The Adjoints of the Schur Functor

0.2 MB·

by  Rebecca Reischuk

THE ADJOINTS OF THE SCHUR FUNCTOR
REBECCAREISCHUK
Abstract. We show that the leftand rightadjoint of the Schur functor can
beexpressedintermsofthemonoidalstructureofstrictpolynomialfunctors.
6 Using this result we give a necessary and sufficient condition for when the
1 tensorproductoftwosimplestrictpolynomialfunctorsisagainsimple.
0
2

n
a
Contents
J

0 1. Introduction 1
2 2. Strict polynomial functors 2

3. Representations of the symmetric group and the Schur functor 4
]
T 4. The left adjoint of the Schur functor 5
R 5. The right adjoint of the Schur functor 7
. 6. Comparing both adjoints 10
h
7. The tensor product of simple functors 10
t
a References 13
m
[

2
v 1. Introduction
3
Inhis dissertation,IssaiSchur defines an algebra,nowadaysknownas the Schur
1
5 algebra, whose module category is equivalent to polynomial representations of the
3 generallineargroup. Hethenusesafunctor,nowcalledtheSchurfunctor,torelate
0 representations of the general linear group and representations of the symmetric
.
1 group.
0 For suitable choices of parameters, another category, namely the category of
6 strict polynomial functors, is equivalent to the category of modules over the Schur
1
algebra. This category, first defined by Friedlander and Suslin in [4], inherits a
:
v tensor product from the category of divided powers. A tensor product for the
i
X category of representations of the symmetric group is given by its Hopf algebra
structure. IthasbeenshownrecentlythattheSchurfunctorpreservesthismonoidal
r
a structure ([1]).
The Schur functor has fully faithful left and right adjoints. These adjoints have
been studied in order to relate the cohomolgy of general linear and symmetric
groups (cf. [3]) and to relate (dual) Specht filtrations of symmetric group modules
to Weyl filtrations of modules over the general linear group (cf. [5]).
Weshowthattheleftresp.rightadjointoftheSchurfunctorcanbeexpressedin
termsoftheinternaltensorproductresp.internalhomofstrictpolynomialfunctors
(Theorem 4.3 resp. Theorem 5.4). We make use of these expressions to relate the
two adjoints. In addition, we will see that the adjoints induce equivalences of
categories when restricting to the subcategories of injective resp. projective strict
polynomial functors.
Inthelastsectionweconsiderthetensorproductoftwosimplestrictpolynomial
functors. Touz´e showed in [10] that in almost all cases such a tensor product is
1
2 REBECCAREISCHUK
not simple. We use the left adjoint of the Schur functor to calculate the remaining
cases. As a consequence we get a necessary and sufficient condition in terms of
Ext-vanishing between certain simple functors for when the tensor product of two
simple strict polynomial functors is simple (Theorem 7.4). In the case n=d=p a
full characterizationis given (Theorem 7.7).
Acknowledgements. IwouldliketothankKarinErdmannforvaluablecomments
and discussions about representations of the symmetric group during a research
visit in Oxford. In particular the results in the last section were completed with
her assistance. I am very grateful to Greg Stevenson for many fruitful discussions
and his continuous advice on (monoidal) categories. I am thankful to Nicholas
Kuhn for comments on an earlier version of this paper.
2. Strict polynomial functors
Inthefollowingwebrieflyrecallthedefinitionsofstrictpolynomialfunctorsand
oftheinternaltensorproductasdescribedin[6,Section2]. Letkbeacommutative
ring and denote by P the category of finitely generatedprojective k-modules. For
k
d∈N denote by S the symmetric grouppermuting d elements and for V ∈P let
d k
S act on the right on V⊗d by permuting the factors.
d
Divided, symmetric and exterior powers. The submodule ΓdV consisting of
theS -invariantpartofV⊗discalledthemoduleofdividedpowers. Thecoinvariant
d
part is the module of symmetric powers, denoted by SdV. The quotient of V⊗d
by the ideal generated by v⊗v are the exterior powers ΛdV. Since the k-modules
ΓdV, SdV and ΛdV are free provided V is free, sending a module V to ΓdV, SdV
resp. ΛdV yields functors Γd,Sd,Λd: P →P .
k k
The category of degree d divided powers. We define the categoryΓdP to be
k
the category with the same objects as P and where the morphisms between two
k
objects V and W are given by
HomΓdPk(V,W):=ΓdHom(V,W)=(Hom(V,W)⊗d)Sd.
The category of strict polynomial functors. Finally we define RepΓd to be
k
the category of k-linear representations of ΓdP , i.e.
k
RepΓd =Fun (ΓdP ,M ),
k k k k
whereM denotesthecategoryofallk-modules. Themorphismsbetweentwostrict
k
polynomial functors X,Y are denoted by Hom (X,Y).
Γd
k
The full subcategoryoffinite representations, i.e. X ∈RepΓd suchthat X(V)∈
k
P for all V ∈ΓdP , is denoted by repΓd.
k k k
The strict polynomial functor represented by V ∈ΓdP is given by
k
Γd,V :=Hom (V,−).
ΓdPk
For X ∈RepΓd, the Yoneda isomorphism yields
k
(2.1) HomΓd(Γd,V,X)∼=X(V).
k
We have Γd,k =HomΓdPk(k,−)=ΓdHom(k,−)∼=Γd(−) and thus Γd ∈RepΓdk. It
is not hard to see that also Sd,Λd ∈RepΓd.
k
THE ADJOINTS OF THE SCHUR FUNCTOR 3
External tensor product. For non-negative integers d,e and X ∈ RepΓd and
k
Y ∈RepΓe we can form the external tensor product
k
X ⊠Y ∈RepΓd+e.
k
It is given on objects by (X ⊠Y)(V) = X(V)⊗Y(V) and on morphisms via the
map
Γd+eHom(V,W)→ΓdHom(V,W)⊗ΓeHom(V,W).
In particular, for positive integers n,d and a composition λ = (λ ,λ ,...,λ )
1 2 n
of d in n parts, i.e. an n-tuple of non negative integers such that λ = d, we
i i
canformrepresentablefunctorsΓλ1,k ∈RepΓλ1,...,Γλn,k ∈RepΓλn andtaketheir
k k P
tensor product to obtain a functor in RepΓd
k
Γλ :=Γλ1 ⊠···⊠Γλn.
In the same way define
Sλ :=Sλ1 ⊠···⊠Sλn
Λλ :=Λλ1 ⊠···⊠Λλn.
RepresentationsofSchuralgebras. Forn,dpositiveintegers,theSchuralgebra
can be defined as
Sk(n,d)=EndSd((kn)⊗d)=EndΓdk(Γd,kn)op.
If n ≥ d there is an equivalence of categories ([4, Theorem 3.2] and [6, Theorem
2.10])
(2.2) RepΓdk −∼=→ModEndΓd(Γd,kn)∼=Sk(n,d)Mod,
k
given by evaluating at kn, i.e. a strict polynomial functor X is mapped to X(kn).
The internal tensor product of strict polynomial functors. For V,W in P
k
denote by V ⊗ W the usual tensor product of k-modules. This induces a tensor
k
product on ΓdP , the category of divided powers. It coincides on objects with the
k
one for P and on morphisms it is given via the following composite:
k
ΓdHom(V,V′)×ΓdHom(W,W′)→Γd(Hom(V,V′)⊗Hom(W,W′))
−∼→ΓdHom(V ⊗W,V′⊗W′).
By Day convolution, this in turn yields an internal tensor product on RepΓd,
k
suchthattheYonedafunctorisclosedstrongmonoidal. Itisgivenforrepresentable
functors Γd,V and Γd,W in RepΓd by
k
Γd,V ⊗ Γd,W :=Γd,V⊗W.
Γd
k
For arbitrary objects it is given by taking colimits, see [6, Proposition 2.4] for
more details.
The tensor unit is given by
1Γdk :=Γd,k ∼=Γ(d).
In the same way, RepΓd is equipped with an internal hom, defined on repre-
k
sentable objects by
Hom (Γd,V,Γd,W):=Γd,Hom(V,W).
Γd
k
Theinternalhomisindeedanadjointtotheinternaltensorproduct,i.e.wehave
a natural isomorphism (cf. [6, Proposition 2.4])
HomΓd(X ⊗Γd Y,Z)∼=HomΓd(X,HomΓd(Y,Z)).
k k k k
4 REBECCAREISCHUK
We will omit the indices and write −⊗− and Hom(−,−) whenever it is clear
which category is considered.
Dualities. The category of strict polynomial functors admits two kinds of dual,
onecorrespondingtothetransposedualityformodulesoverthegenerallineargroup
and the other one using the internal hom structure of RepΓd.
k
The Kuhn dual. For X ∈ RepΓd it is defined by X◦(V) := X(V∗)∗ where (−)∗ =
k
Hom (−,k) denotes the usual dual in P . Taking the Kuhn dual is a contravari-
k k
ant exact functor, sending projective objects to injective objects and vice versa.
Symmetric powers are duals of divided powers, i.e. (Γd)◦ =Sd and more generally
(Γλ)◦ =Sλ. Exterior powers are self-dual, i.e. (Λλ)◦ =Λλ.
The monoidal dual. It is defined for X ∈ RepΓd by X∨ := Hom (X,Γd). This
k Γd
k
functor is left exact, but in general not right exact.
Lemma 2.1. [6, Lemma 2.7 and Lemma 2.8] For all X,Y ∈ RepΓd we have a
k
natural isomorphism
HomΓdPk(X,Y◦)∼=HomΓdPk(Y,X◦).
If X is finitely presented we have natural isomorphisms
X ⊗Γd Y◦ ∼=HomΓd(X,Y)◦
k k
(X ⊗Γd Y)◦ ∼=HomΓd(X,Y◦).
k k
We collect some important calculations:
(2.3) X ⊗Γd ∼=X
(2.4) Sd⊗Sd ∼=Sd
(2.5) (Γd)∨ =Hom(Γd,Γd)∼=(Γd⊗Sd)◦ ∼=Γd
(2.6) Γλ⊗Sd ∼=Sλ
(2.7) Hom(Sd,Sλ)∼=Sλ
3. Representations of the symmetric group and the Schur functor
Recall that S is the symmetric group permuting d elements. The representa-
d
tions of S , i.e. (left) modules over its group algebra will be denoted by kS Mod.
d d
Define kS mod to be the subcategory of modules that are finitely generated pro-
d
jective over k.
Partitions. We will denote by Λ(n,d):={λ=(λ ,...,λ )| λ =d} the set of
1 n i
all compositions of d into n parts. Those compositions that are weakly decreasing,
i.e. λ ≥ λ ≥ ··· ≥ λ ≥ 0, are called partitions and denotePd by Λ+(n,d). The
1 2 n
subsetofp-restrictedpartitions,i.e.λ∈Λ+(n,d)withλ −λ <p,aredenotedby
i i+1
Λ+(n,d). A sequence (i ...i ) belongs to λ, denoted as (i ...i )∈λ, if (i ...i )
p 1 d 1 d 1 d
has λ entries equal to l.
l
Permutation modules. Fixabasise ,...,e ofknandconsiderthed-foldtensor
1 n
product (kn)⊗d. It becomes a left kS -module by defining the module action via
d
σ(v1⊗···⊗vd):=vσ−1(1)⊗···⊗vσ−1(d).
forσ ∈S andv ⊗···⊗v ∈(kn)⊗d. Itdecomposesintoadirectsumoftransitive
d 1 d
permutation modules
(3.1) (kn)⊗d = Mλ,
λ∈MΛ(n,d)
THE ADJOINTS OF THE SCHUR FUNCTOR 5
where Mλ is the k-span of the set {e ⊗···⊗e | (i ...i )∈λ}.
i1 id 1 d
The internal tensorproductofrepresentationsofsymmetricgroups. The
Hopfalgebrastructure ofthe groupalgebrakS endowskS Modwith aninternal
d d
tensor product, the so-calledKroneckerproduct. For N, N′ ∈kS Mod it is given
d
by taking the usual tensor product over k, denoted by N ⊗ N′, together with the
k
following diagonal action of σ ∈S :
d
σ·(n⊗n′)=σn⊗σn′.
The tensor unit is given by M(d) ∼=k, the trivial kSd-module.
From the antipode which is defined by S(σ) = σ−1 for σ ∈ S , we also get an
d
internal hom, denoted by Hom (N,N′). It is given by taking k-linear morphisms
k
Hom (N,N′) togetherwith the following actionforσ ∈S , f ∈Hom (N,N′) and
k d k
n∈N:
σ·f(n)=σf(S(σ)n)=σf(σ−1n).
pDruovailditeys.aBdyuasleNtti∗ng:=NH′o:m= 1(NkS,kd)=. Tkh,etahcetitornivbiaelcokmSeds-module, the internal hom
k
σ·f(n)=σf(σ−1n)=f(σ−1n).
The Schur functor. The Schur functor was originally defined from representa-
tions of the Schur algebra to representationof the symmetric group. Via the equi-
valence (2.2) this translates to a functor from the category of strict polynomial
functors to the representations of the symmetric group.
Let ω =(1,...,1)∈Λ(d,d) be the partition with d entries equal to 1. There is
(cf. [7, Section 4]) an algebra isomorphism
EndΓd(Γω)∼=kSodp,
k
which identifies the module categories kS Mod and ModEnd (Γω). In the fol-
d Γd
k
lowing we will write End(Γω) instead of End (Γω).
Γd
k
Definition 3.1. The Schur functor, denoted by F, is defined as
F :=HomΓd(Γω,−): RepΓdk →ModEnd(Γω)∼=kSdMod.
k
An equivalence of categories. Let Γ = {Γλ} , M = {Mλ} and
λ∈Λ(n,d) λ∈Λ(n,d)
S = {Sλ} . Denote by addΓ resp. addS the full subcategory of RepΓd
λ∈Λ(n,d) k
whose objects are direct summands of finite direct sums of Γλ resp. Sλ. Define
addM similarly as a subcategory of kS Mod. In [1, Lemma 4.3] it is shown that
d
the functor F =Hom (Γω,−) induces an equivalence of categories between addΓ
Γd
k
andaddM. BytakingdualswegetthatF alsoinducesanequivalenceofcategories
between addS and addM.
NotethataddΓisthesubcategoryofconsistingofallfinitelygeneratedprojective
objects and addS is the subcategory of finitely generated injective objects.
4. The left adjoint of the Schur functor
In this section we present the connection between the left adjoint of the Schur
functor and the monoidal structure of strict polynomial functors. In addition we
show that the left adjoint induces an equivalence between some subcategories of
S Mod resp. of RepΓd.
d k
LetN ∈ModEnd(Γω) andX ∈RepΓd. By the usualtensor-homadjunctionwe
k
get the following isomorphism
HomΓd(N ⊗End(Γω)Γω,X)∼=HomEnd(Γω)(N,HomΓd(Γω,X))
k k
=Hom (N,F(X)).
End(Γω)
6 REBECCAREISCHUK
Thus, F has a left adjoint, namely
G⊗: ModEnd(Γω)→RepΓdk
N 7→N ⊗ Γω,
End(Γω)
in terms of modules for the symmetric group algebra this reads
G⊗: kSdMod→RepΓdk
N 7→(−)⊗d⊗kSd N.
We will denote the unit by η⊗: idEnd(Γω) → FG⊗ and the counit by ε⊗: G⊗F →
idRepΓd and omit indices where possible. Note that G⊗ is fully faithful, hence the
k
unit η⊗ is an isomorphism, i.e. FG⊗(X)∼=X for all X ∈RepΓdk.
We are now interested in the composition G⊗F.
Proposition 4.1. There is a natural isomorphism
G⊗F(X)∼=X
for all X ∈addS.
Proof. Let V ∈ ΓdP and X ∈addS. Using Lemma 2.1, the Yoneda isomorphism
k
(2.1), and the equivalence of addS and addM we get the following sequence of
isomorphisms,
(X)◦(V)∼=HomΓd(Γd,V,(X)◦)
k
∼=HomΓd(X,(Γd,V)◦)
k
∼=HomkSd(F(X),F((Γd,V)◦))
∼=HomΓd(G⊗F(X),(Γd,V)◦)
k
∼=HomΓd(Γd,V,(G⊗F(X))◦)
k
∼=(G⊗F(X))◦(V)
and thus G⊗F(X)∼=X. (cid:3)
Corollary 4.2. The functor G⊗ restricted to addM is an inverse of F|addS, i.e.
we have the following equivalences of categories
rr G⊗
addS 22 addM (cid:3)
F
If we do not restrict to the subcategory addS, the composition G⊗F is not
isomorphic to the identity. Though we have the following result:
Theorem 4.3. There is a natural isomorphism
G⊗F(X)∼=Sd⊗Γd X.
k
Proof. By [1, Theorem 4.4] the functor F is monoidal and thus there is a natural
isomorphism
Φ : F(X)⊗ F(Y)→F(X⊗ Y).
X,Y k Γd
k
Using this isomorphism and by adjunction we get a sequence of isomorphisms
HomkSd(F(X)⊗kN,F(X)⊗kN)∼=HomkSd(F(X)⊗kN,F(X)⊗kFG⊗(N))
∼=HomkSd(F(X)⊗kN,F(X ⊗Γdk G⊗(N)))
∼=HomΓd(G⊗(F(X)⊗kN),X ⊗Γd G⊗(N)).
k k
THE ADJOINTS OF THE SCHUR FUNCTOR 7
Thus, the identity on F(X)⊗ N yields a map ϑ which is given by
k X,N
ϑX,N :=(ε⊗)◦G⊗(Φ◦(id⊗kη⊗)): G⊗(F(X)⊗kN)→X⊗Γd G⊗(N).
k
By setting N := , the trivial module, we get a map
1
ϑX,1: G⊗F(X)→X⊗Γdk G⊗(1).
We will show that it is an isomorphism. Since G⊗F(−) and −⊗Γdk G⊗(1) are right
exact functors it is enough to show that ϑ is an isomorphism for X projective.
X,
Thus,let X =Γλ. Since F(Sd)∼= , we know1by Proposition4.1that G⊗( )∼=Sd.
It follows that Γλ⊗Γdk G⊗(1)∼=Sλ1by (2.6) and that 1
(ε⊗)Γλ⊗ΓdkG⊗(1): G⊗F(Γλ⊗Γdk G⊗(1))→Γλ⊗Γdk G⊗(1)
is an isomorphism by Corollary 4.2. Both maps Φ and η⊗ are isomorphisms and
thus G⊗(Φ ◦ (id⊗k η⊗)) is an isomorphism. It follows that ϑΓλ, : G⊗F(Γλ) →
Γλ ⊗Γdk G⊗(1) is an isomorphism. Identifying G⊗(1) with Sd we1get the desired
isomorphism
∼
ϑX, : G⊗F(X)−=→X ⊗Γd Sd ∼=Sd⊗Γd X. (cid:3)
1 k k
Recall from (2.4) that Sd ∼= Sd⊗Sd. Thus, by using the fact that F preserves
the monoidal structure, we get the following
Corollary 4.4. The functor G⊗ is compatible with the tensor product, i.e.
G⊗(N ⊗kN′)∼=G⊗(N)⊗Γd G⊗(N′). (cid:3)
k
theHtoewnseovreru,nnitotientRheaptΓthd.eUtesninsogrLuenmitm1akS2.d1,iswmeagpeptetdheufnodlleorwGin⊗gtdoeSscdriwphtiiocnhoisfnthoet
k
Schur functor composed with its left adjoint:
Corollary 4.5. We can express the endofunctor G⊗F by duals, namely
G⊗F(X)∼=Sd⊗Γd X ∼=Hom(X,Γd)◦ =(X∨)◦ (cid:3)
k
5. The right adjoint of the Schur functor
This section provides analogous results to those in the preceeding section, now
for the right adjoint of the Schur functor. In particular, we will see how the right
adjoint can be expressed in terms of the monoidal structure of strict polynomial
functors.
Let V ∈ ΓdP , X ∈ RepΓd and N ∈ ModEnd(Γω). We write End(Γd,V)
k k
for End (Γd,V) and consider Hom (Γd,V,X) as a right End(Γd,V)-module and
Γd Γd
k k
Hom (Γω,Γd,V) as an End(Γd,V)-End(Γω)-bimodule. By the usual tensor-hom
Γd
k
adjunction we then get the following isomorphism
Hom (Hom (Γd,V,X),Hom (Hom (Γω,Γd,V),N))
End(Γd,V) Γd End(Γω) Γd
k k
∼=HomEnd(Γω)(HomΓd(Γd,V,X)⊗End(Γd,V)HomΓd(Γω,Γd,V),N).
k k
On the other hand, since Hom (Γd,V,Γω) is finitely generated projective over
Γd
k
End(Γd,V), we also have
HomEnd(Γd,V)(Γω(V),X(V))∼=HomEnd(Γd,V)(HomΓd(Γd,V,Γω),HomΓd(Γd,V,X))
k k
∼=HomΓd(Γd,V,X)⊗End(Γd,V)HomΓd(Γω,Γd,V)
k k
and thus
Hom (Hom (Γd,V,X),Hom (Hom (Γω,Γd,V),N))
End(Γd,V) Γd End(Γω) Γd
k k
∼=HomEnd(Γω)(HomEnd(Γd,V)(Γω(V),X(V)),N).
8 REBECCAREISCHUK
Since ModEnd(Γd,V) ∼= RepΓd for V := kn with n ≥ d by (2.2) and X ∼=
k
Hom (Γd,−,X) this isomorphism finally becomes
Γd
k
Hom (X,Hom (Hom (Γω,Γd,−),N))
Γd End(Γω) Γd
k k
∼=HomEnd(Γω)(HomΓd(Γω,X),N).
k
Thus, F =Hom (Γω,−) has a right adjoint, namely
Γd
k
G : ModEnd(Γω)→RepΓd
Hom k
N 7→Hom (Hom (Γω,Γd,−),N)
End(Γω) Γd
k
in terms of modules for the symmetric group algebra this reads
G : kS Mod→RepΓd
Hom d k
N 7→HomkSd(HomkSd((−)⊗d,kSd),N).
We will denote the unit by η : id → G F and the counit by
Hom End(Γω) Hom
ε : FG → id . Note that G is fully faithful, hence the counit ε
Hom Hom RepΓd Hom Hom
k
is an isomorphism, i.e. FG (X)∼=X for all X ∈RepΓd.
Hom k
Again, we are interested in the composition G F. We have the following
Hom
result, dual to Proposition 4.1:
Proposition 5.1. There is a natural isomorphism
G F(X)∼=X
Hom
for all X ∈addΓ.
Proof. LetV ∈ΓdP andX ∈addΓ. DuetothetheYonedaisomorphism(2.1)and
k
theequivalenceofaddΓandaddM wehavethefollowingsequenceofisomorphisms
X(V)∼=Hom (Γd,V,X)
Γd
k
∼=HomkSd(F(Γd,V),F(X))
∼=HomΓd(Γd,V,GHomF(X))
k
∼=G F(X)(V)
Hom
and thus G F(X)∼=X. (cid:3)
Hom
Corollary 5.2. The functor G restricted to addM is an inverse of F| , i.e.
Hom addΓ
we have the following equivalences of categories
F ,,
addΓ ll addM (cid:3)
GHom
Remark 5.3. Supposethatk isafieldofcharacteristic≥5. In[5,Theorem3.8.1.]
it is shown that on Filt(∆), the full subcategory of Weyl filtered modules, G is
Hom
aninversetoF. ThesubcategoryFilt(∆)containsthesubcategoryaddΓ,sointhe
case of a field of characteristic ≥ 5, Corollary 5.2 follows also from [5]. But note
that Corollary 5.2 is independent of any assumption on the commutative ring k.
If we do not restrict to the subcategory addΓ, the composition G F is not
Hom
isomorphic to the identity. Though we have the following result, dual to Theorem
4.3:
Theorem 5.4. There is a natural isomorphism
G F(X)∼=Hom(Sd,X).
Hom
THE ADJOINTS OF THE SCHUR FUNCTOR 9
Proof. ByusingthefactthatF ismonoidal([1,Theorem4.4])andsomeadditional
calculations, one can show that F is also a closed functor, i.e. there is a natural
isomorphism
ΨX,Y : HomkSd(F(X),F(Y))→F(HomRepΓdk(X,Y)).
Using this isomorphism and by adjunction we get a sequence of isomorphisms
HomkSd(Hom(F(X◦),N),Hom(F(X◦),N))
∼=HomkSd(Hom(F(X◦),N),Hom(N∗,F(X◦)∗))
∼=HomkSd(Hom(F(X◦),FGHom(N)),Hom(N∗,F(X)))
∼=HomkSd(F(Hom(X◦,GHom(N))),Hom(N∗,F(X)))
∼=HomΓd(Hom(X◦,GHom(N)),GHom(Hom(N∗,F(X))))
k
∼=HomΓd(Hom(GHom(N)◦,X),GHom(Hom(N∗,F(X)))).
k
Thus, the identity on Hom(F(X◦),N) yields a map
κ : Hom(G (N)◦,X)→G (Hom(N∗,F(X◦))).
N,X Hom Hom
By setting N := , the trivial module, we get a map
1
κ : Hom(G ( )◦,X)→G (Hom( ,F(X))).
,X Hom Hom
1 1 1
Similarly to the case of G⊗ this is an isomorphism. This time, we use the fact
that since Hom(G ( ),−) and G (Hom( ,F(−))) are left exact functors it is
Hom Hom
enoughtoshowthatκ1,X isanisomorphismfo1rX =Sλ injective. ButF(Γd)∼= ,
thus we know by Prop1osition 5.1 that G ( )∼=Γd. It follows that 1
Hom
1
Hom(G ( )◦,Sλ)∼=Γλ
Hom
1
by (2.7) and hence
(ηHom)Hom(GHom(1)◦,Sλ): Hom(GHom(1)◦,Sλ)→GHomF(Hom(GHom(1)◦,Sλ))
is an isomorphism by Corollary 5.2. Similarly as before κ is the composition
,X
1
of this isomorphism and further isomorphisms, hence is itself and isomorphism.
Identifying G ( )◦ with Sd and Hom( ,F(X)) with F(X) we finally get the
Hom
1 1
desired isomorphism
κ : Hom(Sd,X)→G F(X). (cid:3)
,X Hom
1
Corollary 5.5. The functor G preserves the internal hom up to duality, i.e.
Hom
G F(Hom(X◦,Y))∼=Hom(G F(X)◦,G F(Y))
Hom Hom Hom
and
G Hom(N∗,N′)∼=Hom(G (N)◦,G (N′)). (cid:3)
Hom Hom Hom
Remark 5.6. In general, G does not preserve the internal tensor product, e.g.
Hom
defining sign to be the sign-representation in kSdMod we get GHom(sign) ∼= Λd if
2 is invertible in k, but G ( )∼=Γd and thus
Hom
1
GHom(sign⊗ksign)=GHom(1)∼=Γd 6=Sd ∼=Λd⊗Γdk Λd
∼=GHom(sign)⊗Γd GHom(sign).
k
Corollary 5.7. Let X ∈ repΓd, i.e. X◦◦ ∼= X. We can express the endofunctor
k
G F by duals, namely
Hom
G F(X)∼=Hom(Sd,X)∼=Hom(X◦,Γd)=(X◦)∨. (cid:3)
Hom
10 REBECCAREISCHUK
6. Comparing both adjoints
The results in the previous two sections allow us to relate the left and the right
adjoint. In the case of k a field of characteristic p, this has already been done in a
more general setting by N. Kuhn in [8, Theorem 6.10, Lemma 6.11].
In our setting, k is still an (arbitrary) commutative ring and we obtain
Proposition 6.1. The left and the right adjoints of the Schur functor are related
by taking duals, namely
(G⊗◦F(X))◦ ∼=GHom◦F(X◦)
and
G⊗(N)◦ ∼=GHom(N∗)
for all X ∈RepΓd and N ∈kS Mod.
k d
Proof. Using Theorem 5.4 and Theorem 4.3 we get
(G⊗◦F(X))◦ ∼=(Sd⊗X)◦ ∼=Hom(Sd,X◦)∼=GHom◦F(X◦).
By setting N :=F(X) and using the fact that F(X◦)∼=F(X)∗ we get the second
isomorphism. (cid:3)
We have the following commutative diagram
rr G⊗
repΓdk 11kSdmod
F
(−)◦ (−)∗
(cid:15)(cid:15) F .. (cid:15)(cid:15)
(repΓdk)op mm (kSdmod)op
GHom
where the vertical arrows are equivalences of categories. The horizontal arrows
become equivalences when restricted to the following subcategories
rr G⊗
addS 22 addM
F
(−)◦ (−)∗
(cid:15)(cid:15) F -- (cid:15)(cid:15)
(addΓ)op mm (addM)op
GHom
7. The tensor product of simple functors
If k is a field, the isomorphism classes of simple functors in RepΓd are indexed
k
by partitions λ ∈Λ+(n,d). Simple functors are self-dual, i.e. L◦λ ∼=Lλ, see e.g. [7,
Proposition 4.11].
In [8, Theorem7.11]a generalizedSteinberg Tensor Product Theoremis proved
thatstatesthatsimple functorsaregivenby the externaltensorproductoftwisted
simple functors. In our setting, this has been formulated also by Touz´e:
Theorem 7.1 ([10, Theorem 4.8]). Let k be a field of charachteristic p. Let
λ0,...,λr be p-restricted partitions, and let λ = r piλi. There is an isomor-
i=0
phism:
P
Lλ ∼=Lλ0 ⊠L(λ11)⊠···⊠Lλ(rr),
where (−)(i) denotes the i-th Frobenius twist.