Alvis-Curtis Duality for Finite General Linear Groups and a Generalized Mullineux Involution
We study the effect of Alvis-Curtis duality on the unipotent representations of GLn(q) in non-defining characteristic ℓ. We show that the permutation induced on the simple modules can be expressed in terms of a generalization of the Mullineux involution on the set of all partitions, which involves b...
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| citation_txt | Alvis-Curtis Duality for Finite General Linear Groups and a Generalized Mullineux Involution / O. Dudas , N. Jacon // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 27 назв. — англ. |
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| description | We study the effect of Alvis-Curtis duality on the unipotent representations of GLn(q) in non-defining characteristic ℓ. We show that the permutation induced on the simple modules can be expressed in terms of a generalization of the Mullineux involution on the set of all partitions, which involves both ℓ and the order of q modulo ℓ.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 007, 18 pages
Alvis–Curtis Duality for Finite General Linear Groups
and a Generalized Mullineux Involution
Olivier DUDAS † and Nicolas JACON ‡
† Université Paris Diderot, UFR de Mathématiques, Bâtiment Sophie Germain,
5 rue Thomas Mann, 75205 Paris CEDEX 13, France
E-mail: olivier.dudas@imj-prg.fr
‡ Université de Reims Champagne-Ardenne, UFR Sciences exactes et naturelles,
Laboratoire de Mathématiques EA 4535, Moulin de la Housse BP 1039, 51100 Reims, France
E-mail: nicolas.jacon@univ-reims.fr
Received June 17, 2017, in final form January 22, 2018; Published online January 30, 2018
https://doi.org/10.3842/SIGMA.2018.007
Abstract. We study the effect of Alvis–Curtis duality on the unipotent representations
of GLn(q) in non-defining characteristic `. We show that the permutation induced on the
simple modules can be expressed in terms of a generalization of the Mullineux involution on
the set of all partitions, which involves both ` and the order of q modulo `.
Key words: Mullineux involution; Alvis–Curtis duality; crystal graph; Harish-Chandra theo-
ry
2010 Mathematics Subject Classification: 20C20; 20C30; 05E10
1 Introduction
Let Sn be the symmetric group on n letters. It is well known that the complex irreducible
representations of Sn are naturally labelled by the set of partitions of n. Tensoring with the
sign representation induces a permutation of the irreducible characters which corresponds to
the conjugation of partitions. An analogous involution can be considered for representations in
positive characteristic p > 0. In this case, the irreducible representations are parametrized by
p-regular partitions and the permutation induced by tensoring with the sign representation has
a more complicated combinatorial description. The explicit computation of this involution Mp
was first conjectured by Mullineux in [26] and proved by Ford–Kleshchev in [14]. Their result
was later generalized to representations of Hecke algebras at a root of unity by Brundan [3] with
a view to extending the definition of Mp to the case where p is any positive integer.
For representations of a finite group of Lie type G, the Alvis–Curtis duality DG functor
(see [10]) provides an involution of the same nature. For example, the Alvis–Curtis dual of
a complex unipotent character of GLn(q) parametrized by a partition is, up to a sign, the
character parametrized by the conjugate partition. Unlike the case of symmetric groups, the
duality DG does not necessarily map irreducible representations to irreducible representations,
but only to complexes of representations. Nevertheless, Chuang–Rouquier showed in [7] how
to single out a specific composition factor in the cohomology of these complexes, yielding an
involution dG on the set of irreducible representations of G in non-defining characteristic ` ≥ 0.
The purpose of this paper is to explain how to compute this involution dG using the Harish-
Chandra theory and the representation theory of Hecke algebras, see Theorem 2.5. We illustrate
This paper is a contribution to the Special Issue on the Representation Theory of the Symmetric Groups
and Related Topics. The full collection is available at https://www.emis.de/journals/SIGMA/symmetric-groups-
2018.html
mailto:olivier.dudas@imj-prg.fr
mailto:nicolas.jacon@univ-reims.fr
https://doi.org/10.3842/SIGMA.2018.007
https://www.emis.de/journals/SIGMA/symmetric-groups-2018.html
https://www.emis.de/journals/SIGMA/symmetric-groups-2018.html
2 O. Dudas and N. Jacon
our method on the irreducible unipotent representations of GLn(q), which are parametrized by
partitions of n. This yields an explicit involution Me,` on the set of partitions of n which depends
both on ` and on the order e of q modulo ` (with the convention that e = ` if ` | q − 1), see
Theorem 3.7.
Theorem 1.1. Assume that ` - q. Let S(λ) be the simple unipotent module of GLn(q) over F`
parametrized by the partition λ. Then
dG(S(λ)) = S(Me,`(λ)).
The involution Me,` is a generalization of the original Mullineux involution since it is defined
on the set of all partitions, and coincides with Me on the set of e-regular partitions. Such
a generalization already appeared in a work of Bezrukavnikov [2] and Losev [23] on wall-crossing
functors for representations of rational Cherednik algebras, but in the case where `� 0.
Kleshchev showed in [20] that the Mullineux involution Mp can also be interpreted in the
language of crystals for Fock spaces in affine type A, which are certain colored oriented graphs
whose vertices are labeled by partitions. More precisely, the image by Mp of a p-regular parti-
tion λ is obtained by changing the sign of each arrow in a path from the empty partition to λ in
the graph. We propose a definition of several higher level crystal operators on the Fock space
which give a similar description for our generalized involution Me,`, see Proposition 4.8.
The paper is organized as follows. In Section 2 we introduce the Alvis–Curtis duality for
finite reductive groups and show how to compute it within a given Harish-Chandra series using
a similar duality for the corresponding Hecke algebra. Section 3 illustrates our method in the
case of finite general linear groups. We give in Section 3.3 the definition of a generalized version
of the Mullineux involution and show in Theorem 3.7 that it is the shadow of the Alvis–Curtis
duality for GLn(q). The final section is devoted to an interpretation of our result in the context
of the theory of crystal graphs.
2 Alvis–Curtis duality
In this section we investigate the relation between the Alvis–Curtis duality for a finite reductive
group within a Harish-Chandra series and a similar duality in the Hecke algebra associated to
the series.
2.1 Notation
Let G be a connected reductive algebraic group defined over an algebraic closure of a finite
field of characteristic p, together with an endomorphism F , a power of which is a Frobenius
endomorphism. Given an F -stable closed subgroup H of G, we will denote by H the finite
group of fixed points HF . The group G is a finite reductive group.
We will be interested in the modular representations of G in non-defining characteristic. We
fix a prime number ` different from p and an `-modular system (K,O, k) which is assumed to
be large enough for G, so that the algebras KG and kG split. Throughout this section Λ will
denote any ring among K, O and k.
Given a finite-dimensional Λ-algebra A, we denote by A-mod (resp. mod-A) the category of
finite-dimensional left (resp. right) A-modules. The corresponding bounded derived category
will be denoted by Db(A-mod) (resp. Db(mod-A)) or simply Db(A) when there is no risk of
confusion. We will identify the Grothendieck group of the abelian category A-mod with the
Grothendieck group of the triangulated category Db(A). It will be denoted by K0(A). We will
write [M ] for the class of an A-module M in K0(A).
Alvis–Curtis Duality and a Generalized Mullineux Involution 3
2.2 Harish-Chandra induction and restriction
Given an F -stable parabolic subgroup P of G with Levi decomposition P = LU, where L is
F -stable, we denote by
ΛL-mod
RG
L
((
ΛG-mod
∗RG
L
gg
the Harish-Chandra induction and restriction functors. Under the assumption on `, they form
a biajdoint pair of exact functors. The natural transformations given by the adjunction (unit
and counit) are denoted as follows
Id
ηL⊂G
−−−→ RGL
∗RGL
εL⊂G
−−−→ Id and Id
ηL⊂G
−−−→ ∗RGLR
G
L
εL⊂G
−−−→ Id,
where Id denotes here the identity functor. When it is clear from the context, we will usually
drop the superscript L ⊂ G.
Let Q be another F -stable parabolic subgroup with F -stable Levi decomposition Q = MV.
If P ⊂ Q and L ⊂ M then the Harish-Chandra induction and restriction functors satisfy
RGM ◦ RML ' RGL and ∗RML ◦ ∗RGM ' ∗RGL . Together with the counit εL⊂M : RML
∗RML −→ 1, this
gives a natural transformation
ϕL⊂M⊂G : RGL
∗RGL −→ RGM
∗RGM .
If R is any other F -stable parabolic subgroup containing Q with an F -stable Levi complement N
containing M then the natural isomorphism giving the transitivity of Harish-Chandra induction
and restriction can be chosen so that ϕM⊂N⊂G ◦ ϕL⊂M⊂G = ϕL⊂N⊂G (see for example [4,
Section 4] or [12, Section III.7.2]).
2.3 Alvis–Curtis duality functor
We now fix a Borel subgroup B of G containing a maximal torus T, both of which are assumed
to be F -stable. Let ∆ be the set of simple roots defined by B. The F -stable parabolic subgroups
containing B are parametrized by F -stable subsets of ∆. They have a unique Levi complement
containing T. Such Levi subgroups and parabolic subgroups are called standard. Given r ≥ 0,
we denote by Lr the (finite) set of F -stable standard Levi subgroups corresponding to a subset
I ⊂ ∆ satisfying |I/F | = |∆/F | − r. In particular L0 = {G} and L|∆/F | = {T}. Following
[5, 10, 12] we can form the complex of exact functors
0 −→ RGT
∗RGT −→ · · · −→
⊕
L∈L2
RGL
∗RGL −→
⊕
L∈L1
RGL
∗RGL −→ Id −→ 0,
where Id is in degree 0. It yields a functor DG on the bounded derived category Db(ΛG) of
finitely generated ΛG-modules, called the Alvis–Curtis duality functor. Note that the original
definition of the duality by Alvis and Curtis [1, 8] refers to the linear endomorphism on K0(KG)
induced by this functor. The complex above was introduced by Deligne–Lusztig in [10].
Theorem 2.1 (Cabanes–Rickard [5]). The functor DG is a self-equivalence of Db(ΛG) satisfying
DG ◦ RGL ' RGL ◦ DL[r] (2.1)
for every L ∈ Lr.
4 O. Dudas and N. Jacon
Note that any quasi-inverse of DG will also satisfy the relation (2.1), up to replacing r by −r.
Chuang–Rouquier deduced in [7] from (2.1) that the equivalence induced by DG is perverse
with respect to the cuspidal depth (see below for the definition). For the reader’s convenience
we recall here their argument.
Assume that Λ is a field. The simple ΛG-modules are partitioned into Harish-Chandra series,
see [19]. Given a simple ΛG-module S, there exists L in Ls and a simple cuspidal ΛL-module X
such that S appears in the head (or equivalently in the socle) of RGL (X). The pair (L, X) is
unique up to G-conjugation. We say that S lies in the Harish-Chandra series of the cuspidal
pair (L, X) and we call s the cuspidal depth of S. In particular, the cuspidal modules are the
modules with cuspidal depth zero.
Proposition 2.2 (Chuang–Rouquier [7]). Assume that Λ is a field. Let S be a simple ΛG-
module of cuspidal depth s. Then
(i) H i(DG(S)) = 0 for i > 0 and i < −s.
(ii) The composition factors of H i(DG(S)) have depth at most s.
(iii) Among all the composition factors of
⊕
iH
i(DG(S)), there is a unique composition factor
of depth s. It is a submodule of H−s(DG(S)), and it lies in the same Harish-Chandra
series as S.
Proof. We denote by (L,X) a cuspidal pair associated with S. Given any other cuspidal pair
(M,Y ) with M ∈ Lr and n ∈ Z we have
HomDb(ΛG)
(
RGM (Y ),DG(S)[n]
)
' HomDb(ΛG)
(
D−1
G (RGM (Y )), S[n]
)
' HomDb(ΛG)
(
RGM (D−1
M (Y ))[−r], S[n]
)
' HomDb(ΛM)
(
Y, ∗RGM (S)[r + n]
)
, (2.2)
where in the last equality we used that D−1
M (Y ) ' Y since Y is cuspidal. In particular, it is zero
when r < −n or when M does not contain a G-conjugate of L, so in particular when r > s.
Take n to be the smallest integer such that Hn(DG(S)) 6= 0 and consider a simple ΛG-modu-
le T in the socle of Hn(DG(S)). Let (M,Y ) be the cuspidal pair above which T lies. Then the
composition RGM (Y )� T ↪→ Hn(DG(S)) yields a non-zero element in
HomDb(ΛG)
(
RGM (Y ),DG(S)[n]
)
.
From (2.2) we must have −n ≤ r ≤ s which proves (i). Furthermore, if n = −s then r and s are
equal, and in that case
HomDb(ΛG)
(
RGM (Y ),DG(S)[−s]
)
' HomDb(ΛM)
(
Y, ∗RGM (S)
)
' HomΛM
(
Y, ∗RGM (S)
)
.
By the Mackey formula, ∗RGM (S) is isomorphic to a direct sum of G-conjugates of X. Therefore
if T lies in the socle of H−s(DG(S)) then Y is G-conjugate to X which means that T and S lie
in the same Harish-Chandra series.
Now if we replace Y by its projective cover PY in (2.2) we get
HomDb(ΛG)
(
RGM (PY ),DG(S)[n]
)
' HomDb(ΛM)
(
PY , DM
(∗RGM (S)
)
[r + n]
)
,
which again is zero unless r ≤ s or unless r = s and (M,Y ) is conjugate to (L,X). In that
latter case we have
HomDb(ΛG)
(
RGL (PX),DG(S)[n]
)
' HomDb(ΛL)
(
PX ,DL
(∗RGL (S)
)
[n+ s]
)
' HomDb(ΛL)
(
PX ,
∗RGL (S)[n+ s]
)
,
Alvis–Curtis Duality and a Generalized Mullineux Involution 5
since ∗RGL (S) is a sum of conjugates of X. Therefore the composition factors of DG(S) lying in
the Harish-Chandra series of (L,X) can only appear in degree −s. They appear with multiplicity
one since DG is a self-equivalence, which proves (iii). �
Using the property that DG is a perverse equivalence (given in Proposition 2.2) we can
define a bijection on the set of simple ΛG-modules as follows: given a simple ΛG-module S
with cuspidal depth s we define dG(S) to be the unique simple ΛG-module with depth s which
occurs as a composition factor in the cohomology of DG(S). Note that dG(S) and S lie in the
same Harish-Chandra series.
2.4 Compatibility with Hecke algebras
Given a cuspidal pair (L,X) of G, we can form the endomorphism algebra
HG(L,X) = EndG
(
RGL (X)
)
.
By [17, Theorem 2.4], the isomorphism classes of simple quotients of RGL (X) (the Harish-Chandra
series of (L,X)) are parametrized by the simple representations of HG(L,X). The structure
of this algebra was studied for example in [17, Section 3]; it is in general very close to be
a Iwahori–Hecke algebra of a Coxeter group.
Let M be an F -stable standard Levi subgroup of G containing L. Since RGM is fully-faithful,
HM (L,X) = EndG(RML (X)) embedds naturally as a subalgebra ofHG(L,X). To this embedding
one can associate the induction and restriction functors
mod-HM (L,X)
Ind
HG
HM
++
mod-HG(L,X)
Res
HG
HM
kk
between the categories of right modules. The purpose of this section is to compare the Alvis–
Curtis duality functor DG for the group with a similar functor DH of the Hecke algebra. From
now on we shall fix the cuspidal pair (L,X), and we will denote simply HG and HM the
endomorphism algebras of RGL (X) and RML (X) respectively.
Let Y be a (non-necessarily cuspidal) ΛM -module. We consider the natural transforma-
tion ΘM,Y defined so that the following diagram commutes:
HomM (Y, ∗RGM (−))⊗EndM (Y ) HomM (∗RGMRGM (Y ), Y )
mult //
∼
��
HomM (∗RGMRGM (Y ), ∗RGM (−))
∼
��
HomG(RGM (Y ),−)⊗EndM (Y ) EndG(RGM (Y ))
ΘM,Y // HomG(RGM (Y ),RGM
∗RGM (−)).
In other words, ΘM,Y is defined on the objects by
ΘM,Y (f ⊗ h) = RGM
∗RGM (f) ◦ RGM (η
Y
) ◦ h.
Using this description one can check that ΘM,Y is compatible with the right action of
EndG(RGM (Y )). Therefore it is a well-defined natural transformation between functors from
ΛG-mod to mod-EndG(RGM (Y )).
Let N be another standard F -stable Levi subgroup of G with M ⊂ N. Recall from Sec-
tion 2.2 the natural transformation ϕM⊂N⊂G : RGM
∗RGM −→ RGN
∗RGN which was needed for the
construction of DG. By composition it induces a natural transformation
HomG
(
RGM (Y ),RGM
∗RGM (−)
)
−→ HomG
(
RGM (Y ),RGN
∗RGN (−)
)
.
6 O. Dudas and N. Jacon
Proposition 2.3. The following diagram is commutative
HomG
(
RG
M (Y ),−
)
⊗EndM (Y ) EndG
(
RG
M (Y )
) ΘM,Y //
��
HomG
(
RG
M (Y ),RG
M
∗RG
M (−)
)
��
HomG
(
RG
M (Y ),−
)
⊗
EndN
(
RN
M
(Y )
) EndG
(
RG
M (Y )
)
∼
��
HomG
(
RG
M (Y ),RG
N
∗RG
N (−)
)
∼
��
HomG
(
RG
N (RN
M (Y )),−
)
⊗
EndN
(
RN
M
(Y )
) EndG
(
RG
N
(
RN
M (Y )
)) Θ
N,RN
M
(Y )
// HomG
(
RG
N
(
RN
M (Y )
)
,RG
N
∗RG
N (−)
)
.
Proof. Since RNM is faithful one can see EndM (Y ) as a subalgebra of EndN (RNM (Y )). Then the
first vertical map on the left-hand side of the diagram is the canonical projection. We will write
t : RGM
∼−→RGNR
N
M (resp. t∗ : ∗RGM
∼−→ ∗RNM ∗RGN ) for the isomorphism of functors coming from the
transitivity of Harish-Chandra induction (resp. restriction).
Let Z be a ΛG-module, h ∈ End(RGM (Y )) and f ∈ HomG(RGM (Y ), Z). The commutativity of
the diagram is equivalent to the relation
RGN
∗RGN
(
f ◦ t−1
Y
)
◦ RGN
((
ηN⊂G
)
RN
M (Y )
)
◦
(
tY ◦ h ◦ t−1
Y
)
=
(
ϕM⊂N⊂G
)
Z
◦ RGM ∗RGM (f) ◦ RGM
((
ηM⊂G
)
Y
)
◦ h ◦ t−1
Y .
Since
(
ϕM⊂N⊂G
)
Z
◦ RGM ∗RGM (f) = RGN
∗RGN (f) ◦
(
ϕM⊂N⊂G
)
RG
M (Y )
it is enough to show that
(
ϕM⊂N⊂G
)
RG
M (Y )
◦ RGM
((
ηM⊂G
)
Y
)
= RGN
∗RGN
(
t−1
Y
)
◦ RGN
((
ηN⊂G
)
RN
M (Y )
)
◦ tY . (2.3)
This comes from the following commutative diagram
RGM
∗RGMRGM
t //
ϕM⊂N⊂G
++
RGNR
N
M
∗RGMRGM
t∗ // RGNR
N
M
∗RNM
∗RGNR
G
M
εM⊂N
//
t
��
RGN
∗RGNR
G
M
RGM
ηM⊂G
OO
t // RGNR
N
M
ηM⊂G
OO
ηM⊂N
))
RGNR
N
M
∗RNM
∗RGNR
G
NR
N
M
εM⊂N
// RGN
∗RGNR
G
NR
N
M
t−1
OO
RGNR
N
M
∗RNMRNM εM⊂N
//
ηN⊂G
OO
RGNR
N
M .
ηN⊂G
OO
where for simplicity we did not write the identity natural transformations. The only non-trivial
commutative subdiagram is the central one, which comes from the relation(∗RNM · ηN⊂G · RNM) ◦ ηM⊂N = (t∗ · t) ◦ ηN⊂G,
which we assume to hold by our choice of t and t∗. For more on the compatibility of the unit,
counit, t and t∗ see [12, Part 3]. Now the relation (2.3) comes from the equality between two
natural transformations between the functors RGM and RGN
∗RGNR
G
M , given by the top and the
bottom arrows respectively. �
Alvis–Curtis Duality and a Generalized Mullineux Involution 7
In the particular case where Y = RGL (X) with (L,X) being a cuspidal pair, the natural
transformation ΘM,RM
L (X) becomes
IndHG
HM
ResHG
HM
(
HomG(RGL (X),−)
) Θ
M,RM
L
(X)
−−−−−−−→ HomG(RGL (X),RGM
∗RGM (−)).
It can be seen as a way to intertwine the endofunctor IndHG
HM
ResHG
HM
of mod-HG and the endo-
functor RGM
∗RGM of ΛG-mod via the functor HomG(RGL (X),−), as shown in the following diagram
kG-mod
HomG(RG
L (X),−)
//
RG
M
∗RG
M
��
mod-HG
Ind
HG
HM
Res
HG
HM
��
⇐=
kG-mod
HomG(RG
L (X),−)
// mod-HG.
In this diagram the central double arrow represents the natural transformation ΘM,RM
L (X). We
give a condition for this transformation to be an isomorphism. Note that similar results were
obtained by Dipper–Du in [11, Theorem 1.3.2] in the case of general linear groups and by
Seeber [27] with coinduction instead of induction.
Proposition 2.4. Assume that Λ is a field. Let (L,X) be a cuspidal pair of G, and M be
a standard Levi of G containing L. Assume that any cuspidal pair of M which is G-conjugate
to (L,X) is actually M -conjugate to (L,X). Then ΘM,RM
L (X) is an isomorphism.
Proof. By definition of ΘM,RM
L (X) it is enough to show that the multiplication map
HomM
(
RML (X), ∗RGM (−)
)
⊗
EndM
(
RM
L (X)
) HomM
(∗RGMRGL (X),RML (X)
)
��
HomM
(∗RGMRGL (X), ∗RGM (−)
)
is an isomorphism of k-vector spaces. Now by the Mackey formula [4, Proposition 1.5] we have
∗RGMRGL (X) '
⊕
x∈Q\G/P
xL⊂M
RMxL(Xx).
Under the assumption on (L,X), any cuspidal pair (xL,Xx) with xL ⊂M is conjugate to (L,X)
under M . In particular RMxL(Xx) ' RML (X) and we deduce that each of the composition maps
HomM
(
RML (X), ∗RGM (−)
)
⊗
EndM
(
RM
L (X)
) HomM
(
RMxL(Xx),RML (X)
)
��
HomM
(
RMxL(Xx), ∗RGM (−)
)
is an isomorphism. �
As in the case of the finite group G we can form a complex of functors coming from induction
and restriction in HG. Given r ≥ 0, we denote by Lr(L) the subset of Lr of standard Levi
subgroups containing L. The complex of functors
0 −→ IndHG
HL
ResHG
HL
−→ · · · −→
⊕
M∈L1(L)
IndHG
HM
ResHG
HM
−→ Id −→ 0,
8 O. Dudas and N. Jacon
where Id is in degree 0, induces a triangulated functor DHG
in Db(mod-HG) whenever each term
of the complex is exact. For that property to hold we need to assume that HG is flat over each
subalgebra of the form HM . Combining Propositions 2.3 and 2.4 we get
Theorem 2.5. Assume that Λ is a field. Let L be an F -stable standard Levi subgroup of G
and X be a cuspidal ΛL-module. Let HG = EndG(RGL (X)). Assume that for every F -stable
standard Levi subgroup M of G containing L we have:
(i) HG is flat over HM = EndM
(
RML (X)
)
.
(ii) Every cuspidal pair of M which is G-conjugate to (L,X) is actually M -conjugate to (L,X).
Then there is a natural isomorphism of endofunctors of Db(mod-HG)
DHG
(
RHomG
(
RGL (X),−
)) ∼→ RHomG
(
RGL (X),DG(−)
)
.
3 Unipotent representations of GLn(q)
In this section we show how to use Theorem 2.5 to compute dGLn(q)(S) for every unipotent simple
kGLn(q)-module S. This will involve an involution on the set of partitions of n generalizing the
Mullineux involution [26]. So here G = GLn(Fp) will be the general linear group over an
algebraic closure of Fp, and F : G −→ G the standard Frobenius endomorphism, raising the
entries of a matrix to the q-th power.
3.1 Partitions
A partition λ of n ∈ N is a non-increasing sequence (λ1 ≥ λ2 ≥ · · · ≥ λr) of positive integers
which add up to n. By convention, ∅ is the unique partition of 0 and is called the empty
partition. The set of partitions of n will be denoted by P(n), and the set of all partitions by
P := tn∈NP(n). We shall also use the notation Λ = (1r1 , 2r2, . . . , nrn) where ri denotes the
multiplicity of i in the sequence λ.
Given λ and µ two partitions of n1 ∈ N and n2 ∈ N respectively, we denote by λ t µ the
partition of n1 + n2 obtained by concatenation of the two partitions and by reordering the
parts to obtain a partition. If λ is a partition of n and k ∈ N, we denote by λk the partition
λ t λ t · · · t λ︸ ︷︷ ︸
k times
.
Let d ∈ N>1. A partition λ is called d-regular if no part in λ is repeated d or more times.
Each partition λ can be decomposed uniquely as λ = µ t νd where µ is d-regular. Then λ is
d-regular if and only if ν is empty. The set of d-regular partitions of n is denoted by Regd(n).
This set has a remarkable involution Md called the Mullineux involution which will be defined
in the next section (see Section 4.5 for its interpretation in terms of crystals).
More generally we shall decompose partitions with respect to two integers. Recall that ` is
a prime number. If λ = (1r1 , 2r2, . . . , nrn) is a partition of n, we can decompose the integers ri
as
ri = ri,−1 + dri,0 + d`ri,1 + d`2ri,2 + · · ·+ d`nri,n
with 0 ≤ ri,−1 < d and 0 ≤ ri,j < ` for all j ≥ 0. If we define the partition λ(j) =
(1r1,j , 2r2,j , . . . , nrn,j ), then
λ = λ(−1) t (λ(0))
d t (λ(1))
d` t · · · t (λ(n))
d`n ,
where λ(−1) is d-regular and λ(j) is `-regular for all j ≥ 0. This decomposition is called the
d-`-adic decomposition of λ [11].
Alvis–Curtis Duality and a Generalized Mullineux Involution 9
3.2 Hecke algebras of type A and the Mullineux involution
Let q ∈ k× and m ≥ 1. We denote by Hq(Sm) the Iwahori–Hecke algebra of the symmetric
group Sm over k, with parameter q. It has a k-basis {Tw}w∈Sm satisfying the following relations,
for w ∈ Sm and s = (i, i+ 1) a simple reflection:
TwTs =
{
Tws if `(ws) > `(w) (i.e., if w(i) < w(i+ 1)),
qTws + (q − 1)Tw otherwise.
In particular the basis elements corresponding to the simple reflections generate Hq(Sm) as an
algebra, and they satisfy the relation (Ts − q)(Ts + 1) = 0.
Let us consider the integer
e := min
{
i ≥ 0 | 1 + q + q2 + · · ·+ qi−1 = 0
}
∈ N>1.
It is equal to the order of q in k× when q 6= 1, and to ` = char k when q = 1. Then the
set of simple Hq(Sm)-modules is parametrized by the set of e-regular partitions of m. Given
an e-regular partition λ, we will denote by D(λ) the corresponding simple module. Note that
when q = 1, the Hecke algebra Hq(Sm) coincides with the group algebra of Sm over k, whose
irreducible representations are parametrized by `-regular partitions of m.
The map α : Tw 7−→ (−q)`(w)(Tw−1)−1 is an algebra automorphism ofHq(Sm) of order 2. This
follows from the fact that −qT−1
s satisfies the same quadratic equation as Ts. The automor-
phism α induces a permutation α∗ on the set of simple Hq(Sm)-modules, and therefore on the
set of e-regular partitions. In other words, the exists an involution Me on Rege(n), called the
Mullineux involution, such that for any e-regular partition λ
α∗(D(λ)) ' D(Me(λ)).
In the case where q = 1, the involution α is just the multiplication by the sign representation ε
in the group algebra of Sm and D(M`(λ)) ' D(λ)⊗ ε.
The involution M` was introduced by Mullineux, who suggested in [26] a conjectural explicit
combinatorial algorithm to compute it. This conjecture was subsequently proved by Ford–
Kleshchev in [14]. An interpretation in terms of crystals was later given by Kleshchev [20] (for
the case q = 1) and by Brundan [3] for the general case. We will review their result in Section 4.5.
Remark 3.1. When e > m, every partition of m is e-regular. The Hecke algebra is actually
semi-simple in that case and the Mullineux involution Me corresponds to conjugating partitions.
Recall that Sm has a structure of a Coxeter group, where the simple reflections are given
by the transpositions (i, i + 1). Given a parabolic subgroup S of Sm, one can consider the
subalgebra of Hq(Sm) generated by {Tw}w∈S. It corresponds to the Hecke algebra Hq(S)
of S with parameter q, and Hq(Sm) is flat as a module over that subalgebra. It is even free,
with basis given by the elements Tw where w runs over a set of representatives of S\Sm with
minimal length. Therefore, following Section 2.4 (see also [22]) we can use the induction and
restriction functors to define a duality functor DH on the bounded derived category of finitely
generated Hq(Sm)-modules. For Hecke algebras this duality functor is actually a shifted Morita
equivalence.
Theorem 3.2 (Linckelmann–Schroll [22]). Given a finitely generated right Hq(Sm)-module X
we have
DH(X) ' α∗(X)[−m+ 1]
in Db(mod-Hq(Sm)).
In particular when X = D(λ) is simple we get DH(D(λ)) ' D(Me(λ))[−m+ 1].
10 O. Dudas and N. Jacon
3.3 Harish-Chandra series of GLn(q) and a generalized Mullineux involution
From now on G = GLn(Fp) is the general linear group over an algebraic closure of Fp, and
F : G −→ G the standard Frobenius endomorphism, raising the entries of a matrix to the q-th
power. The corresponding finite reductive group is G = GLn(q).
Recall that the unipotent simple kG-modules are parametrized by partitions of n. We will
denote by S(λ) the simple kG-module corresponding to the partition λ. We review now the
results in [11, 16] on the partition of the unipotent representations into Harish-Chandra series
(see also [4, Section 19]). This classification depends on both `, and the integer e > 1 defined
as the order of q modulo ` (with the convention that e = ` if q ≡ 1 modulo `).
By [11, Corollary 4.3.13] S(λ) is cuspidal if and only if λ = 1 or λ = 1e`
i
for some i ≥ 0.
In particular, given n ≥ 1 there is at most one cuspidal unipotent simple kGLn(q)-module, and
there is exactly one if and only if n = 1 or n = e`i for some i ≥ 0. By considering products of
such representations one can construct any cuspidal pair of G. To this end, we introduce the set
N (n) =
{
m = (m−1,m0, . . . ,mn) |n = m−1 + em0 + e`m1 + · · ·+ e`nmn
}
.
Given m ∈ N (n) we define the standard Levi subgroup
Lm = GL1(q)m−1 ×GLe(q)
m0 × · · · ×GLe`n(q)mn .
It has a unique cuspidal unipotent simple module Xm, and all the cuspidal pairs (Lm, Xm) of G
are obtained this way for various m ∈ N (n).
The simple kG-modules lying in the corresponding Harish-Chandra series are parametrized
by the irreducible representations of the endomorphism algebra Hm = EndG
(
RGLm
(Xm)
)
. By
[11, Section 3.4] (see also [4, Lemma 19.24]), there is a natural isomorphism of algebras
Hm ' Hq(Sm−1)⊗ kSm0 ⊗ · · · ⊗ kSmn , (3.1)
where Hq(Sm−1) is the Iwahori–Hecke algebra of Sm−1 introduced in the previous section.
Therefore the simple modules of Hm are parametrized by tuples of partitions
λ = (λ(−1), λ(0), . . . , λ(n)),
where λ(−1) is an e-regular partition of m−1 and each λ(i) for i ≥ 0 is an `-regular partition
of mi.
By the tensor product theorem of Dipper–Du [11, Corollary 4.3.11] (see also [4, Theo-
rem 19.20]), the simple module S(λ) of G attached to this multipartition λ is given by
λ = λ(−1) t (λ(0))
e t (λ(1))
e` t · · · t (λ(n))
e`n . (3.2)
In other words,
HomkG
(
RGLm
(Xm), S(λ)
)
' D(λ) in mod-Hm. (3.3)
Conversely, any partition λ of n can be uniquely decomposed as (3.2) using the e-`-adic de-
composition defined in Section 3.1. The tuple m = (|λ(−1)|, |λ(0)|, . . . , |λ(n)|) will be denoted
by hc(λ), thus defining a map hc : P(n) −→ N (n). With this notation, the simple kG-modules
lying in the Harish-Chandra of (Lm, Xm) are parametrized by hc−1(m).
Motivated by the isomorphism (3.1) and the tensor product theorem of Dipper–Du we define
a version of the Mullineux involution as follows.
Definition 3.3. Let λ be a partition of n, written as λ = λ(−1) t (λ(0))
e t · · · t (λ(n))
e`n where
λ(−1) is e-regular and each λ(i) for i ≥ 0 is `-regular. The generalized Mullineux involution on λ
is defined by
Me,`(λ) = Me(λ(−1)) t (M`(λ(0)))
e t · · · t (M`(λ(n)))
e`n .
Alvis–Curtis Duality and a Generalized Mullineux Involution 11
Example 3.4. It is interesting to note the following particular cases.
(a) If λ is an e-regular partition then Me,`(λ) = Me(λ) is the ordinary Mullineux involution.
(b) If e` > n then λ(i) = ∅ for i > 0. In addition, ` is bigger than the size of λ(0) and therefore
M`(λ(0)) = λt(0) is just the conjugate of λ(0). Consequently we get
Me,`(λ) = Me(λ(−1)) t (λ(0)
t)e.
Quite remarkably, this involution already appears in the work of Bezrukavnikov [2] and
Losev [23, Corollary 5.7] on the wall-crossing bijections for representations of rational
Cherednik algebras.
Let Irrk
(
G|(Lm, Xm)
)
be the set of isomorphism classes of simple kG-modules lying in the
Harish-Chandra series associated with (Lm, Xm). Note that Me,` preserves hc−1(m), there-
fore S(λ) and S(Me,`(λ)) lie in the same Harish-Chandra series. By construction, the involu-
tion Me,` is the unique operation which makes the following diagram commutative:
hc−1(m) oo //
Me,`
��
Irrk
(
G|(Lm, Xm)
) HomG
(
RG
Lm
(Xm),−
)
// IrrHm
α∗
��
hc−1(m) oo // Irrk
(
G|(Lm, Xm)
) HomG
(
RG
Lm
(Xm),−
)
// IrrHm.
3.4 Computation of dG(S)
Fix m = (m−1,m0, . . . ,mn) ∈ N (n) and let (Lm, Xm) be the corresponding cuspidal pair,
as defined in Section 3.3. One cannot apply Theorem 2.5 directly to G and Lm since the
assumption (ii) might not be satisfied for every intermediate Levi subgroup between Lm and G.
For example, if m = (e, 1) then the Levi subgroups (GL1(q))e×GLe(q) and GLe(q)× (GL1(q))e
are conjugate under GL2e(q) but not under GLe(q)×GLe(q). To solve this problem we consider,
instead of G, the standard Levi subgroup
Gm = GLm−1
(
Fp
)
×GLem0
(
Fp
)
×GLe`m1
(
Fp
)
× · · · ×GLe`nmn
(
Fp
)
.
Then Lm is the only standard Levi subgroup of Gm which is conjugate to Lm. In particular,
assumption (ii) of Theorem 2.5 is satisfied.
Lemma 3.5. The functor RGGm
induces an isomorphism of algebras
EndGm
(
RGm
Lm
(Xm)
) ∼→ EndG
(
RGLm
(Xm)
)
= Hm.
Proof. Since RGGm
is fully-faithful, the natural map EndGm(RGm
Lm
(Xm)) −→ EndG(RGLm
(Xm)) is
an embedding of algebras. To conclude it is enough to compute the dimensions using the Mackey
formula. The equality comes from the fact that any element g ∈ G which normalizes Lm and Xm
is in fact in Gm. �
In particular, the simple kGm-modules lying in the Harish-Chandra series of (Lm, Xm) are
also parametrized by multipartitions λ = (λ(−1), λ(0), . . . , λ(n)) of m, where λ(−1) is e-regular,
and each λ(i) for i ≥ 0 is `-regular. We will write Sm(λ) for the simple module corresponding
to λ, which by definition satisfies
HomkGm(RGm
Lm
(Xm), Sm(λ)) ' D(λ) in mod-Hm (3.4)
12 O. Dudas and N. Jacon
(compare with (3.3)). Then it follows from the tensor product theorem [11, Corolary 4.3.11]
that
Sm(λ) = S(λ(−1))� S
(
(λ(0))
e
)
� S
(
(λ(1))
e`
)
� · · ·
and
RGGm
(Sm(λ)) ' S(λ), (3.5)
where as in (3.2) we set λ = λ(−1) t (λ(0))
e t (λ(1))
e` t · · · t (λ(n))
e`n .
The construction of the isomorphism (3.1) given for example in [4, Section 19] is compatible
with induction and restriction. The map M 7−→ EndM (RMLm
(Xm)) gives a one-to-one correspon-
dence between the standard Levi subgroups of Gm containing L and the parabolic subalgebras
of Hm. Since q 6= 0, Hm is flat over each of these subalgebras and Theorem 2.5 can be applied
to get
DHm
(
RHomGm
(
RGm
Lm
(Xm),−
)) ∼−→ RHomGm
(
RGm
Lm
(Xm),DGm(−)
)
. (3.6)
Recall from Section 2.3 that given a simple kG-module S, there is unique composition factor
in the cohomology of DG(S) which lies in the same Harish-Chandra series as S. We denote
this composition factor by dG(S). Combining (3.6) and Theorem 3.2 we can determine dGm
explicitly on the unipotent representations.
Proposition 3.6. Let λ = (λ(−1), λ(0), λ(1), . . . , λ(n)) be a multipartition where λ(−1) is an e-
regular partition of m−1 and each λ(i) for i ≥ 0 is an `-regular partition of mi. Then
dGm(Sm(λ)) ' Sm
(
Me(λ(−1)),M`(λ(0)), . . . ,M`(λ(n))
)
.
Proof. For simplicity we will write S for Sm(λ) throughout this proof. Let r be the cuspidal
depth of the kGm-module S. By Proposition 2.2 and the definition of dGm(S) the natural map
HomGm
(
RGm
Lm
(Xm), dGm(S)
)
−→ HomGm
(
RGm
Lm
(Xm), H−r(DGm(S))
)
is an isomorphism of right Hm-modules. Now let us consider the distinguished triangle
H−r(DGm(S)) −→ DGm(S)[−r] −→ τ>−r(DGm(S))[−r]
in Db(kGm). We apply the functor HomDb(kGm)
(
RGm
Lm
(Xm),−
)
, which, by the properties of
DGm(S) listed in Proposition 2.2, gives an isomorphism
HomGm
(
RGm
Lm
(Xm), H−r(DGm(S))
) ∼−→ HomDb(kGm)
(
RGm
Lm
(Xm),DGm(S)[−r]
)
.
Combining this with Theorem 2.5 gives
HomGm
(
RGm
Lm
(Xm), H−r(DGm(S))
)
' H−r
(
RHomGm
(
RGm
Lm
(Xm),DGm(S)
))
' H−r
(
DH
(
RHomGm(RGm
Lm
(Xm), S)
))
in mod-Hm. Now, by Theorem 3.2 the duality functor DH is induced by a shifted Morita
equivalence, obtained by twisting by the algebra automorphism α defined in Section 3.2. Note
that the corresponding shift equals the rank of the Coxeter group associated with the Hecke
algebra Hm, which also equals the cuspidal depth r of S. In particular, we have
H−r
(
DH
(
RHomGm(RGm
Lm
(Xm), S)
))
' H−r
(
α∗RHomGm
(
RGm
Lm
(Xm), S
)
[r]
)
' α∗H0
(
RHomGm
(
RGm
Lm
(Xm), S
))
' α∗
(
HomDb(kGm)
(
RGm
Lm
(Xm), S
))
' α∗D(λ).
Note that the last isomorphism uses (3.4) and the fact that the natural functor kGm → Db(kGm)
is fully-faithful. Finally, by definition of the Mullineux involution, the module α∗D(λ) is the
simple Hm-module labelled by the multipartition (Me(λ(−1)),M`(λ(0)), . . . ,M`(λ(n))). �
Alvis–Curtis Duality and a Generalized Mullineux Involution 13
We can finally prove the expected relation between the Alvis–Curtis duality and our gene-
ralization of the Mullineux involution.
Theorem 3.7. Let e = min{i ≥ 0 | 1 + q + q2 + · · ·+ qi−1 ≡ 0 mod `} and λ be a partition of n.
Then
dGLn(q)(S(λ)) ' S(Me,`(λ)),
where Me,` is the generalized Mullineux involution defined in Definition 3.3.
Proof. Let λ be the multipartition associated to λ as in (3.2). Given a bounded complex C of
representations (of kG or kGm), recall that [C] denotes its class in the corresponding Grothen-
dieck group (K0(kG) or K0(kGm), see Section 2.1). By (2.1) we have[
DG
(
RGGm
(Sm(λ))
)]
= ±
[
RGGm
(DGm(Sm(λ)))
]
in K0(kG). (3.7)
Let C (resp. Cm) be the sublattice of K0(kG) (resp. K0(kGm)) spanned by the classes of simple
modules with cuspidal depth strictly less than S(λ) (resp. Sm(λ)). It follows from Proposi-
tion 2.2(ii) that these lattices are stable under Alvis–Curtis duality. In addition, the Harish-
Chandra induction functor satisfies RGGm
(Cm) ⊂ C.
By Proposition 2.2, we have [DGm(Sm(λ))] ∈ ±[dGm(Sm(λ))] + Cm. We deduce from (3.5)
and Proposition 3.6 that[
RGGm
(DG(Sm(λ)))
]
∈ ±[S(Me,`(λ))] + C.
On the other hand [DG(S(λ))] ∈ ±[dG(S(λ))] + C, so that again by (3.5) we have[
DG
(
RGGm
(Sm(λ))
)]
∈ ±[dG(S(λ))] + C
and we conclude that [dG(S(λ))] = [S(Me,`(λ))] using (3.7) . �
Remark 3.8. The simple unipotent kGLn(q)-module associated with the trivial partition λ=(n)
is the trivial module k. In that case the complex DG(k) is quasi-isomophic to a module shifted in
degree−n+1, by the Solomon–Tits theorem [9, Theorem 66.33]. This module is a characteristic `
version of the Steinberg representation. By Theorem 3.7, its socle is isomorphic to S(Me(n)),
which is consistent with [15].
4 Interpretation in terms of crystals
The aim of this section is to give an alternative description of the map Me,` using the crystal
graph theory in the same spirit as for the original Mullineux involution [20] (see Proposition 4.8).
4.1 More on partitions
We fix an integer d > 1. Given a partition λ = (λ1 ≥ λ2 ≥ · · · ≥ λr > 0), its Young diagram [λ]
is the set
[λ] =
{
(a, b) | 1 ≤ a ≤ r, 1 ≤ b ≤ λa
}
⊂ N× N.
The elements of this set are called the nodes of λ. The d-residue of a node γ ∈ [λ] is by definition
resd(γ) = b − a + dZ. For j ∈ Z/dZ, we say that γ is a j-node if resd(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. We write µ
j→ λ if [µ] ⊂ [λ] and
[λ] \ [µ] = {γ} for a j-node γ.
14 O. Dudas and N. Jacon
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 wj(λ) be the word obtained by reading all the addable and
removable j-nodes 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 sequence A · · ·AR · · ·R. 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.
4.2 Fock space and Kashiwara operators
Let F := CP be the C-vector space with basis given by the set P of all partitions. There is
an action of the quantum group U(ŝld) on F [25] which makes F into an integrable module of
level 1. The Kashiwara operators Ẽi,d and F̃i,d are then defined as follows:
F̃i,d · λ =
{
µ if λ
i→ µ and [µ] \ [λ] is a good addable i-node of λ,
0 if λ has no good addable i-node,
Ẽi,d · µ =
{
λ if λ
i→ µ and [µ] \ [λ] is a good removable i-node of µ,
0 if µ has no good removable i-node.
Using these operators one can construct the ŝld-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/dZ if and only if F̃i,d · λ = µ, or
equivalently if and only if λ = Ẽi,d · µ.
Note that the definition makes sense for d = ∞. The corresponding sl∞-crystal graph co-
incides with the Young graph, also known as the branching graph of the complex irreducible
representations of symmetric groups.
The following result can be found for example in [21, Section 2.2].
Proposition 4.1. A partition λ is a d-regular partition of n if and only if there exists (i1, . . . , in)
∈ (Z/dZ)n such that
F̃i1,d · · · F̃in,d ·∅ = λ.
In other words, the connected component the ŝld-crystal graph containing the empty partition
is the full subgraph of the ŝld-crystal whose vertices are labelled by d-regular partitions.
The arrows in this component give the branching rule for induction and restriction in the
Hecke algebra of symmetric groups at a primitive d-th root of unity (see [3] for more details).
The partitions λ for which we have Ẽi,d · λ = 0 for all i ∈ Z/dZ are the highest weight vertices.
One can observe that they correspond to partitions of the form λ = µd for some partition µ.
There is also a representation theoretic interpretation of the other components, using the
representation theory of the finite general linear group. Let e be the order of q modulo `,
which we assume to be different from 1. Following [18], one can define a weak Harish-Chandra
theory for unipotent representations of GLn(q) for various n. Recall from Section 3.3 that
these unipotent representations are parametrized by partitions. Consequently, the complexified
Grothendieck group of the category of unipotent representations is naturally isomophic to F .
Under this identification, it follows from [6] that the action of U(ŝle) comes from a truncated
version of Harish-Chandra induction and restriction. As in [13], we deduce that:
• A simple unipotent module S(λ) is weakly cuspidal if and only if λ labels a highest weight
vertex in the ŝle-crystal graph of F .
Alvis–Curtis Duality and a Generalized Mullineux Involution 15
• Two simple modules S(λ) and S(µ) of kGLn(q) lie in the same weak Harish-Chandra
series if and only if there exist a highest weight vertex ν, k ∈ N, (i1, . . . , ik) ∈ (Z/eZ)k and
(j1, . . . , jk) ∈ (Z/eZ)k such that
F̃i1,e . . . F̃ik,e · ν = λ, and F̃j1,e . . . F̃jk,e · ν = µ.
This means that λ and µ are in the same connected component of the associated ŝle-crystal
graph.
4.3 Crystals: the case ` = ∞
Let e ∈ Z>1. Recall from Section 3.1 that any partition λ of n can be decomposed in a unique
way as
λ = λ(−1) t (λ(0))
e,
where λ(−1) is e-regular.
We claim that the entire ŝle-crystal graph structure on the Fock space may be recovered from
the subgraph with vertices labelled by e-regular partitions. Indeed, it follows from the definition
of the Kashiwara operators that for any partition λ
F̃i,eλ = µ ⇐⇒ F̃i,eλ(−1) = µ(−1) and λ(0) = µ(0). (4.1)
One can now define crystal operators F̃i,∞,0 and Ẽi,∞,0 for all i ∈ Z by
F̃i,∞,0λ := µ ⇐⇒ F̃i,∞λ(0) = µ(0) and λ(−1) = µ(−1),
Ẽi,∞,0λ := µ ⇐⇒ Ẽi,∞λ(0) = µ(0) and λ(−1) = µ(−1).
This endows P with an sl∞-crystal structure, which by (4.1) commutes with the ŝle-crystal
structure. Note that the only highest weight with respect to these two structures is the empty
partition. In fact, these constructions already appear in the work of Losev [23] (where the
sl∞-crystal is called the Heisenberg crystal, see also [24, Proposition 4.6]).
4.4 Crystals: the case ` ∈ N
More generally, recall from Section 3.3 that any partition λ of n can be decomposed in a unique
way as
λ = λ(−1) t (λ(0))
e t (λ(1))
e` t · · · t (λ(n))
e`n , (4.2)
where λ(−1) is e-regular and each λ(i) for i > −1 is `-regular. For example, with µ = (22.17),
e = 2 and ` = 3 we obtain µ = (1) t (2)2 t (1)2×3.
As in the previous section, the ŝle-crystal operators act on the e-regular part of partitions,
which makes the definition of other operators possible. Let us fix j ∈ N. We set F̃i,`,jλ = µ
if and only if F̃i,`λ(j) = µ(j) and λ(l) = µ(l) for all l 6= j. Similarly, Ẽi,`,jλ = µ if and only
if Ẽi,`λ(j) = µ(j) and λ(l) = µ(l) for all l 6= j. In other words, F̃i,`,j and Ẽi,`,j are defined
as the usual ŝl`-crystal operators acting on the component λ(j) in the decomposition (4.2), or
equivalently on the (non-necessarily `-regular) partition given by λ(j) t λ`(j+1) t · · · .
As a consequence we obtain an ŝle-crystal structure of level 1 together with many ŝl`-crystal
structures (each of them indexed by an integer j ∈ N, and of level e`j) on the set of partitions.
The following proposition is clear using the decomposition of a partition.
16 O. Dudas and N. Jacon
Proposition 4.2. The above ŝle-crystal structure and the various ŝl`-crystal structures on P
mutually commute.
We can thus define a graph containing all the information on the ŝle-crystal structure and
ŝl`-crystal structures. There is then an obvious notion of highest weight.
Lemma 4.3. The empty partition is the unique highest weight vertex with respect to the above
crystal structure.
In other words, the corresponding crystal graph is connected.
Proof. Assume that λ is a non-empty partition. Then there exists r ∈ Z≥−1 such that λ(r) 6= ∅.
Now
• If r = −1 then λ−1 is e-regular and we have Ẽi,e · λ 6= ∅ for some i ∈ Z.
• If r 6= −1 then λr is `-regular and we have Ẽi,`,r · λ 6= ∅ for some i ∈ Z.
Thus λ is not a highest weight vertex, and ∅ is the only highest weight vertex. �
Example 4.4. Let e = 2 and consider the partition λ = (22.17).
(a) Assume that ` =∞. Then we have λ(−1) = 1 and λ(0) = 2.13. We have
F̃2,∞,0 · λ := (1) t
(
3.13
)2
= 32.17,
F̃0,∞,0 · λ := (1) t
(
22.12
)2
= 24.15,
F̃−4,∞,0 · λ := (1) t
(
2.14
)2
= 22.19.
All the others Kashiwara operators F̃j,∞,0 act by 0 on λ.
(b) Assume that ` = 3. Then we have λ(−1) = 1, λ(0) = 2 and λ(1) = 1.
F̃2,3,0 · λ := (1) t (3)2 t 16 = 32.17
and the action of the other operators F̃j,0,3 are 0. We also have
F̃1,3,1 · λ := (1) t (2)2 t 26 = 28.1,
F̃2,3,1 · λ := (1) t (2)2 t (1.1)6 = 22.113.
One can also consider
F̃1,3,2 · λ = (1) t (3)2 t 16 t 118 = 32.125
or, more generally for k ≥ 2:
F̃1,3,k · λ = (1) t (3)2 t 16 t 13k×2 = 32.12×3k+6.
Remark 4.5. Assume λ is a partition of n such that e` > n. Then it follows from the construc-
tion that λ(k) = ∅ for all k > 0. Note also that λ(0) is a partition of rank strictly less than `,
and therefore it is `-regular. In that case the action of Ẽi,`,0 and Ẽi,∞,0 coincide. In particular,
the ŝl` and sl∞-crystal structures coincide for the partitions of rank less than n.
Alvis–Curtis Duality and a Generalized Mullineux Involution 17
4.5 Crystals and the Mullineux involution
Recall that any d-regular partition λ belongs to the connected component of the empty partition
in the ŝld-crystal graph. The image of λ by the Mullineux involution Md is also in that component
and it can be computed using the following theorem.
Theorem 4.6 (Ford–Kleshchev [14]). Let λ ∈ Regd(n) and let (i1, . . . , in) ∈ (Z/dZ)n such that
F̃i1,d · · · F̃in,d ·∅ = λ
(see Proposition 4.1). Then there exists µ ∈ Regd(n) such that
F̃−i1,d · · · F̃−in,d ·∅ = µ
and µ = Md(λ).
Remark 4.7. When d > n every addable node is good, and Theorem 4.6 implies that Md(λ)=λt,
the conjugate partition (see also Remark 3.1).
From the definition of Me,` (see Definition 3.3) and the construction of the various crystal
operators, Theorem 4.6 generalizes to the following situation.
Proposition 4.8. Let λ be a partition which we write
F̃i1,p1,k1 · · · F̃im,pm,km ·∅ = λ
with for all j = 1, . . . ,m,
• kj ∈ Z≥−1,
• pj = e if kj = −1 and pj = ` otherwise,
• ij ∈ Z/pjZ.
Then
F̃−i1,p1,k1 · · · F̃−im,pm,km ·∅ = Me,`(λ).
Proof. This is clear as the usual Mullineux involution on the set of e-regular partitions is given
in Theorem 4.6. �
Remark 4.9. When ` = ∞, the result remains valid and Me,∞ coincides with the operation
described by Bezrukavnikov in [2] and Losev in [23] (see Example 3.4(b)).
Acknowledgements
The authors gratefully acknowledge financial support by the ANR grant GeRepMod ANR-16-
CE40-0010-01. We thank Gunter Malle, Emily Norton and the referees for their many valuable
comments on a preliminary version of the manuscript.
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1 Introduction
2 Alvis–Curtis duality
2.1 Notation
2.2 Harish-Chandra induction and restriction
2.3 Alvis–Curtis duality functor
2.4 Compatibility with Hecke algebras
3 Unipotent representations of GLn(q)
3.1 Partitions
3.2 Hecke algebras of type A and the Mullineux involution
3.3 Harish-Chandra series of GLn(q) and a generalized Mullineux involution
3.4 Computation of dG(S)
4 Interpretation in terms of crystals
4.1 More on partitions
4.2 Fock space and Kashiwara operators
4.3 Crystals: the case =
4.4 Crystals: the case N
4.5 Crystals and the Mullineux involution
References
|
| id | nasplib_isofts_kiev_ua-123456789-209457 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T16:26:26Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Dudas, O. Jacon, N. 2025-11-21T19:13:05Z 2018 Alvis-Curtis Duality for Finite General Linear Groups and a Generalized Mullineux Involution / O. Dudas , N. Jacon // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 20C20; 20C30; 05E10 arXiv: 1706.04743 https://nasplib.isofts.kiev.ua/handle/123456789/209457 https://doi.org/10.3842/SIGMA.2018.007 We study the effect of Alvis-Curtis duality on the unipotent representations of GLn(q) in non-defining characteristic ℓ. We show that the permutation induced on the simple modules can be expressed in terms of a generalization of the Mullineux involution on the set of all partitions, which involves both ℓ and the order of q modulo ℓ. The authors gratefully acknowledge financial support from the ANR grant GeRepMod ANR-16-CE40-0010-01. We thank Gunter Malle, Emily Norton, and the referees for their many valuable comments on a preliminary version of the manuscript. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Alvis-Curtis Duality for Finite General Linear Groups and a Generalized Mullineux Involution Article published earlier |
| spellingShingle | Alvis-Curtis Duality for Finite General Linear Groups and a Generalized Mullineux Involution Dudas, O. Jacon, N. |
| title | Alvis-Curtis Duality for Finite General Linear Groups and a Generalized Mullineux Involution |
| title_full | Alvis-Curtis Duality for Finite General Linear Groups and a Generalized Mullineux Involution |
| title_fullStr | Alvis-Curtis Duality for Finite General Linear Groups and a Generalized Mullineux Involution |
| title_full_unstemmed | Alvis-Curtis Duality for Finite General Linear Groups and a Generalized Mullineux Involution |
| title_short | Alvis-Curtis Duality for Finite General Linear Groups and a Generalized Mullineux Involution |
| title_sort | alvis-curtis duality for finite general linear groups and a generalized mullineux involution |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209457 |
| work_keys_str_mv | AT dudaso alviscurtisdualityforfinitegenerallineargroupsandageneralizedmullineuxinvolution AT jaconn alviscurtisdualityforfinitegenerallineargroupsandageneralizedmullineuxinvolution |