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Standardly Based Algebras and 0-Hecke Algebras 1. Introduction

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Standardly Based Algebras and 0-Hecke Algebras 1. Introduction 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. 1550141-1 August 22, 2015 9:7 WSPC/S0219-4988 171-JAA 1550141 G. Yang & Y. Li 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) 1550141-2 August 22, 2015 9:7 WSPC/S0219-4988 171-JAA 1550141 Standardly based algebras and 0-Hecke algebras 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 1550141-3 August 22, 2015 9:7 WSPC/S0219-4988 171-JAA 1550141 G. Yang & Y. Li 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. 1550141-4 August 22, 2015 9:7 WSPC/S0219-4988 171-JAA 1550141 Standardly based algebras and 0-Hecke algebras (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.
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