Crystal isomorphisms and Mullineux involution II

We present a new combinatorial and conjectural algorithm for computing the Mullineux involution for the symmetric group and its Hecke algebra. This algorithm is built on a conjectural property of crystal isomorphisms which can be rephrased in a purely combinatorial way.


Introduction
The Mullineux involution is an important map which has been originally defined by Mullineux [22] in the context of the modular representation theory of the symmetric group.More generally, it can be defined for the class of Hecke algebras of the symmetric group [2].Let n ∈ Z >0 and e ∈ Z >1 .Let η be a primitive e root of 1.The Hecke algebra of the symmetric group H n (η) is defined as the associative unital C-algebra with generators T 1 , . . ., T n−1 and the following relations: It is known that the simple modules of this algebra are naturally labelled by the set of e-regular partitions Reg e (n) with rank n (see §2.1 for the definition): There is a C-algebra automorphism ♯ which can be defined on the generators of H n (η) as follows.For all i = 1, . . ., n − 1, we have T ♯ i = −ηT i .This automorphism induces an involution: m e : Reg e (n) → Reg e (n), defined as follows.For all λ ∈ Reg e (n) there exists a unique µ ∈ Reg e (n) such that the module D λ twisted by ♯ is isomorphic to D µ .Then we define m e (λ) := µ.If e is prime, this involution describes the structure of a simple F e S n -module twisted by the sign representation.If e is sufficiently large, or more generally if λ is an e-core, it is easy to see that m e (λ) is just the conjugate partition λ ′ .The study of the Mullineux involution has a long story.A first conjectural and combinatorial description of m e (if e is prime) was first given by Mullineux [22] and proved later by Ford and Kleshchev [11].Before this proof, Kleshchev gave a solution to the computation of the involution [18] (see also [1] and [3]).This solution may be rephrased in terms of the crystal graph theory.Other algorithms were given by Xu [23,24], or more recently by Fayers [6], and by the author [13].We also note that there exist different generalizations in the context of Ariki-Koike algebras [7,16], affine Hecke algebras [21,17], general linear groups [5] or rational Cherednik algebras [20,10] and they are all connected with the above one.We also mention a recent conjecture by Bezrukavnikov on this involution in relation with nabla operators and Haiman's n! conjecture studied in [4].
All the above algorithms for computing the Mullineux involution have a common feature: they are recursive algorithms in n.The algorithms to compute the Mullineux image of a partition λ of rank n requires the computation of the Mullineux involution m e (µ) for |µ| < n.The aim of this paper is to present a conjectural algorithm which is recursive in e.This conjecture is in fact built on the description of the Mullineux involution by Kleshchev in terms of crystal graphs together with the concept of crystal isomorphisms described in [15].The conjecture follows in fact from a purely combinatorial conjecture which can be described without any mention to crystals and in a very simple way.Assuming the conjecture true, it becomes possible to compute m e from the datum of m 2e .As m e corresponds to the conjugation of partitions if e is sufficiently large, the algorithm follows.
The paper is organized as follows.We first recall several elementary combinatorial notions on partitions and crystals.This section ends with a presentation of the Kleshchev' solution to the Mullineux problem.The second section explains the notion of crystal isomorphism.We then give a conjectural combinatorial property, Conjecture 4.5, which can be rephrased in the context of crystal isomorphisms.The last section presents several new results around this notion and states the conjectural algorithm for computing the Mullineux involution.

Acknowledgements:
The authors are grateful to Matt Fayers for useful discussions.The first author is supported by ANR project AHA ANR-18-CE40-0001.Both authors are supported by ANR project COR-TIPOM ANR-21-CE40-0019.

Mullineux involution for Hecke algebras
We first start with the definition of several elementary notions.Then we present the Kleshchev solution to the computation of the Mullineux involution.

Partitions and Young diagrams.
A partition is a non increasing sequence λ = (λ 1 , • • • , λ m ) of nonnegative integers.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|.The unique partition of 0 is the empty partition ∅.We denote by Π(n) the set of partitions of n.For e ∈ Z >1 , we say that λ is an e-regular partition if no non zero part of λ can be repeated e or more times.The set of e-regular partitions of rank n is denoted by Reg e (n).Given a partition λ ∈ Π(n), its Young diagram [λ] is the set: The elements of this set are called the nodes of λ.The e-residue (ore more simply, residue) of a node γ ∈ [λ] is by definition res(γ) = b − a + eZ.For j ∈ Z/eZ, we say that γ is a j-node if res(γ) = j.In addition, γ is called a removable j-node for λ if the set [λ] \ {γ} is the Young diagram of some partition µ.In this case, we also say that γ is an addable j-node for µ.
Let γ = (a, b) and γ ′ = (a ′ , b ′ ) be two addable or removable j-nodes of the same partition λ.Then we write γ > γ ′ if a < a ′ .Let w j (λ) be the word obtained by reading all the addable and removable jnodes in increasing order and by encoding each addable j-node with the letter A and each removable j-node with the letter R. Then deleting as many subwords RA in this word as possible, we obtain a new word The node corresponding to the rightmost A (if it exists) is called the good addable j-node and the node corresponding to the leftmost R (if it exists) is called the good removable j-node.

2.2.
Level 1 Fock space.Let F be the C-vector space with basis given by all the partitions.It is called the (level 1) Fock space.There is an action of U( sl e ) on F which makes F into an integrable module of level 1.For i ∈ Z, the Kashiwara operators e i+eZ,e and f i+eZ,e are then defined as follows.
Using these operators one can construct the sl e -crystal graph of F , which is the graph with • vertices: all the partitions λ of n ∈ N, • arrows: there is an arrow from λ to µ colored by i ∈ Z/eZ if and only if f i+eZ,e •λ = µ, or equivalently if and only if λ = e i+eZ,e • µ.
Note that the definition makes sense for e = ∞.The corresponding graph, sl ∞ -crystal graph, coincides with the Young graph, which describes the branching graph of the complex irreducible representations of symmetric groups.

Mullineux involution.
We can first give an interpretation of the set of e-regular partitions using Kashiwara operators.The following result can be found for example in [19, §2.2].
Proposition 2.1.A partition λ is an e-regular partition of n if and only if there exists: In other words, the vertices in the connected component of the sl e -crystal graph containing the empty partition are exactly the e-regular partitions.We thus have a subgraph of this crystal graph with vertices all these e-regular partitions.
Recall the definition of the Mullineux involution given in the introduction.The following result permits to compute it in a purely combinatorial way thanks to the above results.
Theorem 2.2 (Kleshchev).Let λ be a e-regular partition.Then, there exists (i 1 , . . ., i n ) ∈ Z n such that: Then, there exists an e-regular partition µ such that: Moreover, we have m e (λ) = µ where m e is the Mullineux involution defined in the introduction.
If λ is a partition, every node of its Young diagram has an associated hook, defined as the set of nodes directly below or to its right (including itself).A partition is called an e-core if it has no hook with k.e nodes for every k ∈ N. Of course, if e is sufficiently large comparing to n (e > n), every partition of n is an e-core.If λ is an e-core, it is already contained in Mullineux's original paper [22] that m e (λ) is the conjugate partition of λ (defined as the partition obtained by interchanging rows and columns in the Young diagram of λ) Example 2.3.Let e = 3 and let λ = (5, 2, 1, 1).This is a 3-regular partition.Then we have: ). and thus m 3 (5, 2, 1, 1) = (4, 2, 2, 1).If e = 6 then λ = (5, 2, 1, 1) is a 6-core and we have: , which is the conjugate partition of λ, as expected.
In the following, we will study another way to compute this map without any use of the crystal and the Kashiwara operators.

Crystal isomorphisms for bipartitions
In this section, we quickly summarize the needed results to expose our algorithm.These results mainly concern certain expansions of the above discussion to the case of bipartitions.
3.1.Level 2 Fock space.From now we fix a bicharge, that is a couple s One can define the level 2-Fock space as the C-vector space with basis indexed by all the elements of Π 2 (n) for n ∈ Z ≥0 .There is also a notion of crystal for this 2-Fock space with similar notions of Kashiwara operators f s i+eZ,e and e s i+eZ,e .Importantly, the action of these operators on each bipartition really depends on the choice of s.
To each and the residue res(γ) is by definition the content of the node taken modulo e.We will say that γ is an i + eZ-node of λ when res(γ) ≡ i + eZ (we will sometimes simply called it an i-node).Finally, we say that γ is removable when γ = (a, b, c) ∈ [λ] and [λ]\{γ} is the Young diagram of a bipartition.Similarly, γ is addable when γ = (a, b, c) / ∈ [λ] and [λ] ∪ {γ} is the Young diagram of a bipartition.Let γ, γ ′ be two removable or addable i-nodes of λ.We denote For λ a bipartition and i ∈ Z/eZ, we can consider its set of addable and removable i-nodes.Let w (e,s) i (λ) be the word obtained first by writing the addable and removable i-nodes of λ in increasing order with respect to ≺ s , next by encoding each addable i-node by the letter A and each removable i-node by the letter R. Write w (e,s) i (λ) = A p R q for the word derived from w (e,s) i (λ) by deleting as many of the factors RA as possible.
In the following, we will sometimes write w i (λ) and w i (λ) instead of w (e,s) i (λ) and w (e,s) i (λ) if there is no possible confusion.
If p > 0, let γ be the rightmost addable i-node in w i .The node γ is called the good addable i-node.If r > 0, the leftmost removable i-node in w i is called the good removable i-node.The definition of the Kashiwara operators f s i+eZ,e and e s i+eZ,e follows then exactly as in §2.2.In the same spirit as in the above discussion, one can also define a certain subset of bipartitions Φ (s,e) (n): Definition 3.1.We say that (λ 1 , λ 2 ) is an Uglov bipartition associated with s ∈ Z 2 if there exist (i 1 , . . ., i n ) ∈ Z n such that: ).We denote by Φ (e,s) the set of Uglov bipartitions and by Φ (e,s) (n) the set Φ (e,s) ∩ Π 2 (n).
We make the three important following remarks.Remark 3.2.

Explicit computations and a combinatorial property
In this section, we explain how one can compute the above crystal isomorphisms.Our main conjecture is relied on a combinatorial conjectural property of these maps.This property can in fact be settled in a completely general framework.

4.1.
A combinatorial map.We recall here results from [15].Let e be a positive integer.For r a positive integer, we denote by P r the set of strictly increasing partitions in r parts.Let m 1 and m 2 be two integers such that m 1 ≤ m 2 .
• We repeat this procedure with (a 2 , . . ., a m1 ) and X 2 \ {ϕ(a 1 )} and thus associate to each element of X 1 a unique element in X 2 .
We now define a map: Ψ (e,(m1,m2)) : where we reorder these two sets so that Y 1 ∈ P m1 and Y 2 ∈ P m2+e .
Then if k is odd, we have X 1 ⊂ X 2 .
From the above procedure, the elements of Y 1 are some elements of X + e ∪ {0, 1, . . ., e − 1} and Y 2 is given by {0, 1, . . ., e − 1} together with all the elements of X + e and other elements of X + e ∪ {0, 1, . . ., e − 1} translated by e.We thus have In the following, it will be convenient to write the image of an element (X 1 , X 2 ) ∈ P m1 × P m2 under a map Ψ (e,s) as . This is what we are going to do in the following example.Assume which yet satisfies the inclusion property.Note that in the assumptions of the conjecture, we really need k to be odd.In the case when k is even, the assertion is wrong as we can see in the above example.
Remark 4.6.This conjecture has been checked for all couples (X, X) = (X 0,m (λ), X 0,m (λ)) with λ an arbitrary partition of rank n with n ≤ 40 (and e arbitrary).A proof for the conjecture has already been obtained by M.Fayers when e = 2 [8].

Conjectural consequences on crystal isomorphisms
We first establish some elementary results concerning e-regular partitions and then explain our conjectural algorithm.
We set λ := f i1+Ze,e . . .f in−1+Ze,e .∅.By induction, we have that ( f where for all i = 1, . . ., m, Z i ∈ {A, R} correspond to a node (a i , b i ).Then we have: where T 2i−1 = Z i correspond to the node (a i , b i , 2) for i = 1, . . ., m and T 2i = Z i for i = 1, . . ., m corresponds to the node (a i , b i , 1).It follows that if (a k , b k ) is a good addable i n + eZ-node for λ then (a i , b i , 2) is a good addable i n + eZ-node for ( λ, λ) and (a i , b i , 1) is a good addable i n + eZ-node for ( λ, λ).We conclude that ( f The following result comes from [14, Lemma 3.2.12](see also [12] for an similar result, but for a different realization of the Fock space).Proposition 5.2.Let λ be an e-regular partition and let (i 1 , . . ., i n ) ∈ Z n be such that: f i1+Ze,e . . .f in+Ze,e .∅= λ.
We now use the above proposition together with the following result which rephrases Conjecture 4.5 in terms of crystal isomorphisms.