Two maps on affine type A crystals and Hecke algebras

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Two maps on affine type A crystals and Hecke algebras

N Jacon

To cite this version:

N Jacon. Two maps on affine type A crystals and Hecke algebras. The Electronic Journal of Combi-

natorics, Open Journal Systems, 2021, 5. �hal-03135458v2�

Two maps on affine type A crystals and Hecke algebras

N.Jacon

Abstract

We use the crystal isomorphisms of the Fock space to describe two maps on partitions and multipartitions which naturally appear in the crystal basis theory for quantum groups in affine typeAand in the representation theory of Hecke algebras of typeG(l, l, n).

1 Introduction

Since the works of Lascoux, Leclerc, Thibon and Ariki in the90’s, it is known that the representation theory of Hecke algebras of complex reflection groups is closely related to the crystal basis theory for quantum groups. In particular, the crystal basis for Fock spaces in affine typeAleads to a classification of the simple modules of the Hecke algebra of typeG(l,1, n)(also known as Ariki-Koike algebra) in the modular case by certain combinatorial objects called Uglovl-partitions. This includes the cases of Iwahori-Hecke algebras of typeAandB (see [1, 2]).

A lot of informations on the representation theory of Hecke algebras of typeDnor, more generally, of type G(l, l, n)can be obtained from the G(l,1, n)case. In fact, these latter algebras can be seen as subalgebras of Hecke algebras of type G(l,1, n) and it is possible to produce all the simple modules by studying the restriction of the simple modules of the Hecke algebras of typeG(l,1, n), using Clifford theory. This problem has been studied in various papers using different approaches (see [4, 3] and the references theirin). The one developed in [4] and [5] in particular involves the existence of two maps which are defined using the crystal graph of an irreducible highest weight module in affine type A:

• The first map associates to each Uglov l-partition labelling a vertex of the crystal, another Uglov l-partition.

• The second one associates to eache-regular partition, labelling a vertex of the crystal of the fundamental representation a certain Uglovl-partition.

The existence of such maps is non trivial and based on the structure of the associated crystals and their descriptions are only recursive on the size of the partitions/multipartitions involved. The interests of these maps is that they allow to describe the restriction of the simple modules of the Hecke algebras of type G(l,1, n)to the Hecke algebras of typeG(l, l, n).

The aim of this note is to recover, generalize and explicit these results. The main tools of the proof are the crystal isomorphisms defined and described in [7]. Using them, the proofs become purely combinatorial and quite elementary. They also permits to explain how the approaches developed in [4] and in [3] are related.

2 Crystals

In this part, we quickly recall some basic combinatorial notions, then we focus on the definition and on important properties of the crystals for Fock spaces. In all this section, we setl∈Z_>0 ande∈Z_>1.

2010Mathematics Subject Classification: 20C08,05E10,17B37

2.1 Generalities on Fock spaces and crystals

2.1.1. A partition is by definition a nonincreasing sequenceλ= (λ1, . . . , λm) of nonnegative integers. If P

1≤i≤mλi =n, we say thatλis a partition of n. For j = 1, . . . , l, let λ^j be a partition ofnj ∈Z_≥0 then we say that the l-tuple (λ¹, . . . , λ^l) is an l-partition of nifP

1≤j≤lnj =n. We denote by Π^l(n)the set of l-partitions of rankn. The empty l-partition is by definition the unique partition of0 and it is denoted by

∅:= (∅, . . . ,∅). Whenl= 1, the1-partitions are identified with the partitions in an obvious way.

2.1.2. Lets= (s1, . . . , sl)∈Z^l(we say thatsis amulticharge). Letqbe an indeterminate. TheQ(q)-vector space generated by all thel-partitions:

Fq := M

n∈Z≥0

M

λ∈Π^l(n)

Q(q)λ

is called the Fock space. Let Uq(slc_e) be the quantum group of affine type A⁽¹⁾_e−1. This is an associative Q(q)-algebra with generators ei, fi, ti, t⁻¹_i (for i = 0, . . . , e−1) and∂ and relations given in [2, §6.1]. We denote by U_q^′(csl_e) the subalgebra generated by ei, fi, ti, t⁻¹_i (fori = 0, ..., e−1). For i = 0, . . . , e−1, we denote byΛi the fundamental weights and the simple roots are given by:

αi=−Λi−1+ 2Λi−Λi+1,

where the indices are taken moduloe. There is an action ofU_q^′(slb_e)on the Fock space. This action depends on the choice ofs and the module generated by the empty multipartition is an irreducible highest weight module with weightΛs1+. . .+ Λsl. We do not need the precise definition of this action and we refer to [2, Ch. 6] for details.

2.1.3. To eachλ∈Π^l(n)is associated itsYoung tableau:

[λ] ={(a, b, c)|a≥1, c∈ {0, . . . , l−1},1≤b≤λ^c_a}.

We define thecontentof a nodeγ= (a, b, c)∈[λ]as follows:

cont(γ) =b−a+sc,

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 λwhenres(γ)≡i+eZ(we will sometimes simply called it ani-node). Finally, We say that γ is removable whenγ = (a, b, c)∈[λ] and[λ]\{γ} is the Young diagram of an l-partition. Similarly, γ is addablewhenγ= (a, b, c)∈/ [λ]and[λ]∪ {γ}is the Young diagram of anl-partition.

Letγ,γ^′ be two removable or addablei-nodes ofλ. We denote γ≺sγ^′ ⇐⇒^def

either b−a+sc < b^′−a^′+sc^′,

or b−a+sc =b^′−a^′+sc^′ andc > c^′.

2.1.4. For λan l-partition andi∈ Z/eZ, we can consider its set of addable and removablei-nodes. Let w^(e,_i ^s)(λ)be the word obtained first by writing the addable and removablei-nodes of λin increasing order with respect to≺^s, next by encoding each addable i-node by the letterAand each removablei-node by the letterR. Writewe^(e,s_i (λ) =A^pR^q for the word derived fromw^(e,s)_i (λ)by deleting as many of the factorsRA as possible. In the following, we will sometimes writewei(λ)andwi(λ)instead ofwe_i^(e,s)(λ)and w_i^(e,s)(λ)if there is no possible confusion.

Ifp >0,letγ be the rightmost addablei-node inwei. The node γis called the good addable i-node. If r >0, the leftmost removablei-node inwei is called the good removablei-node.

Example 2.1.5. Forl= 2,s= (0,1) ande= 3. Let us consider the2-partition λ:= ((4),(2,1))of7. We write its Young tableau and the residues of the nodes in the associated boxes:

0 1 2 0 , 1 2 0

We have we0(λ) = RAR and thus(1,4,1) is a good removable 0-node forλ. We have we2(λ) =AAR and thus(2,1,1)is a good addable2-node forλand(1,2,2) is a good removable2-node for it.

2.1.6. We denote by Ge,s the crystal of the Fock space computed using the Kashiwara operatorsee^e,_i^s and fe_i^e,s. Again, we refer to [2] for details. This is the graph with

• vertices : the l-partitionsλ⊢lnwithn∈Z_≥0.

• arrows: λ→ⁱ µif and only iffe_i^e,sλ=µ(or equivalentlyee^e,s_i µ=λ). This means thatµis obtained by adding toλa good addablei-node, or equivalently,λis obtained fromµby removing a good removable i-node.

Example 2.1.7. For l= 3, e= 2and s= (0,0,1) the graph below is the subgraph of Ge,s containing the empty3-partition and with the3-partitions with rank less or equal than4.

(∅,∅,∅)

(1,∅,∅) (∅,∅,1)

(2,∅,∅) (1,1,∅) (∅,∅,2)

(3,∅,∅) (2,∅,1) (2,1,∅) (1,∅,2) (∅,∅,3)

(4,∅,∅) (3,1,∅) (2,∅,2) (2.1,∅,1) (2,2,∅) (1,1,2) (1,∅,3) (∅,∅,4)

✟✟

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0 1

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1 0 0 1

2.1.8. Let Φe,s(n) to be the set ofl-partitions of rank n in the connected component of Ge,s containing the emptyl-partition. This is called the set ofUglovl-partitions. Hence, by definition, an Uglovl-partition is defined by adding successively good nodes to the empty l-partition (with arbitrary residues). It strongly depends on the choice of s. Assume that s is such that 0 < sj−si < e for all 0 < i < j ≤ l then the set Φe,s(n)is known as the set of FLOTW l-partitions and it has a nice non recursive description (see [2,

§6.3.2]). We haveλ= (λ¹, . . . , λ^l)∈Φs,e(n)if and only if:

1. For allj= 1, . . . , l−1 andi∈Z_>0, we have:

λ^j_i ≥λ^j+1_i+sj+1−sj. 2. For alli∈Z>0, we have:

λ^l_i≥λ¹_i+e+s1−sl. 3. For allk∈Z_>0, the set

{λ^j_i −i+sj+eZ|i∈Z_>0, λ^j_i =k, j= 1, . . . , l}, is a proper subset of Z/eZ.

In general, we don’t have such a nice description of the set of Uglovl-partitions.

Example 2.1.9. In the case wherel= 1, the setΦe,(0)(n)is the set ofe-regular partitions ofn, that is, the set of partitions of ranknsuch that no non zero parts are repeatedeor more times.

Example 2.1.10. Following Example 2.1.7, we have

Φ2,(0,0,1)(4) ={(4,∅,∅),(3,1,∅),(2,∅,2),(2.1,∅,1),(2,2,∅),(1,1,2),(1,∅,3),(∅,∅,4)}

2.2 Crystal isomorphisms

In this part, we recall the definition of certain crystal isomorphisms studied in [7]. These maps will be intensively used in the next sections.

2.2.1. LetSb_lbe the (extended) affine symmetric group. This is defined as follows. We denote byPl:=Z^l theZ-module with standard basis{yi|i= 1, . . . , l}. Fori= 1, . . . , l−1, we denote byσi the transposition (i, i+ 1)of S_l. ThenSb_l can be seen as the semi-direct product Pl⋊S_l where the relations are given by σiyj =yjσi forj6=i, i+ 1andσiyiσi=yi+1 fori= 1, . . . , l−1 andj= 1, . . . , l. This group acts onZ^l by setting for anys= (s1, . . . , sl)∈Z^l:

σc.s = (s1, . . . , sc−1, sc+1, sc, sc+2, . . . , sl) forc= 1, . . . , l−1 and yi.s = (s1, s2, . . . , si+e, . . . , sl) fori= 1, . . . , l.

A fundamental domain for this action is given by A^e_l :=

(s1, . . . , sl)∈Z^l |0≤s1≤. . .≤sl< e .

Note that we thus have a description of Φ^s,e(n)when sis in this domain by §2.1.8. Let τ :=ylσl−1. . . σ1

then we see thatSb_lis generated byτ andσi fori= 1, . . . , l−1. In addition, we have:

τ.s= (s2, . . . , sl, s1+e).

2.2.2. Assume thats∈Z^l and s^′ ∈Z^l are in the same orbit moduloSb_l. As explained in [2, §6.2.17], the crystal graph theory allows to construct a combinatorial bijection between the two sets of Uglovl-partitions Φ^s,e(n)andΦ^s′,e(n). Letλ∈Φ^s,e(n)then there exists a sequence(i1, . . . , in)∈(Z/eZ)ⁿ such that:

fe_i^e,₁^s. . .fe_i^e,_n^s∅=λ Then there existsµ∈Φs^′,e(n)such that

fe_i^e,₁^s′. . .fe_i^e,_n^s′∅=µ

We setΨ^es→s^′(λ) :=µ(it does not depends on the choice of the sequence(i1, . . . , in)). This defines a bijection Ψ^es→s^′ : Φs,e(n)→Φs^′,e(n).

A combinatorial description of this map is given in [7]. Let us quickly explain how. There exists w∈ Sb_l such thats^′ =w.s. Then, wis a product of τ and σi’s (i= 1, . . . , l−1). Thus Ψ^es→s′ is a composition of maps of the formΨ^e_v→τ.v andΨ^e_v→σi.v withv∈Z^l explicit by induction.

• For allλ= (λ¹, . . . , λ^l)∈Φe,v(n), we have

Ψ^ev→τ.v(λ) = (λ², . . . , λ^l, λ¹).

• For allλ= (λ¹, . . . , λ^l)∈Φe,v(n), we have

Ψ^ev→σi.v(λ) = (λ¹, . . . , λⁱ⁻¹,eλⁱ⁺¹,λeⁱ, λⁱ⁺², . . . , λ^l),

where (eλⁱ⁺¹,eλⁱ)is obtained from(λⁱ, λⁱ⁺¹)via a purely simple combinatorial process described in [7, th. 5.4.2] (in terms of Lusztig symbols) or in [6, §5.3] (in terms of Young tableaux).

2.2.3. Assume thats = (s1, . . . , sl) satisfies si−sj ≥ n−1−e for alli < j then we say that s is very dominant. If both s and s^′ are very dominant (comparing ton) and in the same orbit then Ψ^es→s′ is the identity and the setΦ^s,e(n)is known as the set of Kleshchevl-partitions (see [2, Ex. 6.2.16]).

If s∈Z^l, one way to compute the set Φs,e(n)of Uglovl-partitions consists in findingw∈Sb_l such that s^′=w.s∈ A^e_l. We can then use the description of the setΦs^′,e(n)in §2.1.8 and then apply the isomorphism Ψ^es^′→s.

Example 2.2.4. For l= 3, e= 2and s= (2,0,3) the graph below is the subgraph of Ge,s containing the empty3-partition.

(∅,∅,∅)

(1,∅,∅) (∅,∅,1)

(2,∅,∅) (1,1,∅) (∅,∅,2)

(3,∅,∅) (2,∅,1) (2,1,∅) (1,∅,2) (∅,∅,3)

(4,∅,∅) (3,1,∅) (3,∅,1) (2.1,∅,1) (2,1,1) (1,∅,2.1)(1,∅,3) (∅,∅,4)

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s ❅

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0 1

0 0

1

0 1

1

0 1

0 0

1 1

1 0 0 1

Looking at example 2.1.7, we see thatΨ²(0,0,1)→(2,0,3)is the identity for the Uglov 3-partitions of ranks

≤3, and we haveΨ²(0,0,1)→(2,0,3)(2,∅,2) = (3,∅,1),Ψ²(0,0,1)→(2,0,3)(2,2,∅) = (2,1,1),Ψ²(0,0,1)→(2,0,3)(1,1,2) = (1,∅,2.1), and Ψ³(0,0,1)→(2,0,3) is the identity for the others Uglov 3-partitions of ranks 4. This formulae can be also obtained without looking at the crystal. Indeed, we have (2,0,3) = (σ1τ)²(0,0,1) and thus Ψ²(0,0,1)→(2,0,3) = Ψ²(0,0,1)→τ(0,0,1)◦Ψ²_{τ(0,0,1)→σ1τ(0,0,1)}◦Ψ³_σ1τ(0,0,1)→τ σ1τ(0,0,1)◦Ψ²_{τ σ1τ(0,0,1)◦σ1τ σ1τ(0,0,1)} and one can use the combinatorial description of the isomorphisms.

3 Two maps on crystals

The above results will allow us to recover and give precisions on two results on crystals in affine typeAthanks to quite elementary proofs. These two results concern a particular choice of multicharge which naturally appears in the context of Hecke algebras, as we will see in the next part. In this section, we thus assume thatl divideseand that s∈Z^lis in the orbit of the multicharge(0, e/l, . . . ,(l−1)e/l)moduloSb_l.

3.1 Hu’s map

The first result that we want ro recover is in fact a direct generalization of a result by Hu [4, Theorem 3.6].

We propose here an elementary proof of this result using our crystal isomorphisms and we will also give a general method to explicitly compute thel-partitions that this results allows to define.

Proposition 3.1.1 (Hu). Assume that λ ∈ Φe,s(n). Then there exists a sequence (i1, . . . , in) ∈ (Z/eZ)ⁿ such that:

fe_i^e,s₁ . . .fe_i^e,s_n ∅=λ.

Then for any such sequences, there existsµ∈Φe,s(n)such that fe_i^e,s

1+e/l. . .fe_i^e,_n^s_+e/l∅=µ

To prove this proposition, we will proceed in two steps, we first prove the proposition for a particular choice of multicharge which is inA^e_l, and then make use of the crystal isomorphisms we have already defined.

Lemma 3.1.2. Let s = (0, e/l, . . . ,(l −1)e/l) Assume that λ ∈ Φe,s(n) and that we have a sequence (i1, . . . , in)∈(Z/eZ)ⁿ such that:

fe_i^e,₁^s. . .fe_i^e,_n^s∅=λ.

Then we have thatµ:= (λ^l, λ¹, . . . , λ^l−1)∈Φe,s(n)and we have fe_i^e,s

1+e/l. . .fe_i^e,s

n+e/l∅=µ.

Proof. We argue by induction onn. The lemma is clear for the emptyl-partition. Assume now thatn >0.

Letλ∈Φe,s(n)and assume that we have a sequence(i1, . . . , in)∈(Z/eZ)ⁿ such that fe_i^e,₁^s. . .fe_i^e,_n^s∅=λ.

Setλ^′ :=fe_i^e,₂^s. . .fe_i^e,_n^s∅ thenλ^′ = (λ^′1, . . . , λ^′l)is inΦe,s(n−1) and by induction,µ^′ := (λ^′l, λ^′1, . . . , λ^′l−1) is inΦe,s(n−1)and we haveµ^′=fe_i^e,s

2+e/l. . .fe_i^e,s

n+e/l∅.

Now, by hypothesis, we have that γ = [λ]/[λ^′] is a good addable in-node for λ^′ (and (e,s)). Set (a, b, c) := γ. Then γ^′ := [µ]/[µ^′] is an addable node for µ^′. We have γ^′ = (a, b, c+ 1) (where the 3rd component is understood modulo l) and, by our choice of multicharge, it is a in+e/l-node. We want to show that this is a good addable node forµ^′.

Assume thatγ1 = (a1, b1, c1)is ain-addable or removable node forλ^′. Thenγ₁^′ := (a1, b1, c1+ 1) is an addable or removable node forµ^′ (and it is removable, resp. addable, if and only ifγ1 is). We have that

cont(γ₁^′) =

cont(γ1) +e/l ifc16=l, cont(γ1) +e/l−e ifc1=l.

Thus, we haveγ1≺e,sγ if and only ifγ₁^′ ≺e,s γ^′.

Reciprocally, ifγ^′₁= (a1, b1, c1)is ain+e/l-addable or removable node forµ^′. Thenγ1:= (a1, b1, c1−1) is an addable or removable forin-node forλ^′ (and it is removable, resp. addable, if and only if γ₁^′ is). We have thatγ1≺e,sγ if and only ifγ₁^′ ≺e,sγ^′. Thus we havew_i^(e,₁^s)(λ^′) =w^(e,_i ^s)

1+e/l(µ^′).

This discussion implies thatγ^′ is good addablei1+e/l-node forµ^′ and thus that fe_i^e,s

1+e/l. . .fe_i^e,s

n+e/l∅=µ, as required.

We can now give a proof of Proposition 3.1.1. Assume thats∈Z^lis in the orbit of(0, e/l, . . . ,(l−1)e/l) moduloSb_l, thatλ∈Φe,s(n)and that we have a sequence(i1, . . . , in)∈(Z/eZ)ⁿ such that:

fe_i^e,₁^s. . .fe_i^e,_n^s∅=λ.

Sets^′:= (0, e/l, . . . ,(l−1)e/l), then by definition, we have fe_i^e,₁^s′. . .fe_i^e,_n^s′∅= Ψ^es→s′(λ).

We can thus use Lemma 3.1.2 to deduce that there existsµ^′ ∈Φe,s^′(n)such that fe_i^e,s′

1+e/l. . .fe_i^e,s′

n+e/l∅=µ^′, and using again our crystal isomorphism, we get that:

fe_i^e,s

1+e/l. . .fe_i^e,s

n+e/l∅= Ψ^es′→s(µ^′),

so the result follows. Note in addition that the l-partition µ^′ may be explicitly described thanks to the explicit description of the crystal isomorphism without the computation of the crystal itself.

Example 3.1.3. Takel= 2and assume thate= 4. We sets= (0,10)which is in the orbit ofs^′= (0,2)∈ A⁴₂. We take λ = (1.1,5.1) ∈ Φ⁴_(0,10)(8). Note that the multicharge is very dominant for 2-partitions of rank 8 so (1.1,5.1) is a Kleshchev bipartition. If we want to find the bipartition µ of Proposition 3.1.1, we first need to find the bipartition λ^′ ∈ Φ⁴_(0,2)(8) such that Ψ⁴s→s^′(λ) = λ^′. Using our description of the isomorphisms, we getλ^′ = (2.1,5). We then haveµ= Ψ⁴s^′→s(5,2.1) = (4,3.1).

3.2 The map ι

We keep the hypothesis thatsis in the orbit of the multicharge(0, e/l, . . . ,(l−1)e/l)moduloSb_l. We now consider another map defined using the crystal. As explained in [5], its existence follows from [8] in the case wherel= 2. Here we will give a general version and we will again give an easy proof using our crystal isomorphism. We will also make things more explicit.

Proposition 3.2.1. Let k be an integer dividing l. Set v := (0, e/l, . . . , e(k−1)/l)∈ Z^k. There exists a unique map

ι^s_k: Φke/l,v(n)→Φe,s(ln/k)

well-defined as follows. For all λ∈Φke/l,v(n), there exists (i1, . . . , in)∈Zⁿ such that fe_i^ke/l,₁ ^v. . .fe_i^ke/l,_n ^v∅=λ,

(the indices are understood modulo ke/l.) Then for all such sequences, we have:

fe_i^e,₁^sfe_i^e,s

1+ke/l. . .fe_i^e,s

1+e−ke/l

| {z }

l/k

. . .fe_i^e,_n^sfe_i^e,s

n+ke/l. . .fe_i^e,s

n+e−ke/l

| {z }

l/k

∅=ι^s_k(λ)

(the indices are understood modulo e.)

In the same spirit as the last result, our strategy consists in proving the result whensis in the fundamental domainA^e_l.

Lemma 3.2.2. Letkbe an integer dividingl. Sets= (0, e/l, . . . ,(l−1)e/l)andv:= (0, e/l, . . . , e(k−1)/l)∈ Z^k. There exists a unique map

ι^s_k: Φke/l,v(n)→Φe,s(ln/k)

well-defined as follows. For all λ∈Φke/l,v(n), there exists (i1, . . . , in)∈Zⁿ such that fe_i^ke/l,₁ ^v. . .fe_i^ke/l,_n ^v∅=λ

Then for all such sequences, we have:

fe_i^e,s₁ fe_i^e,s

1+ke/l. . .fe_i^e,s

1+e−ke/l

| {z }

l/k

. . .fe_i^e,s_n fe_i^e,s

n+ke/l. . .fe_i^e,s

n+e−ke/l

| {z }

l/k

∅= (λ¹, . . . , λ^k

| {z }

l/k

, λ¹, . . . , λ^k

| {z }

l/k

, . . . , λ¹, . . . , λ^k

| {z }

l/k

).

Proof. We again argue by induction onn∈Z≥0. The lemma is clear for the emptyl-partition. Assume now thatn >0. Letλ∈Φke/l,v(n)and assume that we have a sequence(i1, . . . , in)∈Zⁿ such that

fe_i^ke/l,₁ ^v. . .fe_i^ke/l,_n ^v.∅=λ Setλ^′:=fe_i^ke/l,₂ ^v. . .fe_i^ke/l,_n ^v.∅. By induction, we have:

fe_i^e,₁^sfe_i^e,s

1+ke/l. . .fe_i^e,s

1+e−ke/l

| {z }

l/k

. . .fe_i^e,_n^sfe_i^e,s

n+ke/l. . .fe_i^e,s

n+e−ke/l

| {z }

l/k

.∅= (λ^′1, . . . , λ^′k, λ^′1, . . . , λ^′k, . . . , λ^′1, . . . , λ^′k)

Denote

λ[0] := (λ^′1, . . . , λ^′k, λ^′1, . . . , λ^′k, . . . , λ^′1, . . . , λ^′k)

Set γ = [λ]/[λ^′] and let (a, b, c) := γ. This is a good addable i1+ (ke/l)Z-node for λ^′. We have by definition b−a+ (c−1)e/l ≡i1+ (ke/l)Z. So there exists j ∈ {0,1, . . . , l/k−1} such thatb−a+ (c− 1)e/l =i1−j(ke/l) +eZ. We thus haveb−a+ (c−1 + (j−1)k)e/l=i1−(ke/l) +eZ. Let use denote

γj:= (a, b, c+ (j−1)k)(where the3rd component is understood modulol). We have that the residue ofγj

isi1−(ke/l) +eZforλ[0]and the multicharges.

Now assume that η = (a^′, b^′, c^′) is an addable or a removable i1−(ke/l) +eZ-node for λ, different from γ. As above, there exists j^′ ∈ {1, . . . , l} such that ηj^′ := (a^′, b^′, j^′−1) is an addable or removable i1−ke/l+eZ-node forλ[0](and removable if and only ifη is).

In addition, by our definition of ≺., we have η ≺v γ if and only is ηj^′ ≺s γj. Reciprocally, all the i1−ke/l+eZ-nodes are obtained in this way.

This discussion implies thatγj is a good addable i1−ke/l-node for λ[0] because γ is a good one for λ. We denote by λ[1] the l-partition obtained from this one by adding γj to λ[0]. We thus have λ[1] = fe_i^e,_1+e−ke/l^s λ^′6= 0.

Let us now considerγ2:= (a, b,(c−1 + (j−2)k))(where the3rd component is understood moduloe).

It is an addablei1+ (l−2)e/l+eZ-node for λ[1]and by exactly the same argument as above, we see that this is a good addable node. Letλ[2] be thel-partition obtained by adding this node to λ[1]. We obtain fe_i^e,s

1+e−2ke/lfe_i^e,s

1+e−ke/lλ^′ =λ[2]. Continuing in this way we deduce fe_i^e,₁^s. . .fe_i^e,s

1+(l−2)e/lfe_i^e,s

1+(l−1)e/lλ^′ = λ, as required.

One can now give a general proof of the proposition. Assume thats∈Z^lis in the orbit of(0, e/l, . . . ,(l− 1)e/l), thatλ∈Φe,s(n)and that we have a sequence(i1, . . . , in)∈Zⁿ such that

fe_i^ke/l,₁ ^v. . .fe_i^ke/l,_n ^v∅=λ.

Then, by the above lemma, if we sets^′= (0, e/l, . . . ,(l−1)e/l), we have fe_i^e,₁^s′fe_i^e,s′

1+ke/l. . .fe_i^e,s′

1+e−ke/l

| {z }

l/k

. . .fe_i^e,_n^s′fe_i^e,s′

n+ke/l. . .fe_i^e,s′

n+e−ke/l

| {z }

l/k

∅= (λ¹, . . . , λ^k, λ¹, . . . , λ^k, . . . , λ¹, . . . , λ^k)

and thus one can conclude that fe_i^e,₁^sfe_i^e,s

1+ke/l. . .fe_i^e,s

1+e−ke/l

| {z }

l/k

. . .fe_i^e,_n^sfe_i^e,_n^s_+ke/l. . .fe_i^e,_n^s_+e−ke/l

| {z }

l/k

∅= Ψ^es′→s(λ¹, . . . , λ^k, λ¹, . . . , λ^k, . . . , λ¹, . . . , λ^k).

which proves the theorem and also gives en explict way to compute thel-partition involved.

Example 3.2.3. We takel = 2 ande= 4. We set k= 1. Let λ= (4.3.1), this is ae/2-regular partition and thus in Φ2,(0)(8). By Lemma 3.2.2, we have ι^(0,2)₁ (4.3.1) = (4.3.1,4.3.1). Then we obtain for example ι^(0,22)₁ (4.3.1) = Ψ⁴(0,2)→(0,22)(4.3.1,4.3.1) = (3.2.1,4.3.2.1).

4 Hecke algebras of type G(p, p, n)

In this part, we apply the results above to recover and generalize some of the results of [4] and [5] and give precisions on them. We will freely use the results in [3].

4.1 Definition

Letη ∈C^×. Assume that n >2. Lets= (s1, . . . , sl)∈Z^l and letη ∈C^×. The cyclotomic Hecke algebra Hn(s)of typeG(l,1, n)(also known as Ariki-Koike algebra) is theC-algebra with a presentation by:

• generators : T0,T1, ...,Tn−1,

• relations :

(T0−η^s1). . .(T0−η^sl) = 0

(Ti−η)(Ti+ 1) = 0 (1≤i≤n−1) (T0T1)² = (T1T0)²

TiTi+1Ti = Ti+1TiTi+1 (1≤i < n) TiTj = TjTi(j≥i+ 2).

The cyclotomic Hecke algebraH^′_n of typeG(l, l, n)is theC-algebra with a presentation by :

• generators : T0,T1, ...,Tn−1,

• relations :

(Ti−η)(Ti+ 1) = 0 for0≤i≤n−1, TiTi+1Ti=Ti+1TiTi+1 for1≤i≤n−2, T0T2T0=T2T0T2,

(T1T0T2)²= (T2T1T0)²,

T0Tj=TjT0 forj >2,

TiTj=TjTi fori >0etj > i+ 1,

T0T1T0T1...

| {z }

lterms

=T1T0T1T0...

| {z }

lterms

.

From now, we assume thatη is a primitive root of ordere >1. Lets∈Z^l be in the orbit of(0, e/l, . . . ,(l− 1)e/l)then the subalgebra ofHn:=Hn(s)generated by{T0:=Tf0

−1T1Tf0, T1, ..., Tn−1}is isomorphic toH^′_n. MoreoverHn isZ/lZ-graded with respect toH^′_n with gradation

Hn = Ml−1

j=0

fT0 jH^′_n.

As a consequence, one may use Clifford Theory to obtain results for the representation theory ofH^′_n from the one ofHn. To do this, we first need to recall some known results on the representation theory ofHn

4.2 Simple H

-modules

4.2.1. The classification of the simple Hn-modules that we need comes from the theory of basic sets. A complete review of this can be found in [2] but we quickly recall what we need here. One can define a certain set of finite dimensionalHn-modules which are parametrized by the set ofl-partitions, they are called Specht modules

{S^λ |λ∈Π^l(n)}.

These modules are non simple (nor semisimple) in general but we have associated composition series. Let us denote by[S^λ:M]the multiplicity ofM ∈Irr(Hn)in a composition series forS^λ(this is well-defined by the Jordan-Hölder theorem). Then the matrix defined by:

D:= ([S^λ:M])_λ∈Π^l(n),M∈Irr(Hn)

controls a part of the representation theory ofHn.This is called thedecomposition matrix.

4.2.2. We here follow [2, Ch.5, Ch.6]. Then one can define a pre-order≪s on the set ofl-partitions which depends on the choice ofs. We don’t give the definition of this pre-order here, all we need to know is the following theorem (see [2, §6.7]).

Theorem 4.2.3. Under the above hypotheses, for all M ∈Irr(Hn), 1. there existsλM ∈Φs,e(n)such that [S^λM :M] = 1,

2. for all µ⊢ln, if[S^µ:M]6= 0 thenµ≪mλM.

The mapM 7→λM is injective. As a consequence, if for allM ∈Irr(Hn)we denote D^λs,e^M :=M, we have:

Irr(Hn) ={D^µs,e |µ∈Φ^s,e(n)}.

It is thus important to note that this theorem does not give one way to label the simple modules of the algebraHn but in fact several ones: one for each choice of an element in the orbit of s moduloSb_l. It is now natural to ask how all these parametrization are connected. It turns out that the crystal isomorphisms make the links between them.

Proposition 4.2.4([6]). Lets∈Z^lands^′∈Z^lbe two multicharge in the same orbit then for allλ∈Φs,e(n), we have D^λs,e=Ds^Ψ′es,e^→s′(λ).

4.3 Restriction of simple H

-modules

We here sets= (0, e/l, . . . ,(l−1)e/l). There is a natural action of the cyclic groupZ/lZonΠ^l(n)generated by the following map:

(λ¹, λ², ..., λ^l)7→(λ^l, λ¹, ..., λ^l−2).

Forλ∈Π^l(n)we denote byλe the associated equivalence class. Let

r:=r(λ) = l

Cardinality of λe. The following theorem is proved in [3].

Theorem 4.3.1. Letλ∈Φs,e(n) then we have thatRes(D^λs,e)is a direct sum ofr(λ) simpleH^′_n-modules.

It is also possible to show that ifλandµare in the same equivalence class thenRes(Ds^λ,e)andRes(D^µs,e) are isomorphic. In addition, the simple modules appearing in the restriction of the Ds^λ,e’s determined the equivalence class ofλ. As a consequence, one can obtain a classification of the simpleHn-modules knowing the numbersr(λ). Applying Proposition 4.2.4 yields:

Proposition 4.3.2. Let s^′ ∈Z^l in the same class ass= (0, e/l, . . . ,(l−1)e/l). Let λ∈Φ^s′,e(n) then we have thatRes(D^λs^′,e)is a direct sum of r(Ψ^es→,s(λ))simpleH^′_n-modules.

The above proposition gives thus an explicit way to find the number of simple modules in the restriction of the simple Hn-modules without refering to the notion of crystal and for all the known parametrization of the simples. This thus includes the usual parametrization by set the Kleshchev l-partitions using our isomorphisms.

Proposition 4.3.3. Let s^′ ∈ Z^l in the same class as s = (0, e/l, . . . ,(l−1)e/l). Let λ ∈ Φs^′,e(n) then Res(D^λs′,e)splits into a sum of xsimple modules if and only ifλ∈Im(ι^s_l/x^′ )andλ∈/Im(ι^s_l/s^′ )for s > x.

Proof. Take first s^′ =s. Then by Lemma 3.2.2, we have that λ ∈ Im(ι^s_x) if and only if λ is of the form (λ¹, . . . , λ^k, λ¹, . . . , λ^k, . . . , λ¹, . . . , λ^k)wherek=l/xand the result follows from the last proposition.

Now, if s^′ ∈ Z^l in the same class as s = (0, e/l, . . . ,(l −1)e/l), then for all λ ∈ Φs,e(n), we have Ds^λ,e=D^Ψ

e s→s′(λ)

s^′,e and one can conclude noticing that

λ∈Im(ι^s_l/x) ⇐⇒ Ψ^es^′→s(λ)∈Im(ι^s_l/x^′ ).

by the definition of the maps and the properties of crystal isomorphisms.

Example 4.3.4. Takel = 4 ande = 4. Lets= (0,1,2,3). Then we haveλ:= (3.1,2,3.1,2)∈Φe,s(16).

Then we have that r(λ) = 2, this implies thatRes(D^λs,e) splits in two simple H^′_n-modules. Note that we haveλ=ι^s₂(3.1,2). Sets= (0,13,26,39), this multicharge is very dominant and in the same orbit ass. One can computeΨ^es→s^′(λ)and we obtainλ^′= (2.1,1,3.2,2.1). So we have thatRes(D^λs^′′,e)splits in two simple H^′_n-modules.

4.4 The case l = 2

We assume in this part that l= 2and thate is odd. Then one can apply the results above. In particular, for allsin the orbit of(0, e/2)moduloSb₂andλ∈Φe,s(n), theH^′_n-moduleRes(D^λs,e)splits into one or two simple modules.

The aim is to study the set of Uglov bipartitionsλ∈ Φe,s(n)such that Res(Ds^λ,e)splits into a sum of two simple modules. Such bipartitions will be calleddivided bipartitions for the multicharges. This notion strongly depends ons. In the case where s= (0, e/2), by the results above, these bipartitions correspond exactly to the bipartitions of the form(λ, λ)in Φe,s(n). This is exactly the set of bipartitions(λ, λ) where λis ane/2-regular partition ofn/2by §2.1.8.

Proposition 4.4.1. Let N ∈ Z_≥0. We have that (λ¹, λ²)is a divided bipartition for s= (0, e/2 +N e) if and only if we have

Ψ^e(0,e/2+N e)→(N e,e/2)(λ¹, λ²) = (λ², λ¹)

Proof. Lets= (0, e/2 +N e). Assume that(λ¹, λ²) is a divided bipartition. Let(i1, . . . , in)∈(Z/eZ)ⁿ be such that

fe_i^e,s₁ . . .fe_i^e,s_n ∅=λ Then we also have

fe_i^e,₁^s_+e/2. . .fe_i^e,_n^s_+e/2∅=λ Now we have thats^′:=τ.s= (e/2 +N e, e)and by §2.2.2, we obtain:

fe_i^e,₁^s′. . .fe_i^e,_n^s′∅= (λ², λ¹) Then we also have

fe_i^e,s′

1+e/2. . .fe_i^e,s′

n+e/2∅= (λ², λ¹)

But now note thats= (N e, e/2) =s^′−(e/2, e/2)so it is clear that we obtain : fee,(N e,e/2)

i1−e/2 . . .fee,(N e,e/2)

in−e/2 ∅= (λ², λ¹)

and this implies thatΨ^e(0,e/2+N e)→(N e,e/2)(λ¹, λ²) = (λ², λ¹). Reciprocally, assume that Ψ^e(0,e/2+N e)→(N e,e/2)(λ¹, λ²) = (λ², λ¹).

Assume that(i1, . . . , in)∈(Z/eZ)ⁿ is such that

fe_i^e,s₁ . . .fe_i^e,s_n ∅= (λ¹, λ²).

Then we have:

fee,(N e,e/2)

i1 . . .fee,(N e,e/2)

in ∅= (λ², λ¹).

We haveτ.(N e, e/2) = (e/2, N e+e), and thus fee,(e/2,N e+e)

i1 . . .fee,(e/2,N e+e)

in ∅= (λ¹, λ²).

As(0, e/2 +N e) = (e/2, N e+e)−(e/2, e/2), we obtain fee,(0,e/2+N e)

i1+e/2 . . .fee,(0,e/2+N e)

in+e/2 ∅= (λ¹, λ²) which implies that(λ¹, λ²)is a divided partition (for(0, e/2 +N e)).

References

[1] S.Ariki, Representations of quantum algebras and combinatorics of Young tableaux. University Lecture Series, 26. American Mathematical Society, Providence, RI, 2002.

[2] M. Geck and N. Jacon, Representations of Hecke algebras at roots of unity, Algebra and Applications, Vol. 15, Springer,

[3] G. Genet and N.Jacon, Modular representations of cyclotomic Hecke algebras of type G(r,p,n), Inter- national Mathematics Research Notices, Volume 2006, Article ID 93049,

[4] J. Hu, Crystal basis and simple modules for Hecke algebras of typeG(p, p, n), Representation Theory (AMS), Volume 11, Pages 16-44 (March 16, 2007)

[5] H Lin and J. Hu, Crystal of affine slb_l and Hecke algebras at a primitive 2l root of unity, preprint arXiv:1905.07333.

[6] N. Jacon, On the one dimensional representations of Ariki-Koike algebras at roots of unity, Journal of Pure and Applied Algebra Volume 221 (2017),

[7] N. Jacon and C. Lecouvey, Crystal isomorphisms for irreducible highest weightUv(bsl_e)-modules of higher level, Algebras and Representation theory 13, 467-489, 2010.

[8] S. Naito and D. Sagaki, Lakshmibai-Seshadri paths fixed by a diagram automorphism, J. Algebra, 245 (2001), 395-412.

Address:

Nicolas Jacon, Université de Reims Champagne-Ardenne, UFR Sciences exactes et naturelles, Laboratoire de Mathématiques UMR9008 Moulin de la Housse BP 1039, 51100 Reims, FRANCE

nicolas.jacon@univ-reims.fr