly one p-segment. If there was more than one p-segment, since λ is a hook, it would be of the form λ = (p, 1 p , ...) which is not p-regular. So that a l = p = 2r l . But this is not possible since p is odd. Then ε l = 1 and a l = 2r l − 1. The partition µ = (r l , 1 r l −1 ) is self-conjugate, and is a hook of length 2r l − 1 = a l . Since p ∤ a l , then µ ∈ BG p . Its BG-symbol is bg p (µ) = 2r l − 1 r l = a l r l = S.
Remark A.2. The generating function for the cardinality of BG n p is (1 + t 2i+1 ).
For proving the mentioned identity, we need make some remarks about the intersection of conjugacy classes in S n with A n and about Brauer characters of A n .
Splitting of conjugacy classes of S n . Let C be a conjugacy class of S n of even permutations. That is C ⊆ A n . Then one of the two following possibilities holds: • C is a conjugacy class in A n , or • C splits into two conjugacy classes in A n . In the second possibility, say C = C 1 ⊔C 2 , these two conjugacy classes have the same size. Moreover, conjugating such classes by any element of S n \ A n permutes them, that is, if σ ∈ S n \ A n , then σC 1 σ −1 = C 2 and σC 2 σ −1 = C 1 . Furthermore, the conjugacy class C splits if and only if the cycle type of elements in C consists of different odd integers ([JK81, 1.2.10]). We call a conjugacy class p-regular when the order of its elements is not divisible by p.
The set of p-regular conjugacy classes of S n contained in A n is then formed by two types of conjugacy classes: A ⊔ B, where A is set of p-regular conjugacy classes of S n of even permutations which are also conjugacy classes in A n and B is the set conjugacy classes of S n which split into two conjugacy classes in A n . Hence, the set of p-regular conjugacy classes of A n is where B consists of conjugacy classes coming for restriction of those conjugacy classes in B. These conjugacy classes in B come by pairs, in the sense that if σ ∈ S n \ A n and C ∈ B, then σCσ −1 ∈ B and C = C ∪ σCσ −1 is a conjugacy class of S n in B. Hence, a basis for the space of C-valued functions defined on p-regular elements of A n and constant on conjugacy classes is We claim that the set BG n p is in bijection with the set of p-regular conjugacy classes of A n . Notice that the set of conjugacy classes of S n with cycle type consisting of different odd integers is in bijection with self-conjugate partitions of size n. Indeed, a conjugacy class whose cycle type consists of different odd integers is associated to a unique partition λ = (λ 1 , . . . , λ r ) of n with λ 1 , . . . , λ r different odd integers (the lengths of the cycles in the cycle decomposition, in decreasing order). Consider the self-conjugate partition µ defined by the lengths of its diagonal hooks as follows: h µ 11 = λ 1 , h µ 22 = λ 2 , . . . , h µ rr = λ r . The condition of λ i 's being different, then strictly decreasing, ensures that µ is a well defined partition.
Conversely, any self-conjugate partition of n corresponds to a unique finite sequence of different odd integers; the lengths of its hooks ([JK81, 2.5.11]).
Conjugacy classes of S n with cycle type consisting of different odd integers are in particular contained in A n . If we consider those conjugacy classes with the additional condition of being p-regular, which form in fact the set B, they are therefore in bijection with self-conjugate partitions such that p does not divide the length of any diagonal hook, that is, the set BG n p . Hence B is in bijection with BG n p . Brauer characters of A n .
Let D be an irreducible F A n -module. To D we can associate a function χ D which is called the (irreducible) Brauer character of A n afforded by D. This function χ D is a complexed-valued function defined on the set of p-regular elements of A n and it is constant on conjugacy classes. Furthermore, isomorphic F A n -modules are associated to equal Brauer characters. See [Isa06,§15] for the precise definition of Brauer character and for further information.
Theorem [Isa06,15.10] says in particular that the set of irreducible Brauer characters of A n form a basis of the space of C-valued functions defined on p-regular elements of A n and constant on conjugacy classes. This implies that there are as many irreducible Brauer characters of A n as p-regular conjugacy classes of A n , and by Corollary [Isa06,15.11], this is also the number of isomorphism classes of F A nmodules. Therefore to each element µ of the set {λ | λ ⊢ n, λ p-regular and λ = m(λ)} ⊔ {λ + , λ − | λ ⊢ n, λ p-regular and λ = m(λ)}, which parametrizes irreducible F A n -modules (see 1), we can associate an irreducible Brauer character χ [µ] . That way, a basis of the space of C-valued functions defined on p-regular elements of A n and constant on conjugacy classes is {χ [λ] | λ ∈ Reg p (n) and λ = m(λ)} ⊔ {χ [λ + ] , χ [λ − ] | λ ∈ Reg p (n) and λ = m(λ)}, considering only one partition λ for each couple {λ, m(λ)} with λ = m(λ).
Proposition A.3. The sets M n p and BG n p have the same number of elements. Proof. To prove this, we will give two bases of a same space of functions, and the equality of the cardinality of these bases will give the result.
Denote by E the space of C-valued functions defined on p-regular elements of A n and constant on conjugacy classes E = {f : {p-regular elements of A n } −→ C | f is a class function of A n } .
Define an action of S n on E by conjugation as follows: for σ ∈ S n and f ∈ E, f σ is the class function f σ (τ ) := f (στ σ −1 ). For σ ∈ S n \ A n , let E σ be the set of class functions fixed by conjugation by σ: This is a subspace of E. From the above discussion about splitting of conjugacy classes and how conjugation permutes some conjugacy classes, a basis for E σ is We claim that a basis for E σ is {χ [λ] | λ ∈ Reg p (n) and λ = m(λ)} ⊔ {χ [λ + ] + χ [λ − ] | λ ∈ Reg p (n) and λ = m(λ)}.
Indeed, this comes from the fact that, as with usual characters of representations in characteristic zero, conjugation of the character of a representation is the character of conjugation of the representation, here with Brauer characters. And also from the fact that conjugation by σ permutes the modules associated to λ + and λ − above. Now, we have two bases for E and two bases for E σ . On one hand, from the characteristic function basis, the dimension of E is #A + #B = #A + 2(#B), and from the Brauer character basis, the dimension of E is #{{λ, m(λ)} | λ ∈ Reg p (n) and λ = m(λ)} + #{λ + , λ − | λ ∈ Reg p (n) and λ = m(λ)}.
That is #D + 2(#M n p ), where D = {λ, m(λ)} | λ ∈ Reg p (n) and λ = m(λ)}. Hence, #A + 2(#B) = #D + 2(#M n p ). Counting the elements on the two bases for E σ , we obtain that the dimension of E σ is #A + #B = #D + #M n p . These two identities imply that #B = #M n p . Since B is in bijection with BG n p we obtain the result.