Mullineux involution and crystal isomorphisms

We develop a new approach for the computation of the Mullineux involution for the symmetric group and its Hecke algebra using the notion of crystal isomorphism and the Iwahori-Matsumoto involution for the affine Hecke algebra of type A. As a consequence, we obtain several new elementary combinatorial algorithms for its computation, one of which is equivalent to Xu's algorithm (and thus Mullineux' original algorithm). We thus obtain a simple interpretation of these algorithms and a new elementary proof that they indeed compute the Mullineux involution.


Introduction
The Mullineux problem is a long standing problem in the representation theory of the symmetric groups which has been studied by various authors since the end of the 70's.Let S n be the symmetric group on n letters with n > 1.It is known that the irreducible representations of S n over the field of complex numbers are naturally labeled by the partitions of n (the sequences of non increasing positive integers of total sum n.) The characters and the dimensions of these representations may also been easily computed thanks to the combinatorics of partitions.There are exactly two non isomorphic representations of S n with dimension 1: the trivial representation which is labeled by the partition (n) and the sign representation ε, labeled by the partition (1. . . ..1) n times . As a consequence, if λ is a partition of n, there exists another partition µ such that It is natural to ask how one can compute µ from λ.The result is that µ is the conjugate partition of λ which is defined by interchanging rows and columns in the Young diagram of λ (the Young diagram of λ is the finite collection of boxes arranged in left-justified rows, with λ k boxes in the kth row for all k ≥ 1.) Of course, all the above questions and problems arise when we replace C by an arbitrary field k and in particular by a field of characteristic p > 0. In this case, the irreducible representations have first been constructed in [12].They are labeled by a subset of partitions called the set of p-regular partitions the partitions of n where the non zero parts are not repeated p or more times.
We also have two one-dimensional representations: the trivial representation and the sign representation ε and they are non isomorphic if and only if p = 2.By contrast, we still not even know how to compute the dimensions of these representations in general.The other mentioned problem still makes sense in this context.Namely, if λ is a p-regular partition then there exists a unique p-regular partition µ such that ρ µ ≃ ε ⊗ ρ λ .If we set m p (λ) := µ, we thus obtain an involution m p on the set of p-regular partitions.
If p = 2 then it is clear that m p = Id (because then ε is nothing but the trivial representation) but in general, it is difficult to describe m p .In fact, this map may even be defined in the context of Hecke algebras 2010 Mathematics Subject Classification: 20C08,05E10 of type A at a p-root of unity.In this case, p do not need to be a prime but just a positive integer (greater than 2).The associated involution that we obtain coincides with m p if p is prime.A natural problem is thus to find an explicit description of this involution m e on the set of e-regular partitions for all e ∈ N >1 .This is the main subject of the present paper.
In [22], Mullineux has first given a conjectural algorithm for computing this involution (which will be called the Mullineux involution in the sequel).Later, another equivalent algorithm has been given by Xu [23,24].In [20], Kleshchev gave another combinatorial recursive algorithm for computing the Mullineux involution but it was not clear at that time why this algorithm would be equivalent to the Mullineux (and the Xu's) algorithm.Ford and Kleshchev gave a proof of this fact later in [9].Another proof was given in [3] by Bessenrodt and Olsson.In [5], Brundan and Kujawa gave another proof using works by Serganova on the general linear supergroup.We also note that recently, Fayers [8] has given another way for computing the involution.
The aim of this paper is to present several elementary combinatorial (and recursive) algorithms for the computation of the involution using the Kleshchev result.These algorithms are based on the results of [18,19] and on the following points: 1.Each simple module for the Hecke algebra of type A labeled by an e-regular partition of rank n can be seen as a simple module for the affine Hecke algebra of type A.
2. The Mullineux map at the level of Hecke algebra coincide with the so called Iwahori-Matsumoto involution for the affine Hecke algebra of type A.
3. The Iwahori-Matsumoto involution may be computed using an analogue involution at the level of Ariki-Koike algebras associated to a multicharge s ∈ Z l .
4. This later involution may be computed using the Mullineux involution for Hecke algebras of type A on e-regular partitions with rank (strictly) less than n.
As a consequence, to compute the image of an e-regular partition of rank n under the Mullineux involution, we are reduced to compute several images of e-regular partitions of rank strictly less than n under the Mullineux involution.This thus gives a recursive algorithm to solve our problem.In fact, depending on the multicharge, we choose for our Ariki-Koike algebras, we obtain several different algorithms.It turns out that for a particular choice of multicharge, our algorithm is equivalent to Xu's algorithm.This thus gives a new elementary proof for the fact that the Mullineux and the Xu's algorithm give an answer for the Mullineux problem.This also gives a new interpretation of these algorithms (another interpretation is also given in [5]).The paper will be organized as follows.In section 2, we recall some basic facts on the representation theory of affine Hecke algebras of type A and of Ariki-Koike algebras.We also recall several results coming from [18,17] concerning the labelling of the simple modules for these algebras and the relations between them.Section 3 introduces the Mullineux and the Iwahori-Matsumoto involutions and shows how these two maps are related.In section 4, we study combinatorial properties of partitions and multipartitions which will be used in the following sections.Section 5 gives the algorithms we get for computing the Mullineux involution.The last section shows that Xu's algorithm can be seen as one of our algorithm.

Acknowledgement:
The author thanks Cédric Lecouvey for fruitful discussions on the subject of this paper.The author is supported by ANR project JCJC ANR-18-CE40-0001.

Hecke algebras
In this first section, we recall the definitions of the affine Hecke algebra of type A and of the Ariki-Koike algebras.We then give a brief overview of their representation theories.Finally, we explain the relations between the known parametrizations of the simple modules for these algebras.The main references for these parts are [1] and [10].

Affine Hecke algebra of type A
Let n ∈ Z >0 .Let q ∈ C * be a primitive root of unity of order e > 1.The Iwahori-Hecke algebra H n (q) of type A is the unital associative C-algebra generated by T 0 , T 1 , . . ., T n−1 and subject to the relations: The affine Hecke algebra H n (q) is the unital associative C-algebra which is isomorphic to as a C-vector space and such that H n (q) and C[X ±1 1 , . . ., X ±1 n ] are both subalgebras of H n (q) with the following additional relations: for all (i, j) ∈ {1, . . ., n − 1} 2 with i = j.We denote by Mod n the category of finite dimensional H n (q)-modules such that for all j = 1, . . ., n, the eigenvalues of the X j are power of q.The simple objects Irr( H n (q)) in Mod n can be naturally labeled by the set of aperiodic multisegments that we now define: Definition 2.1.1.Let l ∈ N >0 and let i ∈ Z/eZ.The segment of length l and head i is the sequence of consecutive residues (i.e elements of Z/eZ, identified with {0, 1, . . ., e − 1}) [i, i + 1, . . ., i + l − 1] in Z/eZ.The residue i ∈ Z/eZ is then called the head of the segment and the residue i + l − 1 the tail of the segment.A multisegment is a formal sum of segments.A multisegment is said to be aperiodic if for every l ∈ Z >0 , there exists i ∈ Z/eZ such that there is no segment with length l and tail i appearing in the multisegment.We denote by M e the set of aperiodic multisegments.The length of a multisegment is the sum of the lengths of the the segments appearing in it and is denoted by |ψ|.We denote by M e (n) the set of aperiodic multisegments of length n.
Example 2.1.2.For e = 3, the multisegment: is an aperiodic multisegment of length 10 where as is a multisegment of length 10 which is not aperiodic.
By the geometric realization of H n (q) by Chriss and Ginzburg [7], we know that one may naturally label the simple modules in Mod n by the set M e (n) of aperiodic multisegments of length n.We thus have:
Let s be an orbit with respect to the above action and let s := (s 1 , . . ., s l ) ∈ Z l be an element in this orbit.The Ariki-Koike algebra H s n (q) is the quotient H n (q)/I s where I s := 1≤j≤l (X 1 − q sj ) .If l = 1, this is a Hecke algebra of type A (of finite type), and if l = 2 a Hecke algebra of type B (of finite type).One can see that the above algebra is well defined and depends only on the orbit of s modulo the action of S l (and on q).
The representation theory of this algebra has been intensively studied in a number of works.We refer to [1,10] and the references theirin.We will only recall what is needed for the results of the present paper.The analogues of the multisegments in the context of Ariki-Koike algebras are the multipartitions that we now define.For this, let us give some additional combinatorial definitions.
A partition is a nonincreasing sequence λ = (λ 1 , • • • , λ m ) of nonnegative integers.One can assume this sequence is infinite by adding parts equal to zero.The rank of the partition is by definition the number |λ| = 1≤i≤m λ i .We say that λ is a partition of n, where n = |λ|.By convention, the unique partition of 0 is the empty partition ∅.
More generally, for l ∈ Z >0 , an l-partition λ of n is a sequence of l partitions (λ 1 , . . ., λ l ) such that the sum of the ranks of the λ j is n.The number n is then called the rank of λ and it is denoted by |λ|.The set of l-partitions is denoted by Π l and the set of l-partitions of rank n is denoted by Π l (n).Let λ be an l-partition.The nodes or the boxes of λ are by definition the elements of the Young diagram of λ: The content of a node γ = (a, b, c) of λ is the element b − a + s c of Z and the residue is the content modulo eZ.If l = 1 (that is when we consider a partition instead of a multipartition), then the Young diagram is identified with a subset of Z >0 × Z >0 in an obvious way.
Since the works of Ariki and Lascoux-Leclerc-Thibon, it is known that the representation theory of these algebras is closely related to the representation theory of quantum groups.In particular, one can naturally label the simple modules by the crystal basis of a certain integrable representation for the quantum group of affine type A. We will not give the details of all the consequences of this fact but we summarize this below.Again, we refer to [10] for a complete study.For all choice of s ∈ s, we can define a certain subset of l-partitions which are called Uglov l-partitions and which are denoted by Φ e,s (n).These classes of multipartitions, which strongly depends on the choice of s, can all be seen as non trivial generalizations of the set of e-regular partitions: • For all s ∈ Z, we define: This is a fundamental domain for the action of S l on Z l .If s ∈ A l e [s], then the l-partitions in Φ e,s (n) are known as FLOTW l-partitions and they have a non recursive definitions: we have λ = (λ 1 , . . ., λ l ) ∈ Φ s,e (n) if and only if: 1.For all j = 1, . . ., l − 1 and i ∈ Z >0 , we have: 2. For all i ∈ Z >0 , we have: is a proper subset of Z/eZ.
• If s satisfies for all i = 1, . . ., l − 1, s i+1 − s i > n− 1 (we say that s is very dominant, it is also sometimes referred as the "asymptotic case" in the literature) then the set Φ e,s (n) is known as the set Kleshchev l-partitions.If s ′′ satisfy the same property, then the associated set Φ e,s" (n) is the same.
• If l = 1, the set Φ e,(s) (n) is simply the set of e-regular partitions It turns out that each set Φ e,s (n) with s ∈ s gives a natural labelling for the irreducible representations of the Ariki-Koike algebra H s n (q).As a consequence, there are several natural possibilities for the labelling of the simple modules of H s n (q), one for each choice of an element in the orbit s.For more details on these parametrizations, we refer to [10].Thus, one can write: n)}.By [6], each of these labellings has an interpretation in terms of a cellular structure.Last, clearly, if s and s ′ in the same orbit, there is a bijection: which is uniquely defined as follows.For all λ ∈ Φ (e,s) (n) then: This bijection has been explicitly described in [18] in a combinatorial way using crystal isomorphisms (the coincidence of the crystal isomorphisms with these bijections is proved in [14,Prop. 3.7].)We recall this description subsection (a program in GAP3 is available for computing it in all cases [15]).In the next sections, the following particular case: s = (s 1 , s 2 ) and s ′ = (s 1 , s 2 + e) will be of particular interest.
We define the minimal integer

and otherwise the minimal integer
, we associate its s-symbol of length d.This is the following two-rows array.
We will write S(λ 1 , λ 2 ) = L2 L1 where the top row (resp.the bottom row) corresponds to λ 2 (resp.λ 1 ).Of course, it is easy to recover the 2-partition from the datum of its symbol.From this symbol, we define a new symbol L2 L1 as follows.

Aperiodic multisegments and multipartitions
Let s be an orbit of Z l with respect to the action of the affine symmetric group (recall the definition of the action in §2.2).If V is a simple module for the Ariki-Koike algebra then it is also a simple H n (q)-module in the category Mod n .Hence there exists a unique aperiodic multisegment ψ such that V ≃ L ψ (as a H n (q)-module).As a consequence, far any s ∈ s we have a well defined map: χ n e,s : Φ (e,s) (n) → M e (n), which is defined as follows.Let λ ∈ Φ (e,s) (n), then we have a unique χ n (e,s) (λ) ∈ M e (n) such that: . By [2], this map may be described as follows: • Assume first that s ∈ A l e [s] for all non zero part λ c i of λ, we associate the segment By [2], The multisegment χ n e,s (λ) is just the formal sum of all the segments associated to the non zero part of λ.
Given an aperiodic multisegment ψ, It is now natural to try to find the multicharges s such that ψ as an antecedent for the map χ n e,s .This question has been completely solved in [17].There always exist such multicharges (they are non unique in general) which are called admissible multicharges.By [2], χ n e,s is injective so that if s is admissible for ψ there exists a unique λ such that χ n e,s (λ) = ψ.This l-partition will be called admissible (with respect to ψ).By definition, we have the following proposition where we use the following notation.For s and t two multicharges, we denote s ⊂ t if and only if, for all j ∈ Z/eZ, the number of integers conguent to j in s is less or equal to the number of integers conguent to j in t.Proposition 2.4.1.Assume that λ ∈ Φ (e,s) (n) then t is admissible for the multisegment χ n e,s (λ) if and only if s ⊂ t.
Proof.Set s = (s 1 , . . ., s l ) and t = (t 1 , . . ., t m ).Assume that λ ∈ Φ (e,s) (n) then as a H n (q)-module, we have that 1≤j≤l (X 1 −q sj ) acts as 0 on D λ s ≃ L χ n e,s (λ) .As a consequence, as s ⊂ t, we have that 1≤j≤m (X 1 −q tj ) acts as 0 on L χ n e,s (λ) .This implies that it is a well-defined H t n (q)-module and the result follows.
Remark 2.4.2.One can also prove the above proposition combinatorially using the descriptions of the admissible multicharges.

The Mullineux and the Iwahori-Matsumoto involutions
The aim of this section is to introduce the Mullineux involution for the symmetric group and its analogues in the context of Ariki-Koike algebras and affine Hecke algebras.
• The involution ∇ : We have for all x ∈ H n (q): These two involutions thus also induce involutions on the set M e (n) and they have been studied in [17].

Mullineux involution for Ariki-Koike algebras
Assume that s ∈ Z l .Then we have a well-defined algebra automorphism: ), which is defined on the generators as follows: This map naturally induces bijections on the indexing sets of the simple modules of Ariki-Koike algebras.Let s ♯ be the orbit of (−s 1 , . . ., −s l ) modulo the action of the affine symmetric group.Let v ∈ s ♯ then we have a map: This map has been described in [18].If l = 1 and e is prime then it coincides with the usual Mullineux involution of the symmetric group that we have defined in the introduction.If l = 1, then it corresponds to the Mullineux involution of the Hecke algebra of type A of [4] which will simply be denoted by m e (it does not depend on s).In this paper, we will give an algorithm for computing m e .Remark 3.2.1.If λ is a partition and γ a node of its Young diagram, the γ-hook of λ id by definition the set of all the nodes at the right and at the bottom of γ (including γ).The length of the hook is the number of nodes in it.We say that λ is an e-core if all the hooks have length strictly less than e.If λ is an e-core then m e (λ) can be easily described: it is just the conjugation of λ (as in the semisimple case), see [22] (when e is a prime but the results generalizes easily if e is an integer).More generally, it is a natural question to ask how one can describe all the maps m s→v e in general.It turns out that by [16,Prop. 4.2], knowing the map m e , one can describe it quite easily in a particular case: Proposition 3.2.2.Assume that s is very dominant.Let s ♯ := (−s ′ 1 , . . ., −s ′ l ) be a very dominant multicharge such that s ′ i ≡ s i + eZ for all i = 1, . . ., l.Then for all λ ∈ Φ (e,s) (n), we have: (λ) = (m e (λ 1 ), . . ., m e (λ l )).

Relations between the involutions
Now we put all the above results together to deduce relations between the various involutions we have defined.The following result is proved in [17].
Theorem 3.3.1.Let ψ be an aperiodic multisegment and let s ∈ A l e [s] be an admissible multicharge for ψ.Set s t = (−s l , . . ., −s 1 ) ∈ A l e [−s l ] then we have: As a consequence, the Iwahori-Mastumoto involution may be computed as follows.Take an aperiodic multisegment ψ.
• Compute ν := m s→s t e (λ) using the discussion in the last section.
Again, we compute Ψ (0,8)→(0,2) e ((1), ( 6)) = ((1), ( 6)) and thus we get Now, let us explain how one can deduce an algorithm for computing the Mullineux involution for e-regular partitions.This is based on the following elementary remark.Let λ ∈ Φ e,(0) be an e-regular partition and consider the aperiodic multisegment ψ := χ n e,(0) (λ) (recall that this is nothing but the formal sum of the segments given by the rows of the Young diagram of λ).The above theorem shows that: So now we are reduced to compute (χ n e,(0) ) −1 (ψ ♯ ).Take s ∈ A l e [0] such that l > 1 then by Proposition 2.4.1, this is an admissible multicharge.We have: Now µ := (χ n e,s ) −1 (ψ) is the admissible l-partition and the main problem is thus to compute m s→s t e (µ).We have already seen that this can be done in three steps: 1. Compute the crystal isomorphism Ψ s→v e (µ) = (ν 1 , . . ., ν l ) where v is very dominant (recall that this means that v = (s 1 , s 2 + ke) with ke > n − 1) 2. By Proposition 3.2.2,m v→v ♯ e (ν) can be computed by applying the Mullineux map component by component.As |ν| = |λ|, if we assume that at least two components of the l-partition (ν 1 , . . ., ν l ) are non empty, all of the components are of rank < n and we know how to compute the Mullineux involution by induction.
3. Apply again a crystal isomorphism Ψ v ♯ →s t e .
In the next section, we will apply the above algorithm in the case where l = 2 and in particular show that the condition for applying our induction in step 2 is always satisfied (except in the case where s = (s 1 , s 2 ) and s 1 = s 2 .)

Combinatorial properties
In this section, we will try to find simple combinatorial ways to compute several objects that we have already defined: this concerns the admissible multicharges and multipartitions and the crystal isomorphisms.

On admissible multipartitions
If λ and µ are two partitions, we denote by λ ⊔ ν the partition obtained by concatenation (and reordering the parts if necessary).
Assume that we have an e-regular partition λ = (λ 1 , . . ., λ r ) (that is λ ∈ Φ e,(s) (n) for any s ∈ Z).Let s ∈ A l e [s].By Proposition 2.4.1, s is an admissible multicharge.The aim of this subsection is to show that one can easily construct the associated admissible l-partition λ ∈ Φ e,s (n) such that χ n e,s (λ) = χ n e,(s) (λ) (recall that χ n e,s is always injective).To do this, one can use the algorithm developed in [17] from the datum of the multisegment χ n e,(s) (λ) or we can argue as follows.Let l ′ ∈ {1, . . ., l} be minimal such that s l ′ = s l .We construct λ by induction as follows.
Proposition 4.1.1.With this construction, we have λ ∈ Φ e,s (n) and χ n e,s (λ) = χ n e,(s) (λ).Proof.We prove the proposition by induction.The result is trivial when n = 0. Keeping the above notations, one can assume that ν ∈ Φ (e,s ′ ) (n − m).First one can perform exactly the same procedure as in §2.4 for the description of the map χ n e,s to associate to λ a multisegment (even if we have -not already -proved that λ is in Φ e,s (n)).By construction, this multisegment is nothing but χ n e,(s) (λ).It is thus an aperiodic multisegment.This proves condition 3 of FLOTW l-partition for λ (see the definition in §2.2).Hence, we just need to show that the l-partition satisfies the two first points.
Using this, we have thus constructed a map θ n e,s : Φ e,(0) (n) → Φ e,s (n) which associates to λ the l-partition λ constructed above (we will sometimes omit the subscript n).

Crystal isomorphisms
In this second subsection, we study in details the crystal isomorphisms restricted to the multipartitions in the image of θ e,s . in the case where l = 2.The first aim is to implify the procedure to compute it, the second is to show certain crucial properties which will show that our algorithm run.Let λ be an e-regular partition and assume that s = (0, s).We also assume that λ is non empty and that r is maximal such that λ r = 0. Let (λ 1 , λ 2 ) := θ (e,(0,s)) (λ) and consider the associated symbol with length te with t sufficiently large.It is thus of the following form : By definition of the symbol, we here have α j := λ 2 j −j+s for j = 1, . . ., ke and β j := λ 2 j −j for j = 1, . . ., ke−s.We denote (µ 1 , µ 2 ) := Ψ (0,s)→(0,s+k.e)e (λ 1 , λ 2 ) (so that, as usual, ke > n − 1 and thus so that the multicharge (0, s + ke) is very dominant) Assume that λ = ∅ and that µ 2 = ∅ then the algorithm for the computation of Ψ (0,s)→(0,s+k.e)e easily shows that that this can happen if and only if Ψ (0,s)→(0,s+k.e)e is the identity.This thus implies that In this case, we also need to have r ≤ e − s.Now we have for all i = 1, . . ., ke, α i = −i + s and also β j ≤ α j for all j = 1, . . ., ke − s.As a consequence, we have and thus λ 2 2 ≤ s.We conclude Proposition 4.2.1.Under the above notations, assume that µ 2 = ∅ then λ = λ 1 is an e-core.
Proof.The above discussion shows that λ has at most e − s non empty rows and at most s columns.This implies that the hooks of λ has at most length e − 1 and thus that λ is an e-core.Now let us see what we can say if µ 1 = ∅.Before this, we show below that the image of λ under a crystal isomorphism can be quite easily computed in the case where λ is in the image of θ e,s which is the case we are interested in here.
These calculations show that one can perform our crystal isomorphism step by steps in the "blocks" of the symbol separated by vertical lines below.First recall in Example 2.3.1 how the crystal isomorphisms Ψ (0,s ′ )→(0,s ′ +e) e can be described.and we see that the properties above are always satisfy.In particular, with the notations above, we have.
4. We have m e (λ) = θ −1 (e,(0,e−s)) (κ 1 , κ 2 ).Note that in principle, one can choose an arbitrary multicharge s instead of (0, s) (as soon as the second point at the end of subsection 3.3 is satisfied) but the complexity of the algorithm for the computation of the crystal isomorphism from s to a very dominant multicharge increases.However, It is not unreasonable to expect that some particular multicharge can lead to interesting fast new algorithms.

Steps 1 and 2
It follows from Section 4.2 that the first two steps can be both implemented by the process below.Let 0 < s < e and set s = (0, s).We set λ[1] = (λ 1 , . . ., λ e−s ) and λ [2] = (λ e−s+1 , . . ., λ r ), we write the Young tableau of λ [1] with the associated contents and just below, the Young tableau of λ [2] with the associated contents with respect to the multicharge (0, s).Now, starting with the first part of λ [1], consider the content of the rightmost box, say c.In λ [2], we consider the rightmost boxes and we take the one with the greatest content which is less than c, say c ′ .Then we remove the boxes of the first part of λ [1] with content greater than c ′ into this part in λ [2] (in other words, we move the "truncated first row" containing the boxes grater than c to the row in λ [2]).
It is clear that we still have a partition.Then, we do the same for the second part of λ [1] and so on until we reach the last part of λ [1].If this is not possible we switch to the second part of λ [1], and we continue this process until we reach the last part of λ [1].
In our example, we must remove the boxes in bold in the first partition above, and add the boxes in bold in the second partition below.We then collect all the parts of λ [2] that are above the smallest part we have modified, in a partition µ.So here µ = (9,7,6).The new partition λ [2] is given by the remaining parts and we add e to the contents of all the boxes in it.We then move the step above and continue the process until we cannot do anything.The remaining parts of λ [2] are added to µ.Then the partition λ [1] is the first component of Ψ (0,s)→(0,s+k.e)e (λ 1 , λ 2 ) and µ is the second.
We write the Young tableau of ν 1 with the associated contents for each box, and just below, the Young tableau of ν 2 with the associated contents charged by ke − s where k is sufficiently large (that is, the content of the box (a, b) is b − a + (ke − s)).Keeping the above example, we have by induction m 4 (4, 3, 3) = (10) and m 4 (9, 7, 6, 4, 3, 3, 2, 1) = (14, 7, 7, 3, 3, 1).So we consider the bipartition (( 10), (14,7,7,3,3,1) and the multicharge is (0, 3).At each step, starting from the bottom of ν 2 , we see if one can remove boxes from ν 2 to add it to ν 1 as in the subsection above (except that we remove the box from the other partition).Note that ν 1 need to always have the same number of rows so we only add the possible boxes in the e − s rows of ν 1 .Then we remove e from all the contents of the boxes of ν 2 .In the example, we have nothing to do so we remove e from all the contents of the second partitions and again one more time.At the end, the concatenation (and reordering the parts if necessary) of the two partitions we get must be m e (λ).In our example, we obtain ((17), (9, 7, 6, 3, 3)) so that m e (λ) = (17, 9, 7, 6, 3, 3).

Example
Let us keep our running example λ = (10, 8, 7, 5, 4, 4, 3, 2, 1, 1), l = 2 and e = 4 but this time, we take s = 2.The first two steps will give: Assume that the cardinality of the e-rim of λ is m.The truncated e-rim of λ is by definition the set of nodes (i, j) in the e-rim of λ such that (i, j − 1) is also in the e-rim of λ.If e does not divide m, we add also the node (r, x) in the e-rim of λ such that (r, x − 1) is not in the e-rim.We now define λ to be the partition obtained by removing the truncated e-rim from λ.It is easy to see that this partition is e-regular with rank strictly less than the rank of λ.Example 6.1.2.Let e = 3 and λ = (8,5,3,3).The truncated e-rim is given by the nodes marked by a star.
We here give a new proof of this Theorem using the crystal isomorphisms.

Relation with crystal isomorphisms
We will see in this subsection that Xu's algorithm is equivalent to ours in the case where we choose s = e − 1.
For λ an e-regular partition, we denote by λ the partition obtained by removing the truncated p-rim as in Xu's algorithm.We denote by r the number of boxes in the truncated p-rim.Proposition 6.2.1.We have Ψ (0,e−1)→(0,e−1+ke) e • θ e,(0,e−1) (λ) = (r, λ) (k >> 0) Proof.We denote λ[2] = (λ 2 , . . ., λ e ).We begin with the two first steps of our algorithm which are described in §5.1.Assume first that one cannot add any "truncated row" of λ 1 in λ [2].This means that there exists k > 0 such that λ k+1 − k + e − 1 = λ 1 − 1 and we have the following partitions: Then the partition (λ 2 , . . ., λ k+1 ) corresponds to the partition (λ 1 , . . ., λ k ) with the very first truncated e-rim removed.If λ k+1 = 0 then we are done and λ 1 is the number of nodes in the truncated p-rim minus 1.In this case the number of elements in the associated e-rim is not e.Otherwise we get e boxes in the associated rim and we must go to the second step of our algorithm.
Assume that one can add a truncated row of length r.Assume that the row is added in the part λ k+1 .Then the partition (λ 2 , . . ., λ k+1 ) corresponds to the partition (λ 1 , . . ., λ k ) with a truncated e-rim removed.with the partition λ [1] − (e − k) to find λ [2] and take into account that we must add e − k (the length of the truncated p-rim) to the partition we obtain at the end of our algorithm.Note that the content of the leftmost node in our first partition will be now e − k and the contet of the leftmost node of the second partition (e − k) − e − 1 so the induction can be done.
Let us now explain in which way our two algorithms are equivalent in the case where we choose s = (0, e − 1).Let λ be an e-regular partition and recall the 4 steps of our algorithm at the beginning of §5.
1.By Proposition 6.2.1, after the two first steps of our algorithm, we obtain (r, λ) where r is the number of boxes in the truncated rim.
2. By induction, we know m e ( λ) and the third step of our algorithm consists in the computation of the image of (m e (r), m e ( λ)) with respect to Ψ (0,1+ke)→(0,1) e (for k >> 0).
The above result thus shows that Xu's algorithm indeed computes the Mullineux involution.Remark 6.2.3.In [5], Brundan and Kujawa gave another interpretation of the Xu's algorithm using the representation theory of the supergroup GL(n|n).It would be interesting to understand the connection wof this work with ours.
[1]λ k+1 is non zero.Note that the length of the truncated p-rim is e − k.By induction, the first e − k nodes of the partition λ[1]will not moved in our algorithm We can thus just argue by induction by replacing λ[1] − 3 . ..λ e−1 − (e − 1) e − 2 . ..λ 2e−1 − (e − 1) . . . . . . . ..and now, we have to perform the algorithm for the following configuration of partitions:Now, we come to the last step, assume that s is maximal such that λ ke−k+s = 0 (so that ke − k + s is the length of the first column of λ).Then, we are in the following configuration where we have an addable k + 1-node in the second partition.