Canonical basic sets for Hecke algebras

0.17 MB·English

by  Nicolas Jacon

5
0
0
2 Canonical basic sets for Hecke algebras

n
a
Nicolas Jacon
J

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1 Abstract. We give an explicit description of the “canonical basic set” for
allIwahori-Heckealgebras offiniteWeyl groups in“good”characteristic. We

] obtainacompleteclassificationofsimplemodulesforthistypeofalgebras.
T
R
.
h
t 1. Introduction
a
m Let W be a finite Weyl group with set of simple reflections S ⊂ W and let
[ H be the generic Iwahori-Hecke algebra of W over A = Z[v,v−1], where v is an
indeterminate. Letu:=v2. H hasabasis{Tw |w∈W}andwehavethefollowing
1
multiplication rules. Let s∈S and w∈W, then:
v
5
T if l(sw)>l(w),
sw
2 T T =
s w
2 (uTsw+(u−1)Tw if l(sw)<l(w),
1
where l is the usual length function. Let K be the field of fractions of A and let
0
θ :A→k be a homomorphism into a field k such that k is the field of fractions of
5
0 θ(A) and such that θ(v) has finite order.
/ Let H := K ⊗ H and let H := k⊗ H. It is known that H is a split
h K A k A K
t semi-simplealgebra,isomorphictothegroupalgebraK[W]andthatthesimpleHK-
a
modulesareinnaturalbijectionwiththesimplemodulesofK[W]. Theproblemof
m
determiningaparametrizationofthesimpleH -modulesismuchmorecomplicated.
k
v: To solve this problem, it is convenient to use the notion of decomposition map,
Xi which relates the simple HK-modules with the simple Hk-modules via a process
of modular reduction. We obtain a well-defined decomposition map between the
r
Grothendieck groups of finitely generated H -modules and H -modules:
a K k
d :R (H )→R (H ).
θ 0 K 0 k
Assumethatthecharacteristicofkiseither0oragoodprimeforW. Then,in[G3]
and in [GR2], M.Geck and R.Rouquier have defined a canonical set B ⊂ Irr(H )
K
by using Lusztig’s a-function. This set is called the “canonical basic set” and it is
in natural bijection with Irr(H ). Hence, it gives a way to parametrize the simple
k
H -modules. Moreover, the existence of the canonical basic set implies that the
k
decompositionmatrixofd has alowertriangularshape with1alongthe diagonal.
θ
In characteristic 0, the canonical basic set has been completely described for
typeA in[G3],fortypeB in[J3]andfortypeD in[G2]and[J1]. Moreover,
n−1 n n
1991 Mathematics Subject Classification. Primary20C08;Secondary20C20.
Key words and phrases. Modularrepresentationtheory,Heckealgebras.
1
2 NICOLASJACON
this set can be easily deduced for the exceptional types from the explicit tables of
decomposition numbers obtained by M.Geck, K.Lux and J.Mu¨ller. The aim of
this paper is to report these results and to show that the parametrization of B
in characteristic 0 holds in “good” characteristic. We note that the existence of
“basicsets”has beenalsoprovedfor the classofcyclotomic Heckealgebrasoftype
G(d,1,n) in [J3] and recently, for the class of cyclotomic Hecke algebras of type
G(d,p,n) in [GJ].
The paper is organized as follows. In the first part, we recall the definition of
the canonicalbasic set. Next, in the second part,we give some useful properties of
thedecompositionmapandweshowthattheproblemofdeterminingthecanonical
basic set can be reduced to the case of characteristic 0. We finally give an explicit
description of the canonical basic set for all Hecke algebras of finite Weyl groups
and for all specializations.
2. Existence of canonical basic sets
2.1. Decomposition maps. Let H be an Iwahori-Hecke algebra of a finite
WeylgroupW overA:=Z[v,v−1]asitisdefinedintheintroduction. LetK =Q(v)
and let H be the corresponding Hecke algebra. Then A is integrally closed in K
K
and H is a split semi-simple algebra. Let θ : A → k be a specialization into a
K
field k such that k is the field of fractions of θ(A) and such that θ(v) has finite
order. We assume that the characteristic of k is 0 or a good prime number for W.
Then, there exists a discrete valuation ring O with maximal ideal J(O) such that
A ⊂ O and J(O)∩A = kerθ. By [GP, Theorem 7.4.3], we obtain a well-defined
decomposition map
d :R (H )→R (H ).
θ 0 K 0 k
This is defined as follows: let V be a simple H -module. Then, by [GP, section
K
7.4], there exists a H -module V such that K⊗ V =V. By reducing V modulo
O O
the maximal ideal of O, we obtain a H -module where k is the residue field of
kO O
O. We obtain a map: b b b
d′ :R (H )→R (H ).
θ 0 K 0 kO
Since H is split and since k can be seen as an extension field of k, we can
k O
identify R (H ) with R (H ). We obtain the desired decomposition map d be-
0 kO 0 k θ
tween R (H ) and R (H ). Moreover, for V ∈ Irr(H ), there exist numbers
0 K 0 k K
(d ) such that:
V,M M∈Irr(Hk)
d ([V])= d [M].
θ V,M
M∈XIrr(Hk)
The matrix (d ) is called the decomposition matrix. For more details
V,M V∈Irr(HK)
M∈Irr(Hk)
about the construction of decomposition maps, see [G4].
2.2. Canonical basic sets. In this part, we recall the results of [G3] and
[GR2] which show that the above decomposition matrix has always a lower uni-
triangular shape. First, we need to attach non negative integers which are called
“a-values” to the simple modules of H and H as follows.
K k
Let {C } be the Kazhdan-Lusztig basis of H. For x,y ∈W, the multipli-
w w∈W
cation between two elements of this basis is given by:
C C = h C
x y x,y,z z
z∈W
X
CANONICAL BASIC SETS FOR HECKE ALGEBRAS 3
where h ∈ A for all z ∈ W. For any z ∈ W, there is a well-defined integer
x,y,z
a(z)≥0 such that
va(z)h ∈Z[v] for all x,y ∈W,
x,y,z
va(z)−1h ∈/ Z[v] for some x,y ∈W.
x,y,z
We obtain a function which is called the Lusztig’s a-function:
a: W → N
z 7→ a(z)
Now, following [L, Lemma 1.9], to any M ∈ Irr(H ), we can attach an a-value
k
a(M) by the requirement that:
C .M =0 for all w∈W with a(w)>a(M),
w
C .M 6=0 for some w ∈W with a(w)=a(M).
w
We can also attach an a-value a(V) to any V ∈Irr(H ), in an analogous way.
K
We cannowgivethe theoremofexistenceofthe canonicalbasic set. The main
tool of the proof is the Lusztig’s asymptotic algebra.
Theorem 2.1 (M.Geck [G3], M.Geck-R.Rouquier [GR2]). Recall that we as-
sumethatthecharacteristicofkis0oragoodprimeforW. Wedefinethefollowing
subset of Irr(H ):
K
B :={V ∈Irr(H ) | d 6=0 and a(V)=a(M) for some M ∈Irr(H )}.
K V,M k
Then there exists a unique bijection
Irr(H ) → B
k
M 7→ V
M
such that the following two conditions hold:
(1) For all V ∈B, we have d =1 and a(V )=a(M).
M VM,M M
(2) If V ∈ Irr(H ) and M ∈ Irr(H ) are such that d 6= 0, then we have
K k V,M
a(M)≤a(V), with equality only for V =V .
M
The set B is called the canonical basic set with respect to the specialization θ.
Hence, to find the elements of the canonical basic set, for each M ∈ Irr(H ),
k
we have to search for V ∈Irr(H ) such that a(V )=a(M) and d 6=0.
M K M VM,M
Note thata descriptionofthe setB wouldleadto anaturalparametrizationof
the set of simple H -modules. If H is semi-simple, we know by Tits deformation
k k
theorem that the decomposition matrix is just the identity. Hence, we obtain the
following result.
Proposition2.2. Assumethatθ issuchthatH isasplit semi-simplealgebra.
k
Then, we have:
B =Irr(H ).
K
We now want to give an explicit description of B in the non semi-simple case.
By [GP, Theorem 7.4.7], H is semi-simple unless θ(u) is a root of unity. Thus,
k
we can restrict ourselves to the case where θ(u) is a root of unity. In the next
section, we will see that it is sufficient to know the canonical basic set when the
characteristic of k is 0.
4 NICOLASJACON
3. Canonical basic sets in positive characteristic
In this section, we assume that p is a “good” prime number for W. Let θ :
p
A→k beaspecializationintoafieldk ofcharacteristicpsuchthatk isthefield
p p p
of fractions of θ (A). We obtain a decomposition map
p
d :R (H )→R (H ).
θp 0 K 0 kp
Let Bp be the canonical basic set associated to θ as it is defined in Theorem 2.1.
p
By Proposition 2.2, we can assume that θ (u) is a root of unity. We put:
p
e:=min{i≥2 | 1+θ (u)+θ (u)2+...+θ (u)i−1 =0}∈N.
p p p
Wefirstshowthatthedecompositionmatrixofd canbeobtainedintwosteps:
θp
one step from u to a eth-root of unity over C and another step from characteristic
0 to characteristic p.
Following [G5], we denote p:=kerθ . Let Φ (u)∈Z[u] be the eth cyclotomic
p e
polynomial. We have Φ (u)∈p and :
e
Φ (u)= Φ (v) if e is even,
e 2e
Φ (u)= Φ (v)Φ (v) and Φ (v)=±Φ (−v) if e is odd.
e e 2e 2e e
Thus,choosingasuitablesquarerootofθ (u)ink ,wecanassumethatΦ (v)∈p.
p p 2e
Letq⊂AbetheprimeidealgeneratedbyΦ (v). WehaveA/q≃Z[η ]whereη
2e 2e 2e
isaprimitive2eth rootofunity. Then,sinceA/qisintegrallyclosedink :=Q(η )
0 2e
andsince H is split,the naturalmapθ :A→A/q induces adecompositionmap
k0 0
d :R (H )→R (H ).
θ0 0 K 0 k0
Similary, the canonical map π :A/q→A/p induces a decomposition map
d :R (H )→R (H ).
π 0 k0 0 kp
The following result shows that d is entirely determined by d and d .
θp θ0 π
Proposition 3.1 (Factorization of decomposition maps, M.Geck-R.Rouquier
[GR1]). The following diagram is commutative:
d
R (H ) θp -R (H )
0 K 0 kp
@ (cid:0)(cid:18)
@ (cid:0)
d d
θ0 @ (cid:0) π
@R (cid:0)
R (H )
0 k0
If D , D and D are the decomposition matrices associated to θ , θ and π
θ0 θp π 0 p
respectively, we have:
D =D D .
θp θ0 π
Now, by Theorem 2.1, we have a canonical basic set Bp ⊂ Irr(H ) associated
K
to the specialization θ and a canonical basic set B0 ⊂Irr(H ) associated to θ .
p K 0
Theorem 3.2 (M.Geck-R.Rouquier[GR1],[G5]). Assume that the character-
istic of k is good. Then, we have:
p
|Irr(H )|=|Irr(H )|.
kp k0
CANONICAL BASIC SETS FOR HECKE ALGEBRAS 5
Hence, we have:
|Bp|=|B0|
Wecannowgivethemainresultofthissection. Notethatanotherproofofthis
theoremcanbe foundin[J2, chapter3]. Theauthorwantstothank therefereefor
suggesting him this more elementary proof.
Theorem 3.3. Assume that the characteristic of k is good. Then, we have:
p
Bp =B0
Proof. LetM ∈Irr(H ) andlet V :=V ∈Bp ⊂Irr(H ) be the associated
kp M K
element of the canonical basic set as in Theorem 2.1. In R (H ), we have:
0 kp
d ([V])=[M]+lower terms with respect to a−value
θp
with a(V)= a(M). Then, by the factorization of the decomposition map, there is
a simple H -module M such that:
k0
• [M]∈R (H ) appears in d ([V]) with non zero coefficient,
0 k0 θ0
• [M]∈R (H ) appears in d ([M]) with non zero coefficient.
0 kp π
Then
a(M)=a(V)≥a(M)≥a(M),
togetherwiththecharacterizationofV inTheorem2.1impliesthatV =V ∈B0.
M M
By Theorem 3.2, we have |B|=|Bp|. Thus, we obtain:
B =Bp.
(cid:3)
Hence, it is sufficientto determine the canonicalbasic setin characteristic0 to
describe the canonical basic set in “good” characteristic.
4. Description of the canonical basic sets
4.1. FLOTW multipartitions. The canonical basic set has been explicitly
computed for all types and all specializations in characteristic 0. In this part, we
recall the parametrizations of these sets and we use Theorem 3.3 to compute the
canonical basic set in “good” characteristic.
In[J3],theexistenceofa“canonicalbasicset”hasbeenprovedforanothertype
of algebras known as Ariki-Koike algebras (or cyclotomic Hecke algebras of type
G(d,1,n)). This type of algebras can be seen as an analogue of Hecke algebras for
complex reflection groups. Moreover, this canonical basic set has been explicitely
described and can be parametrized by some FLOTW multipartitions arising from
a crystal graph studied by Foda et al. [FL] and Jimbo et al. [JM]. The key of
the proof is the Ariki’s theory [A] relating decomposition numbers with canonical
basis of Fock space.
Let v , v ,..., v be integers such that 0 ≤ v ≤ ... ≤ v < e. For d ∈ N,
0 1 d−1 0 d−1
we say that λ is a d-partition of rank n if:
• λ = (λ(0),..,λ(d−1)) where, for i = 0,...,d−1, λ(i) = (λ(i),...,λ(i)) is a
1 ri
partition of rank |λ(i)| such that λ(i) ≥...≥λ(i) >0,
1 ri
d−1
• |λ(k)|=n.
k=0
X
6 NICOLASJACON
We denote by Πd the set of d-partitions of rank n. To define the FLOTW d-
n
partitions, we must introduce some notations. Let λ = (λ(0),...,λ(d−1)) be a d-
partition of rank n. The diagram of λ is the following set:
[λ]= (a,b,c) | 0≤c≤d−1, 1≤b≤λ(c) .
a
n o
The elements of this diagram are called the nodes of λ. Let γ = (a,b,c) be a
node of λ. The residue of γ associated to the set {e;v ,...,v } is the element of
0 d−1
Z/eZ defined by:
res(γ)=(b−a+v )(mod e).
c
Definition 4.1. We say that λ = (λ(0),...,λ(d−1)) is a FLOTW d-partition
associated to the set {e;v ,...,v } if and only if:
0 d−1
(1) for all 0≤j ≤d−2 and i=1,2,..., we have:
λ(j) ≥λ(j+1) ,
i i+vj+1−vj
λ(d−1) ≥λ(0) ;
i i+e+v0−vd−1
(2) forallk >0,amongtheresiduesappearingattherightendsofthelength
k rows of λ, at least one element of {0,1,...,e−1} does not occur.
We denote by Λ1 the set of FLOTW d-partitions associated to the set
{e;v0,...,vd−1}
{e;v ,...,v }.
0 d−1
Now,weareabletogivethe parametrizationsofthecanonicalbasicsetsforall
Hecke algebras of finite Weyl group and for all specializations. Let H be a Hecke
algebra of a finite Weyl group W, let θ : A → k be a specialization into the field
of fractions k of θ(A) such that the characteristic of k is 0 or a good prime for W.
Let K be the field of fractions of A.
4.2. Type A . Assume that W is a Weyl group of type A and that
n−1 n−1
θ(u) is a primitive eth-root of unity. Let λ be a partition of rank n, then, we
can construct an H-module Sλ, free over A which is called a Specht module (see
the construction of “dual Specht modules” in [A, Chapter 13] in a more general
setting). Moreover,we have:
Irr(H )={Sλ | λ∈Π1}.
K K n
Now,HeckealgebrasoftypeA arespecialcasesofAriki-Koikealgebras. Hence,
n−1
we can use the results in [J3] to find the canonical basic sets. We note that we
can also find this set using results of Dipper and James as it is expained in [G3,
Example 3.5]. Note also that there is no bad prime number for W =A .
n−1
Proposition 4.2. Assumethat W is a Weyl group of type A and that θ(u)
n−1
is a primitive eth-root of unity. Then, we have:
B ={Sλ | λ∈Λ1 , |λ|=n}.
K {e;0}
Note that:
λ∈Λ1 ⇐⇒ λ=(λ ,...,λ ) is e−regular.
{e;0} 1 r
⇐⇒ For all i∈N,we can’t have λ =...=λ 6=0.
i i+e−1
CANONICAL BASIC SETS FOR HECKE ALGEBRAS 7
4.3. Type B . Assume thatW is aWeylgroupoftypeB andthatθ(u)is a
n n
primitive eth-rootof unity. Let (λ(0),λ(1)) be a 2-partitionof rank n, then, we can
construct an H-module S(λ(0),λ(1)), free over A which is called a Specht module.
Moreover,we have:
Irr(H )={S(λ(0),λ(1)) | (λ(0),λ(1))∈Π2}.
K K n
Now, Hecke algebras of type B are special cases of Ariki-Koike algebras. Hence,
n
we can use the results in [J3] to find the canonical basic sets. The only bad prime
for type B is p=2.
n
Proposition 4.3. Assume that W is a Weyl group of type B and that θ(u)
n
is a primitive eth-root of unity. Then, we have:
• if e is odd:
B ={S(λ(0),λ(1)) | λ(0),λ(1) ∈Λ1 , |λ(0)|+|λ(1)|=n}.
K {e;0}
• if e is even:
B ={S(λ(0),λ(1)) | (λ(0),λ(1))∈Λ1 , |λ(0)|+|λ(1)|=n}.
{e;1,e}
2
Recall that (λ(0),λ(1))∈Λ1 if and only if:
{e;1,e}
2
(1) for all i=1,2,..., we have:
λ(0) ≥λ(1) ,
i i+e−1
2
λ(1) ≥λ(0) ;
i i+e+1
2
(2) for all k > 0, among the residues appearing at the right ends of the
length k rows of (λ(0),λ(1)), at least one element of {0,1,...,e−1}
does not occur.
4.4. Type D . Assume that W is a Weyl group of type D and that θ(u) is
n n
a primitive eth-root of unity. Then, H can be seen as a subalgebra of an Hecke
algebra H of type B with the following diagram (see [G6]).
1 n
1 u u u
t t t ` ` ` t
The specialization θ induces a decomposition map for H :
1
d1 :R (H )→R (H ).
θ 0 1,K 0 1,k
Similary to the equal parameter case, for all (λ(0),λ(1))∈Π2, we can construct an
n
H -module S(λ(0),λ(1)), free over A which is called a Specht module. We have:
1
Irr(H )={S(λ(0),λ(1)) | (λ(0),λ(1))∈Π2}.
1,K K n
Now,wehaveanoperationofrestrictionResbetweenthe setofH -modules and
1,K
the set of H -modules. For (λ(0),λ(1))∈Π2, we have:
K n
• if λ(0) 6= λ(1), we have Res(S(λ(0),λ(1))) ≃ Res(S(λ(1),λ(0))) and the H -
K K K
module V[λ(0),λ(1)] :=Res(S(λ(0),λ(1))) is a simple H -module.
K K
• if λ(0) = λ(1), we have Res(S(λ(0),λ(1))) = V[λ(0),+] ⊕ V[λ(0),−] where
K
V[λ(0),+] and V[λ(0),−] are non isomorphic simple H -modules.
K
8 NICOLASJACON
Moreover,we have:
Irr(H )= V[λ,µ] | λ6=µ, (λ,µ)∈Π2 V[λ,±] | λ∈Π1 .
K n n
2
Hecke algebras onf type Bn with unequal paraom[etenrs are special casoes of Ariki-
Koike algebras. Hence, we can also define a canonical basic set for these algebras
(the existence has been previously provedin [G6]). Furthermore, in [G6], M.Geck
has shown that the simple H -modules in the canonical basic set for type D are
K n
those which appear in the restriction of the simple H -modules of the canonical
1,K
basicsetfortype D . We obtainthe followingdescriptionofB. Note thatthe only
n
bad prime for type D is p=2.
n
Proposition 4.4. Assume that W is a Weyl group of type D and that θ(u)
n
is a primitive eth-root of unity. Then:
• if e is odd, we have:
B = V[λ(0),λ(1)] | λ(0) 6=λ(1), λ(0),λ(1) ∈Λ1 , |λ(0)|+|λ(1)|=n
{e;0}
n V[λ(0),±] | λ(0) ∈Λ1 , 2|λ(0)|=n . o
{e;0}
• if e is even,Swne have: o
B = V[λ(0),λ(1)] | λ(0) 6=λ(1), (λ(0),λ(1))∈Λ1 , |λ(0)|+|λ(1)|=n
{e;0,e}
2
n V[λ(0),±] | (λ(0),λ(0))∈Λ1 , 2|λ(0)|=n . o
{e;0,e}
2
n o
Recall tShat (λ(0),λ(1))∈Λ1 if and only if:
{e;0,e}
2
(1) for all i=1,2,..., we have:
λ(0) ≥λ(1) ,
i i+e
2
λ(1) ≥λ(0) ;
i i+e
2
(2) for all k > 0, among the residues appearing at the right ends of the
length k rows of (λ(0),λ(1)), at least one element of {0,1,...,e−1}
does not occur.
4.5. Exceptional types. The decomposition matrices are explicitely known
for type G , F , E and E (see [Mu], [G1], [G2] and [GL]) for all specializations
2 4 6 7
in characteristic 0. Hence, in this case, it suffices to study these matrices and to
use Theorem 2.1 to obtain the canonical basic in characteristic 0. Next, Theorem
3.3 gives the canonical basic sets in “good” positive characteristic.
For type E , Mu¨ller has given in [Mu] a set of projective H -modules which
8 O
isin bijectionwith the setofprojectiveindecomposableH -modules (whichcorre-
O
sponds to the columns of the decomposition matrices, see [GP, Section 7.5]). It is
easy to obtain the canonical basic set by studying these projective H -modules.
O
An explicit description of the canonical basic set for all exceptional types and
for all specializations can be found in [J2, Chapter 3].
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Laboratoire de Math´ematiques Nicolas Oresme, Universit´e de Caen, BP 5186, F
14032Caen Cedex, France.
E-mail address: [email protected]