August 22, 2015 9:7 WSPC/S0219-4988 171-JAA 1550141

Journal of Algebra and Its Applications Vol. 14, No. 10 (2015) 1550141 (13 pages)
c World Scientific Publishing Company DOI: 10.1142/S0219498815501418

Standardly based algebrasand0-Heckealgebras

Guiyu Yang School of Science Shandong University of Technology Zibo 255049, P. R. China [email protected]

Yanbo Li∗ School of Mathematics and Statistics Northeastern University at Qinhuangdao Qinhuangdao, 066004, P. R. China [email protected]

Received 8 April 2014 Accepted 20 November 2014 Published 27 August 2015

Communicated by D. Passman

In this paper we prove that standardly based algebras are invariant under Morita equiva- lences. As an application, we prove 0-Hecke algebras and 0-Schur algebras are standardly based algebras. From this point of view, we give a new way to construct the simple modules of 0-Hecke algebras, and prove that the dimension of the center of a symmetric 0-Hecke algebra is not less than the number of its simple modules.

Keywords: Standardly based algebras; 0-Hecke algebras.

Mathematics Subject Classification: 16E99, 20C08, 16G30

1. Introduction Standardly based algebras were introduced by Du and Rui in [7]. They are gen- eralizations of cellular algebras defined by Graham and Lehrer in [9]. Standardly based algebras can be obtained from cellular algebras by omitting the involutions in the definition. The representation theory of Standardly based algebras and the relations between standardly based algebras and cellular algebras are investigated in [7]. The 0-Hecke algebra H0(W ) of a finite Coxeter group W is the degenerate form of the Hecke algebra Hq(W ) by specializing q = 0. The structure and representation

∗Corresponding author.

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theory of H0(W ) has been extensively studied in [3, 4, 10, 14, 17]. Let S = {si | i ∈ I} be the generating set for the finite Coxeter group W .Itisshownin[17] that the number of simple modules of H0(W ) equals the number of subsets of S. Dipper and James have shown in [6] that for a Hecke algebra H of type A over a domain, the dimension of the center Z(H) equals the number of partitions of n +1, where n is the cardinal number of s, if the parameter q is invertible. Moreover, Z(H) is the set of symmetric polynomials in the Murphy operators. Geck and Rouquier prove that the center and simple modules of Iwahori–Hecke algebras have generic features in [8]. Under the framework of cellular algebras, bases of the centers of semi- simple Hecke algebras are described in [11]. In case of 0-Hecke algebras, Brichard has given in [2] a formula to compute the dimension of the center of 0-Hecke algebras of type A by using a calculus of diagrams on the M¨obius band. The aim of this paper is to investigate the simple modules and the dimensions of the centers of 0-Hecke algebras under the framework of standardly based alge- bras. To be precise, we first prove that standardly based algebras are invariant under Morita equivalences. As an application, we prove that 0-Hecke algebras and 0-Schur algebras are standardly based algebras. Then we give a new account on the construction of simple modules of 0-Hecke algebras through this point of view. And we investigate the dimensions of the centers of 0-Hecke algebras under this framework. The paper is organized as follows. In Sec. 2 we introduce the definitions of stan- dardly based algebras and 0-Hecke algebras. In Sec. 3 we prove that standardly based algebras are invariant under Morita equivalences. In Sec. 4 we give a descrip- tion on the dimension of the center of a symmetric standardly based algebra. Then in Sec. 5 we use the theory of standardly based algebras to investigate the simple modules and the dimensions of the centers of 0-Hecke algebras.

2. Preliminaries

Definition 2.1 ([7]). Let R be a commutative ring with identity. An associative R-algebra A is called a standardly based algebra if the following conditions are satisfied:

(1) The algebra A has an R-basis

λ B = {CS,T | S ∈ I(λ),T ∈ J(λ),λ∈ Λ},

where Λ is a poset, I(λ)andJ(λ) are index sets. λ (2) For any a ∈ A, CS,T ∈ B,wehave λ λ aCS,T ≡ fS
,λ(a, S)CS
,T (mod A(>λ)), S
∈I(λ) λ λ CS,T a ≡ fλ,T
(T,a)CS,T
(mod A(>λ)), T
∈J(λ)

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where fS
,λ(a, S),fλ,T
(T,a) ∈ R are independent of T and S, respectively. And A(>λ)istheR-submodule of A generated by µ {CU,V | µ>λ,U∈ I(µ),V ∈ J(µ)}.

Remark 2.2. In this paper we assume that R is a noetherian integral domain if not pointed out. Moreover, we assume that Λ is a finite poset and I(λ),J(λ)are finite index sets for λ ∈ Λ. Under the above assumptions, we have an equivalent definition of standardly based algebras as follows.

∼  Definition 2.3. An ideal J of A is said to be standard if J = ∆ ⊗R ∆ as A-bimodules, where ∆ ⊆ J is a left ideal of A and ∆ ⊆ J is a right ideal of A, and both ∆ and ∆ are free of finite rank over R. R A An associative -algebra is called a standardly based algebra if there is an   j  R-module decomposition A = J1 ⊕···⊕Jn such that setting Jj = l=1 Jl gives a chain of two-sided ideals of A:0=J0 ⊂ J1 ⊂ J2 ⊂ ··· ⊂ Jn = A,andforeachj,  the quotient Jj = Jj/Jj−1 is a standard ideal of A/Jj−1.WecallthischainofA a standard ideal chain. When A is a finite-dimensional standardly based algebra over a field, [7]con- struct a complete set of simple A-modules (up to isomorphism). Now we introduce a sketch of their theory. For λ ∈ Λ, let ∆(λ)(resp.∆(λ)) be the left (resp. right) A-module with R-basis λ λ {CS | S ∈ I(λ)} (resp. {CT | T ∈ J(λ)}), and λ λ aCS = fS
,λ(a, S)CS
, for a ∈ A S
∈I(λ)    λ λ  resp.CT a = fλ,T
(T,a)CT
. T
∈J(λ)

Define a bilinear pairing  λ λ λ βλ :∆(λ) × ∆(λ) → R, βλ(CS ,CT )=aS,T , (2.1) λ where aS,T is defined by the following equation:

λ λ λ λ CU,T CS,V ≡ aS,T CU,V (mod A(>λ)). Proposition 2.4 ([7]). Let A be a finite-dimensional standardly based algebra over afieldR and let Λ1 = {λ ∈ Λ | βλ =0 }.Then{L(λ)=∆(λ)/rad(∆(λ)) | λ ∈ Λ1} is a complete set of non-isomorphic simple A-modules. Now we recall the definition of 0-Hecke algebras.

Definition 2.5. Let W be a Coxeter group with the generating set S = {si | i ∈ I}. Let A = Z[q] be the polynomial ring with indeterminate q. By definition, the Hecke

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algebra (or Iwahori–Hecke algebra) Hq(W )ofW is the A -algebra with generators Ti,fori ∈ I,andrelations 2 Ti =(q − 1)Ti + q for i ∈ I,

TiTjTiTj ··· = TjTiTjTi ··· for i, j ∈ I with i = j,    

mi,j factors mi,j factors

where mi,j denotes the order of sisj.Ifw = si1 ···sir = sj1 ···sjr are reduced

expressions, then Ti1 ···Tir = Tj1 ···Tjr . So the element Tw := Ti1 ···Tir is well defined. It follows that for given i ∈ I and w ∈ W ,

Tsiw,(siw) >(w), TiTw = (q − 1)Tw + qTsiw,(sw) <(w), where  : W → N is the length function. The degenerate form of Hq(W ) obtained by specializing q = 0 is called 0-Hecke algebra. More precisely, if R is a field, then the 0-Hecke algebra is defined to be the R-algebra

H0(W )=H0(W )R = Hq(W ) ⊗A R, where R is viewed as A -module with the action of q being zero. In other words, H0(W )ofW is the R-algebra with the generators Ti and relations 2 Ti = −Ti for i ∈ I,

TiTjTiTj ···= TjTiTjTi ··· for i, j ∈ I with i = j.    

mi,j factors mi,j factors In this paper, we assume that W is a finite irreducible Coxeter group. Thus, W has type An (n ≥ 1), Bn = Cn (n ≥ 2), Dn (n ≥ 4), E6, E7, E8, F4, H3, H4 or I2(l)(l ≥ 5). If W is of type X,wealsouseH0(X)todenoteH0(W ).

3. Standardly Based Algebras and Morita Equivalences In this section we prove that standardly based algebras are invariant under Morita equivalences. To be precise, suppose that algebras A and B are Morita equivalent. Then A is a standardly based algebra if and only if B is a standardly based algebra. First we recall the definition of Morita equivalence. Denote by A-mod or mod-A the category of finitely generated left or right A-modules, respectively. Then A and B are said to be Morita equivalent if there are two functors F : A-mod → B-mod,G: B-mod → A-mod ∼ ∼ such that FG = idB-mod and GF = idA-mod. Proposition 3.1 ([1]). The following statements are equivalent: (1) A and B are Morita equivalent under the functors F and G as above.

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(2) There are two bimodules APB and BQA and a pair of surjective bimodule homo- morphisms:

θ : P ⊗B Q → A, ϕ : Q ⊗A P → B

such that for p1,p2 ∈ P and q1,q2 ∈ Q,

θ(p1,q1)p2 = p1ϕ(q1,p2),q1θ(p1,q2)=ϕ(q1,p1)q2. In this case,Pand Q are projective both as A-modules and as B-modules, and there are the following isomorphisms: ∼ ∼ F = Q ⊗A − = HomA(P, −), ∼ ∼ G = P ⊗B − = HomB(Q, −), ∼ ∼ A = End(PB ),B= End(AP ). Let J be an ideal of A. Define

ψ(J)=ϕ(QJ ⊗A P ). (3.1) Proposition 3.2 ([13]). Let algebras A and B be Morita equivalent. (1) Let P, Q be the projective modules as above and J be an ideal in A.Then Pψ(J)=JP, QJ = ψ(J)Q. (2) Let J be an ideal in A.ThenA/J and B/ψ(J) are Morita equivalent. Lemma 3.3. Let A be a standardly based algebra and J be a standard ideal of A. Suppose that A and B are Morita equivalent. Then ψ(J) is a standard ideal in B.

Proof. Since J is a standard ideal of A, there is a left ideal ∆ ⊆ J and a right ideal ∆ ⊆ J such that  α :∆⊗R ∆ → J is an A-bimodule isomorphism. Let   Λ=Q ⊗A ∆andΛ=∆ ⊗A P, where P, Q are the projective bimodules defined as in Proposition 3.1.ThenΛisa left ideal of B and Λ is a right ideal of B. Define  β :Λ⊗R Λ → Q ⊗A J ⊗A P, β(q ⊗ δ ⊗ δ ⊗ p)=q ⊗ α(δ ⊗ δ) ⊗ p, where q ∈ Q, δ ∈ ∆,δ ∈ ∆,p∈ P. Then it is easy to check that β is a B-bimodule isomorphism. Note that QA is a projective right A-module, we can identify Q ⊗A J with QJ .By(3.1), ψ(J)= ϕ(QJ ⊗A P ). Then we get the following bimodule isomorphism:  ϕβ :Λ⊗R Λ → ψ(J).

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This proves that ψ(J) is a standard ideal in B.

Theorem 3.4. Suppose that algebras A and B are Morita equivalent and A is a standardly based algebra. Then B is a standardly based algebra.

Proof. Let

0=J0 ⊆ J1 ⊆···⊆Jm = A (3.2)

be a standard ideal chain of A. It suffices to prove that

0=ψ(J0) ⊆ ψ(J1) ⊆···⊆ψ(Jm)=B

is a standard ideal chain of B. By Proposition 3.2, A/Ji is Morita equivalent to B/ψ(Ji)for0≤ i ≤ m.The equivalence can be induced from the following surjective bimodule homomorphisms:

θ P/J P ⊗ Q/QJ → A/J , i : i B/ψ(Ji) i i

p ⊗ q → θ(p, q)+Ji, ϕ Q/QJ ⊗ P/J P → B/ψ J , i : i A/Ji i ( i)

q ⊗ p → ϕ(q, p)+ψ(Ji),

where p ∈ P , q ∈ Q and p, q denote the corresponding elements in quotient algebras. If m = 1, then by Lemma 3.3 B is a standardly based algebra. If m>1, the theorem can be proved by using induction on m.

Proposition 3.5. Let A be a standardly based algebra with an idempotent e ∈ A. Then eAe is a standardly based algebra.

Proof. It suffices to prove that if J is a standard ideal in A then eJe is a standard ∼   ideal in eAe.LetJ = ∆ ⊗R ∆ , where ∆ is a left ideal of A and ∆ is a right ideal  ∼ of A.Thene∆ ⊗R ∆ e = eJe as eAe-bimodules. This proves the proposition.

4. Centers of Symmetric Standardly Based Algebras Throughout this section, we assume that R is a field. Let A be a symmetric standardly based algebra with a standard basis

λ B = {CS,T | S ∈ I(λ),T ∈ J(λ),λ∈ Λ}.

Given a non-degenerate bilinear form f, denote the dual basis by

λ D = {DT,S | S ∈ I(λ),T ∈ J(λ),λ∈ Λ},

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which satisfies

µ λ f(DU,V ,CS,T )=δλµδSV δTU.

Then the so-called Higman ideal Hig(A) in the center of A is as follows:        A Cλ aDλ  a ∈ A . Hig( )= S,T T,S   S∈I(λ),T ∈J(λ),λ∈Λ

For any λ, µ ∈ Λ, S ∈ I(λ),T ∈ J(λ), U ∈ I(µ),V ∈ J(µ), write λ µ ν CS,T CU,V = r(S,T,λ),(U,V,µ),(X,Y,ν)CX,Y . ν∈Λ,X∈I(ν),Y ∈J(ν)

The following lemma gives some formulas on the multiplication between a stan- dard basis and a dual standard basis of A.

Lemma 4.1. For arbitrary λ, µ ∈ Λ and S, P ∈ I(λ),T,Q∈ J(λ),U∈ I(µ), V ∈ J(µ), the following conditions hold:  λ µ ν (1) CS,T DV,U = ν∈Λ,X∈I(ν),Y ∈J(ν) r(X,Y,ν),(S,T,λ),(U,V,µ)DY,X. λ λ (2) CS,T DQ,P =0if T = Q. λ µ (3) CS,T DV,U =0if µ λ. λ λ λ λ (4) CS,T DT,P = CS,QDQ,P .

Proof. It can be proved by using the same way as that of Lemma 3.1 in [16].  λ λ By (4) of Lemma 4.1, S∈I(λ) CS,T DT,S is independent of T . So we can define λ λ eλ = CS,T DT,S, for λ ∈ Λ. S∈I(λ)

Let L(A)betheR-submodule of A spanned by eλ, i.e.     L A r e  r ∈ R . ( )= λ λ  λ λ∈Λ

Then by a same way as that of Theorem 3.2 in [15], we get the following lemma.

Lemma 4.2. Let A be a symmetric standardly based algebra. Then

Hig(A) ⊆ L(A) ⊆ Z(A),

where Z(A) is the center of A.

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Theorem 4.3. Let A be a finite-dimensional symmetric standardly based algebra over a field R.ThendimR Z(A) ≥|Λ1|, where |Λ1| is the number of non-isomorphic simple modules of A.

Proof. By Lemma 4.2, L(A) ⊆ Z(A). It suffices to prove that

dimR L(A) ≥|Λ1|. (4.1)

Let {L(λ) | λ ∈ Λ1} be a complete set of simple A-modules as shown in Proposi- tion 2.4. We will prove (4.1) by showing that {eλ | λ ∈ Λ1} is a linearly independent set of L(A). λ By Proposition 2.4, βλ =0foreach λ ∈ Λ1.SothereareCS ∈ ∆(λ)and λ  λ λ CT ∈ ∆ (λ) such that βλ(CS ,CT ) = 0. By Lemma 4.1, it is easy to check that λ λ λ βλ(CS ,CT ) is just the coefficient of DT,S in the expansion of eλ.Andthecoefficient Dλ e µ λ of T,S in the expansion of µ is zero, where . rλeλ λ0 1 rλ Suppose λ∈Λ1 =0and is a minimal element in Λ .Then 0 =0by the above analysis. By induction on the order of Λ1, we can prove that rλ =0for each λ ∈ Λ1. This proves the theorem.

5. 0-Hecke Algebras In this section we first prove that 0-Hecke algebras and 0-Schur algebras are stan- dardly based algebras over R. As an application we give a new way to classify simple modules of the 0-Hecke algebra H0(W ). We investigate the dimension of the center of H0(W ) in the last part of this section.

Theorem 5.1. Let W be a finite irreducible Coxeter group with generating set I. Then the 0-Hecke algebra H0(W ) is a standardly based algebra over R. Moreover, if R is a field, then the simple modules of H0(W ) are indexed by subsets of I.

Proof. By Definition 2.5, H0(W ) has an R-basis {Tw | w ∈ W }. Fix a total ordering on the elements of W such that wi >wj whenever (wi) >(wj). Then H0(W )is a standardly based algebra with standard basis {Tw | w ∈ W },Λ=W and each I(w), J(w) introduced in Definition 2.1 consists only one element for w ∈ W . In order to classify the simple modules of H0(W ), it suffices to determine the set  Λ1 by Proposition 2.4.Foreachw ∈ Λ, ∆(w)and∆(w) are both of one-dimensional over R with basis Tw. And the bilinear pairing in (2.1) is as follows:

 βw :∆(w) × ∆ (w) → R,

0,(w2) >(w), (5.1) βw(Tw,Tw)= (−1)(w),(w2) <(w).

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For each subset J ⊆ I,letWJ be the Young subgroup of W which is generated by sj for j ∈ J. Suppose that wJ is the longest element in WJ .Itiseasytocheck that

(w ) (−1) J ,w= wJ , βw(Tw,Tw)= 0, otherwise.

Then by Proposition 2.4, {∆(wJ ) | J ⊆ I} forms a complete set of non-isomorphic simple modules of H0(W ).

Corollary 5.2. Let H0(r) be the 0-Hecke algebra of type Ar−1 and S0(n, r) be the 0-Schur algebra corresponding to H0(r).ThenS0(n, r) is a standardly based algebra.

Proof. We have shown in Remark 4.3 of [5] that for arbitrary n, r, S0(n, r)is Morita equivalent to eH0(r)e,forsomeidempotente ∈ H0(r). Then by Theo- rem 3.4,Proposition3.5 and Theorems 5.1, S0(n, r) is a standardly based algebra.

Now we turn our attention to the dimension of the center of H0(W ). We always assume R is a field in this part. First recall a lemma from [10].

Lemma 5.3 ([10]). Let W be a finite irreducible Coxeter group. Then H0(W ) is a symmetric algebra over R if and only if W has one of the following types:

A1,Bn = Cn (n ≥ 2),D2n (n ≥ 2),E7,E8,F4,H3,H4,I2(2n)(n ≥ 3). (5.2)

By combining Theorem 4.3,Theorem5.1 and Lemma 5.3, we get the following theorem.

Theorem 5.4. Let W be a finite irreducible Coxeter group which has one of the types listed in (5.2). Then dimR Z(H0(W )) ≥|Λ1|, where Z(H0(W )) is the center of H0(W ), and |Λ1| is the number of simple modules of H0(W ) up to isomorphism.

Example 5.5. By Lemma 5.3, H0(B2) is a symmetric algebra. Let Q be the fol- lowing quiver:

α1 Q rr- :  . α2

By [4, Theorem 3.1], ∼ H0(B2) = R × R × RQ/J,

where J is the ideal of the path algebra RQ generated by {α1α2α1,α2α1α2}.Itcan be checked directly that: dimRZ(H0(B2)) = 4; three central primitive idempotents

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and α1α2 + α2α1 form a basis of Z(H0(B2)). Note that H0(B2)hasfoursimple modules. Then dimRZ(H0(B2)) equals the number of its simple modules.

Remark 5.6. If H0(W ) is not a symmetric algebra, dimRZ(H0(W )) may be equal or smaller than the number of simple modules of H0(W ). We can show this by two examples. Let Q and Q be the following quivers: α 1 - Q: rr α2

α1-  α3 123qq - q α α 2 6 4 Q: β2 β1

δ1 - ? δ3 qq - q . 456δ2 δ4

By [4, Theorem 3.1], we have the following algebra isomorphisms: ∼ H0(A2) = R × R × RQ/J,

where J is the ideal of the path algebra RQ generated by {α1α2,α2α1},and ∼   H0(A3) = R × R × RQ /J , where J  is the ideal of the path algebra RQ generated by

{α2α1,α4α3,δ2δ1,δ4δ3,α2β1,α4β1,β1δ1,β1δ3,δ2β2,δ4β2,β2α1,β2α3,

α1α2 − α3α4,β1β2 − α1α2,δ1δ2 − δ3δ4,β2β1 − δ1δ2}.

The indecomposable projective modules of H0(4) are as follows:

P1 : 1 P3 : 3 P4 : 4 P6 : 6     2 2 5 5     3164

P ? P ? 2 : 2 ?? 5 : 5 ??  ??  ??   ?   ? ? ? . 1 ?? 5 3 4 ?? 2 6 ??  ??  ?  ?  25

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Then it can be checked directly that:

(1) dimRZ(H0(A2)) = 3; three central primitive idempotents form a basis of Z(H0(3)); (2) dimRZ(H0(A3)) = 5; three central primitive idempotents together with α1α2,δ1δ2 form a basis of Z(H0(4)).

By Theorem 5.1, H0(A2) has four simple modules and H0(A3) has eight simple modules.

If W is of type I2(l)(l ≥ 5), then H0(W ) has four simple modules. In this case we can compute dimRZ(H0(W )) precisely in the following proposition. This proposition shows that the dimension of the center of H0(W )maybemuchlarger than the number of its simple modules.

Proposition 5.7. Suppose W has type I2(l)(l ≥ 5).Then  l +3  ,lis odd, Z H W 2 dimR ( 0( )) =  l +4  ,lis even. 2

Proof. Let Q be the following quiver: α 1 - Q:1rr 2. α2

If W is of type I2(l), then by [4, Theorem 3.1] ∼ H0(W ) = R × R × RQ/J, where RQ is the path algebra of Q and J is the ideal of RQ generated by all paths of length l − 1. Let ρ1,i be the path starting and ending at vertex 1 of length i,andletρ2,i be the path starting and ending at vertex 2 of length i,wherei is an even number. For example, ρ1,4 = α2α1α2α1 and ρ2,2 = α1α2.Wesetρ1,0 = e1 and ρ2,0 = e2,where e1,e2 are the primitive idempotents corresponding to vertex 1 and 2, respectively. Then it can be checked directly that the set

{ρ1,i + ρ2,i | 0 ≤ i ≤ l − 2,iis a even number} formsabasisofthecenterofRQ/J. Then  l − 1  ,lis odd, Z RQ/J 2 dimR ( )=  l  ,lis even. 2

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Since ∼ H0(W ) = R × R × RQ/J, we get that  l +3  ,lis odd, Z H W 2 dimR ( 0( )) =  l +4  ,lis even. 2

Acknowledgments The authors would like to thank the referee for helpful comments on the paper. The first author is supported by the Natural Science Foundation of China (Grant no. 11201269 and 11126122), the Natural Science Foundation of Shandong Province (Grant no. ZR2011AQ004), and Young Scholars Research Fund of Shandong Univer- sity of Technology. The second author is supported by the Natural Science Founda- tion of Hebei Province (A2013501055), Fundamental Research Funds for the Central Universities (N130423011) and the National Natural Science Foundation of China (Grant no. 11301195).

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Standardly based algebras and 0-Hecke algebras

[14] D. Krob and J. Y. Thibon, Noncommutative symmetric functions. IV. Quantum linear groups and Hecke algebras at q =0,J. Algebraic Combin. 6 (1997) 339–376. [15] Y. Li, Centers of symmetric cellular algebras, Bull. Aust. Math. Soc. 82 (2010) 511– 522. [16] Y. Li, Radicals of symmetric cellular algebra, Colloq. Math. 133 (2013) 67–83. [17] P. N. Norton, 0-Hecke algebras, J. Aust. Math. Soc. 27 (1979) 337–357.

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