Schur-Weyl reciprocity for the q-analogue of the alternating group

0.27 MB·English

by  Hideo Mitsuhashi

Schur-Weyl reciprocity for the q-analogue of the alternating group
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0
Hideo Mitsuhashi
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n
a
J
Department of Information Technology

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2 KanagawaPrefectural Junior College for Industrial Technology

2–4–1 Nakao, Asahi–ku, Yokohama–shi, Kanagawa–ken241–0815,Japan
]
T
R Abstract
.
h In this paper, we establish Schur-Weyl reciprocity for the q-analogue of the alternating group.
t
a We analyze the sign q-permutation representation of the Hecke algebra H (q) on the rth tensor
m Q(q),r
product of Z -graded Q-vector space V = V ⊗V in detail, and examine its restriction to the q-
[ 2 0 1
analogue of the alternating group H1Q(q),r(q). In consequence, we find out that if dimV0 = dimV1,
1
v then the centralizer of H1Q(q),r(q) is a Z2-crossed product of the centralizer of HQ(q),r(q) and obtain
7 Schur-WeylreciprocitybetweenH1 (q)anditscentralizer. Thoughthestructureofthecentralizer
0 Q(q),r
6 ismorecomplicated forthecasedimV06=dimV1,weobtainsomeresultsaboutthecase. Whenq=1,
1
Regev hasproved Schur-Weylreciprocity for alternating groups in [12]. Therefore, ourresult can be
0
6 regarded as an extension of Regev’s work.
0
/
h
t
a 1 Introduction
m
:
v The purpose of this study is to research Schur-Weyl reciprocity for the q-analogue of the alternating
i
X group. In our previous paper [10], we established Schur-Weyl reciprocity between the Hecke algebra
r H (q)andthequantumsuperLiealgebraUσ gl(m,n) . Inthatpaper,wedefinedtheq-permutation
a Q(q),r q
(cid:0) (cid:1)
representation of H (q), and showed that the image of the q-permutation representation is the cen-
Q(q),r
tralizer of the image of the vector representation of the quantum super Lie algebra Uσ gl(m,n) on the
q
(cid:0) (cid:1)
rth tensor product of a Z -graded(m+n)-dimensional Q(q)-vector space V =V ⊕V . In this paper, we
2 0 1
find out the centralizerof the q-analogueof the alternating groupas the restrictionof the q-permutation
representation. When q = 1, Regev has already shown Schur-Weyl reciprocity for the alternating group
in [12]. Hence our result is regarded as an extension of Regev’s work.
1
Let q be an indeterminate and K = Q(q). Let (π ,V⊗r) be the q-permutation representation of
r
H (q)(definitionoftheq-permutationrepresentationisat(4.1))and(ρ ,V⊗r)thevectorrepresentation
K,r r
of Uσ gl(m,n) (definition of the vector representation is at (4.4)). We have proved in [10] that A =
q q
π H(cid:0) (q) an(cid:1)d B =ρ Uσ gl(m,n) are full centralizers of each other, namely:
r K,r q r q
(cid:0) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1)
B =End V⊗r and A =End V⊗r. (1.1)
q Aq q Bq
Let R be a commutative domain which includes an invertible element q. We further assume that 2
0
and q+q−1 are invertible elements of R . Then we can define the q-analogue of the alternating group
0
H1 (q) (see Definition 3.1 and Proposition 3.6) in H (q). In [9], we defined the q-analogue of the
R0,r R0,r
alternatinggroupasasubalgebraofIwahori-HeckealgebraoftypeAandobtaineddefiningrelations(see
Proposition 3.7) for the first time. In this paper, we show that H (q) is isomorphic to the Z -crossed
R0,r 2
product which is obtained from the crossed system (H1 (q),Z ,ψ ,α ) (definition of ψ and α are at
R0,r 2 0 0 0 0
(3.2) and (3.3) respectively).
Theorem 3.9. H (q) is isomorphic to the Z -crossed product H1 (q)ψ0[Z ] as R -algebras.
R0,r 2 R0,r α0 2 0
Let C = π H1 (q) and D = End V⊗r. Our main subject is to solve the relation between D
q r K,r q Cq q
(cid:0) (cid:1)
andB . From(1.1), onecanimmediately seethatB ⊆D . Butthe structure ofD is nottrivial. Indeed,
q q q q
the structure of D depends on the dimensions m = dim V and n = dim V . In this paper, we show
q K 0 K 1
that if m = n, then D is isomorphic to Z -crossed product which is obtained from the crossed system
q 2
(B ,Z ,ψ ,α ) (definition of ψ and α are at (5.4) and (5.5) respectively).
q 2 1 1 0 0
Theorem 5.6. If m=n, then D is isomorphic to the Z -crossed product B ψ1[Z ] as K-algebras.
q 2 qα1 2
Fromthistheorem,weimmediatelyobtainthatdim D =2dim B . Moreover,weshowSchur-Weyl
K q K q
reciprocity for H1 (q).
K,r
Theorem 5.8. End V⊗r =D and End V⊗r =C hold.
Cq q Dq q
Inthegeneralcase,thematterismorecomplicated,butwecanfindouttosomeextentifweexchange
the base field fromK to its algebraicclosure K¯. LetU¯σ gl(m,n) =Uσ gl(m,n) ⊗ K¯, A¯ =A ⊗ K¯,
q q K q q K
B¯ = B ⊗ K¯ and C¯ = C ⊗ K¯. Then we have the(cid:0)following(cid:1)theore(cid:0)m by the(cid:1) theory of semisimple
q q K q q K
algebras.
Theorem 6.1. A¯ and C¯ have direct sum decompositions A¯ = A¯0⊕A¯1 and C¯ = C¯0⊕C¯1 respectively,
q q q q q q q q
which are satisfy the following relations.
(1) C¯q0⊆A¯0q and dimK¯ A¯0q =2dimK¯ C¯q0
2
(2) C¯q1 =A¯1q. Especially dimK¯ A¯1q =dimK¯ C¯q1.
As a corollary, we obtain an anomalous phenomenon for non-super case as follows.
Corollary 6.2. Let n=0. If m2 <r, then A¯q =C¯q and EndC¯qV¯⊗r =EndA¯qV¯⊗r.
The details of the decompositions of A¯ andC¯ are describedin section6. We alsoobtain the similar
q q
result to Theorem 6.1 about the endomorphism algebras EndA¯qV¯⊗r and EndC¯qV¯⊗r; there exist two
HK¯,r(q)-submodules W0 and W1 which satisfy the following properties.
Corollary 6.3. EndC¯qW0⊇EndA¯qW0 and EndC¯qW1 =EndA¯qW1.
Although the relation between A¯q and C¯q is made clear by (6.4)-(6.7), that between EndA¯qW and
EndC¯qW is not clear except for Corollary 6.2 at this point.
2 Preliminaries
Let R be a commutative ring with 1 and G a group. In this section, we shall review the definition
and some properties about G-crossed products. A full account about G-graded algebras and G-crossed
products is given in [11].
Definition 2.1 (G-graded algebra). An R-algebra A is said to be G-graded if there exist a family of
R-submodules {A |σ∈G} of A indexed by elements of G which satisfies the following two conditions:
σ
(G1) A= A ,
σ∈G σ
L
(G2) A A ⊆A for σ,τ∈G.
σ τ στ
Moreover,A is said to be strongly G-graded when (G2) is replaced by the following condition:
(G’2) A A =A for σ,τ∈G.
σ τ στ
We notice that if A = A is a G-graded algebra, then A (1 means the identity element of
σ∈G σ 1G G
L
G) is a subalgebra of A and 1 ∈A .
A 1G
Definition 2.2 (G-crossed product). A G-graded R-algebra A = ⊕ A is said to be a G-crossed
σ∈G σ
product if each A has an invertible element.
σ
We notice that a G-crossed product is a strongly G-graded algebra. Indeed, if A is a G-crossed
product, then for an invertible element uσ∈Aσ, u−σ1∈Aσ−1 and 1A = uσu−σ1∈AσAσ−1. So, we have
Aστ =1AAστ⊆(AσAσ−1)Aστ =Aσ(Aσ−1Aστ)⊆AσAτ.
3
Definition 2.3 (crossed system). Let A be an algebra and G a group. Suppose that there exist two
maps
ψ :G−→Aut(A),
(we denote ψ(σ)(a) by σa for brevity), and
α:G×G−→A×,
where A× is the multiplicative group of units of A , which satisfy the relations:
1
σ(τa) = α(σ,τ)(στ)aα(σ,τ)−1 (2.1)
σ1α(σ ,σ )α(σ ,σ σ ) = α(σ ,σ )α(σ σ ,σ ) (2.2)
2 3 1 2 3 1 2 1 2 3
α(σ,1) = α(1,σ)=1, (2.3)
for σ,τ,σ ,σ ,σ ∈G,a∈A. (A,G,ψ,α) is said to be a crossed system. ψ is called a weak action of G on
1 2 3
A, and α is called a ψ-cocycle.
WedenotebyAψ[G]thefreeleftA-modulewiththebasis{u |σ∈G}andthefollowingmultiplication:
α σ
(a u )(a u )=a σa α(σ,τ)u , (2.4)
1 σ 2 τ 1 2 στ
for a ,a ∈A,σ,τ∈G.
1 2
Proposition 2.4 ([11]Proposition 1.4.1). Aψ[G] is a G-crossed product.
α
Proposition 2.5 ([11]Proposition 1.4.2). Every G-crossed product is of the form Aψ[G] for some
α
algebra A, some weak action ψ and some ψ-cocycle α.
When G is finite and a strongly G-graded algebra A = A is finitely generated over R as
σ∈G σ
L
modules, A is said to have a G-graded Clifford system {A |σ∈G} if A satisfies (C1).
σ
(C1) For each σ∈G, there exists an invertible element a ∈A such that A =a A =A a .
σ σ σ 1G 1G σ
It is clear that such a is in A . An exposition about groupgraded Clifford systems can be found in [1],
σ σ
section 11C.
3 The q-analogue of the alternating group and its representation
Let (W,S = {s ,...,s }) be a Coxeter system of rank r. Let R be a commutative domain with 1, and
1 r 0
letq (i=1,...,r)beanyinvertibleelementsofR suchthatq =q ifs isconjugatetos inW. Further
i 0 i j i j
4
we assume that 2 and q +q−1(i=1,2,...,r) are invertible elements of R . The Iwahori-Hecke algebra
i i 0
H (W,S) is an R -algebra generated by {T |s ∈S} with the defining relations:
R0 0 si i
(H1) T2 =(q −q−1)T +1 if i=1,2,...,r,
si i i si
(H2) (TsiTsj)kij =(TsjTsi)kij if mij =2kij,
(H3) (TsiTsj)kijTsi =(TsjTsi)kijTsj if mij =2kij +1,
where m is the order of s s in W. We write T =T for brevity.
ij i j i si
If(W,S)isoftypeAandofrankr−1,thenW isisomorphictothesymmetricgroupS . Furthermore,
r
alltheelementsofS areconjugatetoeachother,hencewemayassumeq =···=q =q. TheIwahori-
1 r−1
Hecke algebra H (q)=H (W,S) of type A has the defining relations:
R0,r R0
(A1) T2 =(q−q−1)T +1 if i=1,2,...,r−1,
i i
(A2) T T T =T T T if i=1,2,...,r−2,
i i+1 i i+1 i i+1
(A3) T T =T T if |i−j|>1.
i j j i
Let ˆ be the Goldman involution. This is an involution on H (q) defined by
R0,r
Tˆ =(q−q−1)−T .
i i
Definition 3.1. We define H±1 (q) to be the eigenspaces of H (q) corresponding to the eigenvalues
R0,r R0,r
±1 of ˆ respectively.
We notice that H1 (q) is a subalgebra of H (q). Let T′ (i = 1,2,...,r−1) be the elements of
R0,r R0,r i
H (q) defined by
R0,r
T −Tˆ 2T −(q−q−1)
T′ = i i = i for i=1,2,...,r−1.
i q+q−1 q+q−1
Then one can immediately check Tˆ′ =−T′.
i i
Proposition 3.2. T′(i=1,2,...,r) generate H (q) and satisfy the following defining relations:
i R0,r
(A’1) T′2 =1 if i=1,2,...,r−1,
i
q−q−1 2
(A’2) T′T′ T′ =T′ T′T′ − (T′−T′ ) if i=1,2,...,r−2,
i i+1 i i+1 i i+1 q+q−1 i i+1
(cid:16) (cid:17)
(A’3) T′T′ =T′T′ if |i−j|>1.
i j j i
5
Proof. From the equations
1
T = {(q+q−1)T′+(q−q−1)} for i=1,2,...,r−1,
i 2 i
which are obtained from the definition of T′, we see that T′(i = 1,2,...,r) generate H (q). The
i i R0,r
defining relations are obtained from a direct computation.
Consider the following sets of monomials:
C ={1,T }
1 1
C ={1,T ,T T }
2 2 2 1
C ={1,T ,T T ,T T T }
3 3 3 2 3 2 1
:
C ={1,T ,T T ,...,T T ···T }
r−1 r−1 r−1 r−2 r−1 r−2 1
WeshallsaythatM M ···M isamonomialinT -normalforminH (q)ifM ∈C fori=1,2,...,r−
1 2 r−1 i R0,r i i
1. The following fact is well-known in the theory of the Iwahori-Hecke algebra.
Proposition 3.3. rank H (q)=r! and
R0 R0,r
H (q)= R M M ···M
R0,r 0 1 2 r−1
MMi∈Ci
We derive from this fact that all monomials in T′-normal form also constitute a basis of H (q).
i R0,r
Proposition 3.4. Let C′ ={1,T′,T′T′ ,...,T′T′ ···T′} for i=1,2,...,r−1. Then we have
i i i i−1 i i−1 1
H (q)= R M′M′···M′ .
R0,r 0 1 2 r−1
MM′∈C′
i i
Proof. Consider the map f from {T } to {T′} which is defined by f(T )=T′. This
i i=1,2,...,r−1 i i=1,2,...,r−1 i i
map induces the R -endomorphism f¯of H (q). f¯is an R -isomorphism because the inverse g¯ which
0 R0,r 0
is induced from the map g from {T′} to {T } defined by
i i=1,2,...,r−1 i i=1,2,...,r−1
1
g(T′)= {(q+q−1)T +(q−q−1)}
i 2 i
exists. Hence we conclude that M′M′···M′ (M′∈C′), which are images of M M ···M (M ∈C ), are
1 2 r−1 i i 1 2 r−1 i i
linearly independent and constitute a basis of H (q).
R0,r
LetE (respectivelyO )bethesetofallmonomialsinT′-normalforminH (q)whichareproducts
r r i R0,r
of even (respectively odd) numbers of T′’s. Then the following holds.
i
6
Lemma 3.5. |E |=|O |=2−1r! for r >1.
r r
Proof. The proof is done by induction on r. It is trivial for r = 2. Let M′M′···M′ ∈E with r > 2
1 2 r−1 r
and M′∈C′. Then M′M′···M′ is considered as a monomial in T′-normal form in H (q). Let
i i 1 2 r−2 i R0,r−1
(C′ )e (respectively (C′ )o)be the subsetofC′ whichconsistsofproductsofeven(respectivelyodd)
r−1 r−1 r−1
numbersofT′’s. WecanreadilyseethatM′M′···M′ ∈E ifandonlyifM′ ∈(C′ )e. Byinduction,
i 1 2 r−2 r−1 r−1 r−1
|E |=|O |=2−1(r−1)! and hence we obtain the following
r−1 r−1
|E |=|(C′ )e||E |+|(C′ )o||O |=r×2−1(r−1)!=2−1r!
r r−1 r−1 r−1 r−1
as desired.
Now we characterize H1 (q) as a q-analogue of the alternating group.
R0,r
Proposition 3.6. rank H (q) = 2rank H1 (q). Moreover H1 (q) is the subalgebra which
R0 R0,r R0 R0,r R0,r
consists of all the products of even numbers of T′’s.
i
Proof. Let He (q) = ⊕ R M and Ho (q) = ⊕ R M. Then we can see immediately that
R0,r M∈Er 0 R0,r M∈Or 0
H (q) = He (q)⊕Ho (q). Furthermore we obtain He (q) = H1 (q) and Ho (q) = H−1 (q)
R0,r R0,r R0,r R0,r R0,r R0,r R0,r
from the property Tˆ′ =−T′ . Combined with Lemma 3.5 we have rank H (q)=2rank H1 (q).
i i R0 R0,r R0 R0,r
Let H¯1 (q) be the set of all the linear combinations of products of even numbers of T′’s. Obviously
R0,r i
H1 (q)⊆H¯1 (q). Fromthedefiningrelations(A’1)-(A’3),onecanseethatifamonomialinT′-normal
R0,r R0,r i
formwhichconsistsofeven(respectivelyodd)numberofT′’sisexpressedinalinearcombinationofother
i
expressions, then each term consists of even (respectively odd) number of T′’s. Hence if we express an
i
element of H¯1 (q) by a linear combination of monomials in T′-normal form, each term is in H1 (q).
R0,r i R0,r
Consequently H¯1 (q)=H1 (q).
R0,r R0,r
When we suppose that R = C and take a limit q→1, H1 (1) is isomorphic to the group algebra
0 C,r
C[A ] of the alternating group A .
n n
Theorem 3.7 ([9]). H1 (q) is isomorphic to the R -algebra which is generated by r − 2 elements
R0,r 0
X ,X ,...,X with the defining relations:
1 2 r−2
q−q−1 2
(B1) X3 =− (X2−X )+1,
1 q+q−1 1 1
(cid:16) (cid:17)
(B2) X2 =1 for i>1,
i
q−q−1 2
(B3) (X X )3 =− (X X )2−X X +1 for i=2,3,...,r−2,
i−1 i q+q−1 i−1 i i−1 i
(cid:16) (cid:17) n o
7
(B4) (X X )2 =1 whenever |i−j|>1.
i j
An isomorphism is given by X −→T′T′ .
i 1 i+1
Next, we shall show that H (q) is a Z -crossed product. Since T′ has an inverse as itself, we can
R0,r 2 i
readily see from Proposition 3.6 that H1 (q)T′ is an R -submodule of H (q) which consists of all
R0,r 1 0 R0,r
linearcombinationsofproductsofoddnumbersofT′’s. Therefore,weobtainadirectsumdecomposition
i
of left H1 (q)-modules:
R0,r
H (q)=H1 (q)⊕H1 (q)T′. (3.1)
R0,r R0,r R0,r 1
Let Z =h1,−1i be a multiplicative group. We define two maps ψ and α to be
2 0 0
ψ :Z −→Aut(H1 (q)), ψ (1)(T)=T,ψ (−1)(T)=T′TT′, T∈H1 (q). (3.2)
0 2 R0,r 0 0 1 1 R0,r
and
α :Z ×Z −→(H1 (q))×, α (σ,τ)=1 for all σ,τ∈Z . (3.3)
0 2 2 R0,r 0 2
Then we have the following immediately.
Lemma 3.8. ψ and α satisfy (2.1)–(2.3).
0 0
Thus, we obtain a Z -crossed product H1 (q)ψ0[Z ] from the crossed system (H1 (q),Z ,ψ ,α ),
2 R0,r α0 2 R0,r 2 0 0
where ψ and α are given by (3.2) and (3.3) respectively.
0 0
Theorem 3.9. HR0,r(q) is isomorphic to HR10,r(q)ψα00[Z2] as R0-algebras.
Proof. Since both HR0,r(q) and HR10,r(q)ψα00[Z2] are free left HR10,r(q)-modules, we may define an isomor-
phism of H1 (q)-modules
R0,r
ι :H1 (q)ψ0[Z ]−→H (q), ι (u )=1,ι (u )=T′
0 R0,r α0 2 R0,r 0 1 0 −1 1
From (2.4) and (3.2) and (3.3), we can determine the multiplication law as follows.
a a u if σ =1,
(a u )(a u )= 1 2 στ
1 σ 2 τ a T′a T′u if σ =−1,
1 1 2 1 στ

where a ,a ∈H1 (q) and σ,τ∈Z . Therefore, we get four formulas in H (q)
1 2 R0,r 2 R0,r
(a 1)(a 1)=a a 1, (a 1)(a T′)=a a T′, (a T′)(a 1)=a T′a T′T′, (a T′)(a T′)=a T′a T′1,
1 2 1 2 1 2 1 1 2 1 1 1 2 1 1 2 1 1 1 1 2 1 1 1 2 1
which derive the conclusion that ι is an isomorphism of R -algebras.
0 0
8
WedenotebyK¯ analgebraicclosureofafieldK. Letq beanindeterminateandK =Q(q). We shall
show (split) semisimplicity of HK1¯,r(q) and the branching rule from HK¯,r(q) to HK1¯,r(q). The manner
of proof given here is credited to K.Uno, who sent me a letter enclosing the outline of this proof. It is
well-knownthatHK¯,r(q)issplitsemisimpleandthatisomorphismclassesofsimpleleftHK¯,r(q)-modules
areparametrizedby Youngdiagramsoftotalsizen. LetΛ be the setofallYoung diagramsoftotalsize
r
r. Then, provided that {Mq,λ|λ∈Λr} is a set of all isomorphism classes of simple left HK¯,r(q)-modules
and that d =degM , we may write
λ q,λ
HK¯,r(q)= Iq,λ Iq,λ∼=Matdλ(K¯) , (3.4)
λM∈Λr (cid:0) (cid:1)
whereeachI isthehomogeneouscomponentcorrespondingtoλ. I isisomorphictoad ×d matrix
q,λ q,λ λ λ
algebraMatdλ(K¯)whoseentrieslieinK¯. Since ˆ isaninvolutionofHK¯,r(q),foreachλ∈Λr thereexists
µ∈Λ such that Iˆ = I . Especially, d = d follows. Dipper and James defined Specht modules
n q,λ q,µ λ µ
Sλ for Hecke algebras as irreducible submodules of regular modules in [4], and improved the theory of
K
representations of Hecke algebras in the series of articles such as [4, 5, 6]. In particular, they showed in
[6] that if K is a field and H (q) is semisimple, then Sˆλ∼= Sλ′ where λ′ denotes the transpose of
K,r K HK,r(q) K
λ. In this case, M is equivalent to Sλ, thus µ = λ′ follows. We divide into two cases depending on
q,λ K¯
whether λ is self-conjugate or not.
(Case1) λ6=λ′:
In this case, ˆ induces an involution on Iq,λ⊕Iq,λ′. Let I˜q,λ = X +Y∈Iq,λ⊕Iq,λ′|(X +Y) =X +Y
for λ6=λ′. Then we have (cid:8) (cid:9)
b
I˜q,λ = X +Xˆ∈Iq,λ⊕Iq,λ′|X∈Iq,λ ∼=Iq,λ ∼=Matdλ(K¯) , (3.5)
(cid:8) (cid:9) (cid:0) (cid:1)
Thereby,the imageofthe regularrepresentationofH1 (q)onSλ isisomorphictoI ,soresHK¯,r(q)Sλ
K¯,r K¯ q,λ H1 (q) K¯
K¯,r
isa simpleleft HK1¯,r(q)-module. Ifg∈HK¯,r(q)satisfiesgˆ=g,thenthe matrixcoefficientswithrespectto
the basis x ,x ,...,x of Sλ is the same as those with respect to the basis xˆ ,xˆ ,...,xˆ of Sˆλ∼=Sλ′.
1 2 dλ K¯ 1 2 dλ K¯ K¯
Hence we have the isomorphism of simple left H1 (q)-modules as follows.
K¯,r
resHK¯,r(q)Sλ∼=resHK¯,r(q)Sλ′.
H1 (q) K¯ H1 (q) K¯
K¯,r K¯,r
(Case2) λ=λ′:
In this case, ˆ induces an involution on I ∼=Mat (K¯). By Skolem-Noether Theorem, there exists an
q,λ dλ
invertible element P of Mat (K¯) such that Xˆ = PXP−1 for all X∈Mat (K¯). Since eigenvalues of P
dλ dλ
9
are ±1, We may also assume that
1
 
...
 
 
 1 
P = .
 −1 
 
 
 ... 
 
 
 −1
 
Then we may write
X X X −X
P  1 2P−1 = 1 2,
X X −X X
3 4 3 4
   
for some submatrices X ,X ,X ,X . Therefore, we obtain
1 2 3 4
X 0
X∈Matdλ(K¯)|Xˆ =X =(cid:26)X∈Matdλ(K¯)|X = 01 X (cid:27).
(cid:8) (cid:9) 4
 
Assuming that 1 appears m times in P. Then dimK¯ X∈Matdλ(K¯)|Xˆ = X = m2 +(dλ −m)2. We
(cid:8) (cid:9)
easily see that
d2 d
m2+(d −m)2≥ λ equality holds iff m= λ.
λ
2 2
1
Combining this with (3.5) and the fact that dimK¯ HK1¯,r(q) = 2dimK¯ HK¯,r(q) (Proposition 3.6), we
deduce that m=d /2. Thereby,
λ
X∈Mat (K¯)|Xˆ =X ∼=Mat (K¯)⊕Mat (K¯) (3.6)
dλ dλ/2 dλ/2
(cid:8) (cid:9)
holds. This means that resHK¯,r(q)Sλ decomposes into two simple left H1 (q)-modules S¯λ+ and S¯λ−
H1K¯,r(q) K¯ K¯,r K¯ K¯
which are mutually non-isomorphic. Let I˜ = X∈I |Xˆ = X and I˜+ (resp. I˜− ) be the homoge-
q,λ q,λ q,λ q,λ
neous component corresponding to S¯λ+ (resp. S¯(cid:8)λ−) for λ6=λ′. T(cid:9)hen (3.6) implies that
K¯ K¯
I˜ =I˜+ ⊕I˜− , (3.7)
q,λ q,λ q,λ
Summarizing our argument, we conclude that H1 (q) is isomorphic to the direct sum of minimal two-
K¯,r
sided ideals as follows,
H1 (q)= I˜ I˜+ ⊕I˜− ,
K¯,r q,λ q,λ q,λ
(cid:8)λ∈ΛMr,λ>λ′ (cid:9)Mhλ∈ΛMr,λ=λ′(cid:8) (cid:9)i
where < denotes the lexicographic order on Λ ; λ = (λ ,λ ,...) < µ = (µ ,µ ,...) iff λ < µ for the
r 1 2 1 2 k k
smallest k such that λ 6=µ . Consequently we have proved the following result.
k k
10