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Schur-Weyl reciprocity for the q-analogue of the alternating group 6 0 0 Hideo Mitsuhashi 2 n a J Department of Information Technology 5 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 isomorphism 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