DOCSLIB.ORG

On Τ-Tilting Finiteness of the Schur Algebra

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Τ-Tilting Finiteness of the Schur Algebra On τ-tilting finiteness of the Schur algebra Qi Wang∗ Abstract We determined the τ-tilting finiteness of Schur algebras over an algebraically closed field of arbitrary characteristic, except for a few small cases. Keywords: Schur algebras, τ-tilting finite, representation-finiteness. 1 Introduction Throughout this paper, we always assume that A is a finite-dimensional basic algebra over an algebraically closed field F of characteristic p> 0. In particular, the representa- tion type of A is divided into representation-finite, (infinite-)tame and wild. In recent years, τ-tilting theory introduced by Adachi, Iyama and Reiten [AIR] has drawn more and more attention. Here, τ is the Auslander-Reiten translation. The crucial concept of τ-tilting theory is the notion of support τ-tilting modules (see Definition 2.1), which can be regarded as a generalization of classical tilting modules from the viewpoint of mutation, that is, any basic almost complete support τ-tilting module is a direct summand of exactly two basic support τ-tilting modules. Moreover, support τ-tilting modules are in bijection with many important objects in representation theory, such as two-term silting complexes, functorially finite torsion classes, left finite semibricks, etc. We refer to [AIR] and [Asa] for more details. Similar to the representation finiteness, a modern notion named τ-tilting finiteness is introduced by Demonet, Iyama and Jasso [DIJ]. We recall that an algebra A is said to be τ-tilting finite if it has only finitely many pairwise non-isomorphic basic support τ-tilting modules. (There are several equivalent conditions in Definition 2.2 and Proposition 2.3.) A typical example of τ-tilting finite algebras is the class of representation-finite algebras. arXiv:2010.05206v2 [math.RT] 9 May 2021 We remark that the τ-tilting finiteness for several classes of algebras is determined, such as algebras with radical square zero [Ada], preprojective algebras of Dynkin type [Mi], Brauer graph algebras [AAC], biserial algebras [Mo] and minimal wild two-point algebras [W1]. In addition, it has been proved in some cases that τ-tilting finiteness coincides with representation-finiteness, including cycle-finite algebras [MS], gentle algebras [Pl], cluster-tilted algebras [Zi], simply connected algebras [W2], quasi-tilted algebras, locally hereditary algebras, etc. [AHMW]. In this paper, we focus on (classical) Schur algebras, which play an important role in both Representation theory and Lie theory. This class of algebras arose in connection with ∗Q.W. is supported by the JSPS Grant-in-Aid for JSPS Fellows 20J10492. 1 the theory of polynomial representations over the general linear group GLn(F) and then received widespread attention. So far, the representation theory of Schur algebras has developed very well and many derivatives appeared, such as q-Schur algebras, infinitesimal Schur algebras, Borel-Schur algebras and so on. In particular, the representation type of Schur algebras is completely determined by various authors, including Erdmann [Er], Xi [Xi], Doty-Nakano [DN] and Doty-Erdmann-Martin-Nakano [DEMN]. We summarize these results in Proposition 3.1. Let n, r be two positive integers and V an n-dimensional vector space over F. We ⊗r denote by V the r-fold tensor product V ⊗F V ⊗F ···⊗F V . Then, the symmetric group ⊗r Gr has a natural action (by permutation) on V which makes it to be a module over ⊗r the group algebra FGr of Gr. Then, the endomorphism ring EndFGr (V ) is called the Schur algebra (see [Ma, Section 2]) and we denote it by S(n, r). Our initial motivation is to check whether τ-tilting finiteness and representation- finiteness coincide in the class of Schur algebras. It is clear that representation-finite Schur algebras are τ-tilting finite. Moreover, we have already determined the number of pairwise non-isomorphic basic support τ-tilting modules blockwise in Theorem 3.2. Therefore, we start with the tame cases. Theorem 1.1 (Theorem 3.4). If the Schur algebra S(n, r) is tame, it is τ-tilting finite. In order to prove the above result, we may show that any block algebra of tame Schur algebras is τ-tilting finite. Let S(n, r) be a tame Schur algebra and A := eS(n, r)e an indecomposable block algebra with a central idempotent e of S(n, r). It is known from [Er] and [DEMN, Section 5] that the block algebra A is Morita equivalent to one of F, A2, D3, D4, R4 and H4 (see Section 3 for the definitions). Then, we have Theorem 1.2 (Theorem 3.2 and Lemma 3.3). Let sτ-tilt A be the set of pairwise non- isomorphic basic support τ-tilting A-modules. Then, A A2 D3 D4 R4 H4 . #sτ-tilt A 6 28 114 88 96 Since the above results give a negative answer to our initial motivation, we next ask whether all wild Schur algebras are τ-tilting infinite. There are a few cases in (⋆) that we currently do not have a suitable method to deal with. Since the number of pairwise non-isomorphic basic support τ-tilting modules for them is huge, we cannot confirm the τ-tilting finiteness of these exceptions by direct calculation. So, we leave these cases here and expect to find a new reduction method (for example, Conjecture 4.13) to show the τ-tilting finiteness. p =2, n =2,r =8, 17, 19; p =2, n =3,r = 4; (⋆) p =2, n > 5,r = 5; p > 5, n =2,p2 6 r 6 p2 + p − 1. 2 Now, we are able to determine the τ-tilting finiteness for most wild Schur algebras. Theorem 1.3 (Table 1, Table 2 and Table 3). Let p be the characteristic of F. Except for the cases in (⋆), the wild Schur algebra S(n, r) is τ-tilting finite if and only if p =2, n =2,r =6, 13, 15 or n =3,r =5 or n =4,r =4. We have the following answers for our motivations. Corollary 1.4. Let p =3. Then, S(n, r) is τ-tilting infinite if and only if it is wild. Corollary 1.5. Let p > 5. Except for the cases in (⋆), S(n, r) is τ-tilting finite if and only if it is representation-finite. Actually, except for the cases in (⋆), we have observed from the proof process that the Schur algebra S(n, r) is τ-tilting infinite if and only if it has one of τ-tilting infinite quivers Q1, Q2 and Q3 (see Lemma 2.15) as a subquiver. This leads us to conjecture that all cases in (⋆) are τ-tilting finite, because all the cases in (⋆) do not have one of Q1, Q2 and Q3 as a subquiver. This paper is organized as follows. In Section 2, we first recall some basic materials on τ-tilting theory and the Schur algebra S(n, r). Then, we give several reduction theorems on the τ-tilting finiteness of Schur algebras, such that we only need to consider small n and r. At the end of the section, we explain our strategy to prove S(n, r) to be τ-tilting infinite. In Section 3, we show that all tame Schur algebras are τ-tilting finite. In Section 4, we determine the τ-tilting finiteness of most wild Schur algebras. Acknowledgements. This paper is inspired by another on-going joint work with Susumu Ariki, Ryoichi Kase, Kengo Miyamoto and Euiyong Park, where the authors are using τ-tilting theory to classify τ-tilting finite block algebras of Hecke algebras of classical type. I am very grateful to them, especially, Prof. Ariki, for giving me a lot of kind guidance in writing this paper. I am also very grateful to the anonymous referee for giving me many useful suggestions. 2 Preliminaries We recall that any finite-dimensional basic algebra A over an algebraically closed field F is isomorphic to a bound quiver algebra FQ/I, where FQ is the path algebra of the Gabriel quiver Q = QA of A and I is an admissible ideal of FQ. We refer to [ASS] for more background materials on quiver and representation theory of algebras. 2.1 τ-tilting theory In this subsection, we review some background of τ-tilting theory. However, we only review the background needed for this paper, so that many aspects related to τ-tilting theory are omitted. One may look at [AIR], [Asa] and [AIRRT] for more materials. 3 Let mod A be the category of finitely generated right A-modules and proj A the category of finitely generated projective right A-modules. For any M ∈ mod A, we denote by add(M) (respectively, Fac(M)) the full subcategory of mod A whose objects are direct summands (respectively, factor modules) of finite direct sums of copies of M. We denote by Aop the opposite algebra of A and by |M| the number of isomorphism classes of indecomposable direct summands of M. In particular, we often describe A- modules via their composition factors. For example, each simple A-module Si is written S1 1 as i, and then S2 ≃ 2 is an indecomposable A-module M with a unique simple submodule S2 such that M/S2 = S1. We recall that the Nakayama functor ν := D(−)∗ is induced by the dualities op ∗ op D := HomF(−, F): mod A ↔ mod A and (−) := HomA(−, A): proj A ↔ proj A . Then, for any M ∈ mod A with a minimal projective presentation f1 f0 P1 −→ P0 −→ M −→ 0, the Auslander-Reiten translation τM is defined by the exact sequence: νf1 0 −→ τM −→ νP1 −→ νP0. Definition 2.1.
Load more

Recommended publications
  • On Schur Algebras, Doty Coalgebras and Quasi-Hereditary Algebras
    CORE Metadata, citation and similar papers at core.ac.uk Provided by White Rose E-theses Online ON SCHUR ALGEBRAS, DOTY COALGEBRAS AND QUASI-HEREDITARY ALGEBRAS Rachel Ann Heaton University of York Department of Mathematics Thesis submitted to the University of York for the degree of Doctor of Philosophy December 2009 1 Abstract Motivated by Doty's Conjecture we study the coalgebras formed from the coefficient spaces of the truncated modules Tr λE. We call these the Doty Coalgebras Dn;p(r). We prove that Dn;p(r) = A(n; r) for n = 2, and also that Dn;p(r) = A(π; r) with π a suitable saturated set, for the cases; i) n = 3, 0 ≤ r ≤ 3p − 1, 6p − 8 ≤ r ≤ n2(p − 1) for all p; ii) p = 2 for all n and all r; iii) 0 ≤ r ≤ p − 1 and nt − (p − 1) ≤ r ≤ nt for all n and all p; iv) n = 4 and p = 3 for all r. The Schur Algebra S(n; r) is the dual of the coalgebra A(n; r), and S(n; r) we know to be quasi-hereditary. Moreover, we call a finite dimensional coal- gebra quasi-hereditary if its dual algebra is quasi-hereditary and hence, in the above cases, the Doty Coalgebras Dn;p(r) are also quasi-hereditary and thus have finite global dimension. We conjecture that there is no saturated set π such that D3;p(r) = A(π; r) for the cases not covered above, giving our reasons for this conjecture. Stepping away from our main focus on Doty Coalgebras, we also describe an infinite family of quiver algebras which have finite global dimension but are not quasi-hereditary.
    [Show full text]
  • The Partition Algebras and a New Deformation of the Schur Algebras
    JOURNAL OF ALGEBRA 203, 91]124Ž. 1998 ARTICLE NO. JA977164 The Partition Algebras and a New Deformation of the Schur Algebras P. MartinU Department of Mathematics, City Uni¨ersity, Northampton Square, London, EC1V 0H8, View metadata, citation and similar papers at core.ac.ukUnited Kingdom brought to you by CORE provided by Elsevier - Publisher Connector and D. Woodcock² School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London, E1 4NS, United Kingdom Communicated by Gordon James Received August 30, 1996 THIS PAPER IS DEDICATED TO THE MEMORY OF AHMED ELGAMAL P Ž. Z The partition algebras PZw¨ x n are algebras defined over the ring wx¨ of integral polynomials. They are of current interest because of their role in the transfer matrix formulation of colouring problems and Q-state Potts models in statistical mechanicswx 11 . Let k be a Zwx¨ -algebra which is a P Ž.P Ž. field, and let Pk n be the algebra obtained from PZw¨ x n by base change to k, a finite dimensional k-algebra. If char k s 0 the representation theory of PkŽ.n is generically semisimple, and the exceptional cases are completely understoodwxwx 11, 12, 14 . In 15 we showed that if char k ) 0 and the image of ¨ does not lie in the prime subfield of k, PkŽ.n is Morita equivalent to kS 0 = ??? = kS n, a product of group algebras of symmetric groups. In the present paper we show that in the remaining cases in positive characteristic the representation theory of PkŽ.n can be obtained by localisation from that of a suitably chosen classical Schur algebraŽ.
    [Show full text]
  • Presenting Affine Q-Schur Algebras
    Mathematische Zeitschrift manuscript No. (will be inserted by the editor) S. R. Doty R. M. Green · Presenting affine q-Schur algebras Received: date / Revised: date Abstract We obtain a presentation of certain affine q-Schur algebras in terms of generators and relations. The presentation is obtained by adding more relations to the usual presentation of the quantized enveloping algebra of type affine gln. Our results extend and rely on the corresponding result for the q-Schur algebra of the symmetric group, which were proved by the first author and Giaquinto. Mathematics Subject Classification (2000) 17B37, 20F55 Introduction r Let V ′ be a vector space of finite dimension n. On the tensor space (V ′)⊗ we have natural commuting actions of the general linear group GL(V ′) and the symmetric group Sr. Schur observed that the centralizer algebra of each action equals the r image of the other action in End((V ′)⊗ ), in characteristic zero, and Schur and Weyl used this observation to transfer information about the representations of Sr to information about the representations of GL(V ′). That this Schur–Weyl dual- ity holds in arbitrary characteristic was first observed in [4], although a special case was already used in [2]. In recent years, there have appeared various applica- tions of the Schur–Weyl duality viewpoint to modular representations. The Schur algebras S(n,r) first defined in [9] play a fundamental role in such interactions. Jimbo [13] and (independently) Dipper and James [6] observed that the ten- r sor space (V ′)⊗ has a q-analogue in which the mutually centralizing actions of GL(V ′) and Sr become mutually centralizing actions of a quantized enveloping algebra U(gln) and of the Iwahori-Hecke algebra H (Sr) corresponding to Sr.
    [Show full text]
  • Arxiv:1203.6370V2 [Math.RT]
    YOUNG MODULE MULTIPLICITIES AND CLASSIFYING THE INDECOMPOSABLE YOUNG PERMUTATION MODULES CHRISTOPHER C. GILL Abstract. We study the multiplicities of Young modules as direct summands of permutation modules on cosets of Young subgroups. Such multiplicities have become known as the p-Kostka numbers. We classify the indecomposable Young permutation modules, and, applying the Brauer construction for p-permutation modules, we give some new reductions for p-Kostka numbers. In particular we prove that p-Kostka numbers are preserved under multiplying partitions by p, and strengthen a known reduction corresponding to adding multiples of a p-power to the first row of a partition. The symmetric group permutation modules on the cosets of Young subgroups are known as the Young permutation modules. They play a central role in the representation theory of the symmetric group, and also in Schur algebras, relating representations of symmetric groups to polynomial representations of general linear groups. The indecomposable modules occurring in direct sum decompositions of Young permutation modules were parametrized by James, and have become known as the Young modules. In the semisimple case, the Young modules are the familiar Specht modules, and their multiplicities as direct summands of permutation modules (the Kostka numbers) have long been known by Young’s rule. However, when the group algebra of the sym- metric group is not semisimple, the multiplicities remain undetermined. It is well-known that the determination of these multiplicities is equivalent to determining the decomposition numbers of the symmetric groups and the Schur algebras. Let Sr be the symmetric group on r letters, and let k be a field of characteristic p.
    [Show full text]
  • On Category O for the Rational Cherednik Algebra of the Complex Reflection Group
    On category for the rational O Cherednik algebra of the complex reflection group (Z/!Z) Sn ! by Richard Thomas Robert Vale A thesis submitted to the Faculty of Information and Mathematical Sciences at the University of Glasgow for the degree of Doctor of Philosophy December 2006 c Richard Vale 2006 " 2 Acknowledgements I would like to thank my supervisors Prof. K. A. Brown and Prof. I. Gordon for giving me a vast amount of help and suggesting many problems and their solutions. I would also like to thank Prof. T. Lenegan of the University of Edinburgh for helping me to obtain funding to visit Chicago in April 2006, and for arranging for me to visit Edinburgh in July 2006. I would like to thank Q. Gashi, V. Ginzburg and I. Gordon for their hospitality during my visit to Chicago. I would like to thank those mathematicians with whom I have had valuable conversations or correspondence, in particular S. Ariki, O. Chalykh, and C. Stroppel. I would like to thank many of my colleagues for their friendship. I acknowledge that my studies were funded by the Engineering and Physical Sciences Research Council. i Statement This thesis is submitted in accordance with the regulations for the degree of Doctor of Philosophy in the University of Glasgow. Chapters 1 and 2 cover notation, definitions and known results (apart from Theorem 2.1). Chapters 3, 4 and 5 are the author’s original work except where stated otherwise. Some of the results therein will appear in [70] and [69]. ii Summary The topic of this thesis is the rational Cherednik algebra of the complex reflection group (Z/!Z) Sn, ! and in particular the category of modules over the Cherednik algebra.
    [Show full text]
  • The Coefficient Coalgebra of a Symmetrized Tensor Space
    THE COEFFICIENT COALGEBRA OF A SYMMETRIZED TENSOR SPACE RANDALL R. HOLMES DAVID P. TURNER Abstract. The coefficient coalgebra of r-fold tensor space and its dual, the Schur algebra, are generalized in such a way that the role of the symmetric group Σr is played by an arbitrary subgroup of Σr. The dimension of the coefficient coalgebra of a symmetrized tensor space is computed and the dual of this coalgebra is shown to be isomorphic to the analog of the Schur algebra. 0. Introduction Let K be the field of complex numbers. The vector space E = Kn is naturally viewed as a (left) module for the group algebra KΓ of the general ⊗r linear group Γ = GLn(K). The r-fold tensor product E is in turn a module for KΓr, where Γr = Γ × · · · × Γ(r factors). Let G be a subgroup of the symmetric group Σr and let χ : G ! K be an irreducible character χ ⊗r of G. The \symmetrized tensor space" associated with χ is E = E tχ, where tχ is the central idempotent of the group algebra KG corresponding to χ (with the action of G on E⊗r being given by place permutation). In this paper, we study the coefficient coalgebra Aχ of the R-module Eχ, where R is the subalgebra of KΓr consisting of those elements fixed r by G under the action of Σr on Γ given by entry permutation. (If V is an R-module with finite K-basis fvj j j 2 Jg, then for each κ 2 R, we have P κvj = i αij(κ)vi for some αij : R ! K.
    [Show full text]
  • Polynomial Representations of the General Linear Group
    Imperial College London MSci Thesis Polynomial Representations of the General Linear Group Author: Supervisor: Misja F.A. Steinmetz Dr. John R. Britnell A thesis submitted in fulfilment of the requirements for the degree of Master in Science in the Algebra Section Department of Mathematics \This is my own work unless otherwise stated." Name: Date: June 2014 Misja F.A. Steinmetz CID: 00643423 KNSM-laan 50 1019 LL Amsterdam the Netherlands [email protected] [email protected] i Abstract The main goal of this project will be to describe and classify all irreducible characters of polynomial representations of GLn(K) for an infinite field K. We achieve this goal in Theorem 4.5.1 at the end of Chapter 4. Our journey towards this big theorem takes us past many interesting topics in algebra and representation theory. In Chapter 1 we will do some of the necessary groundwork: we will introduce the concepts of coalgebras and bialgebras. In Chapter 2 we will introduce finitary functions and coefficient functions (following Green's nomenclature [9]). We will use results from Chapter 1 to deduce some initial consequences from these definitions. In Chapter 3 we will introduce the category MK (n) of finite-dimensional left KGLn(K)-modules which ‘afford’ polynomial representations. This category will be the main object of study in this and the next chapter. Next we introduce the Schur algebra SK (n) and prove that left SK -modules are equivalent to left modules in MK . In Chapter 4 we introduce weights, weight spaces and formal characters. We use these results to prove our big theorem.
    [Show full text]
  • Pp. 131–140. General Linear Supergroups And
    Lecture Notes of Seminario Interdisciplinare di Matematica Vol. 8(2009), pp. 131–140. General linear supergroups and Schur superalgebras Roberto La Scala and Alexander Zubkov Abstract1. In this paper we present a review of some recent results and open prob- lems about the category of polynomial supermodules of general linear supergroups. Once introduced the Schur superalgebra S(m n, r) for the general linear supergroup GL(m n), | | we consider the problem of what properties can and cannot be generalized from the ones satified, in the classical case, by the Schur algebra S(n, r) and the general linear group GL(n). For instance, one proves that in general S(m n, r) is not a quasi-hereditary al- | gebra even if the category of GL(m n)-supermodules is highest weight. Finally, we give | an explicit description of the bases of a class of costandard modules (λ)ofS(m n, r)in r | terms of pairs of superstandard tableaux. 1. Introduction The concept of a Schur algebra originated in two pioneering articles of Schur [25, 26]. The Schur algebra S(n, r) associated to any integers n, r > 0isafinite dimensional quotient algebra of the group algebra KG of the general linear group G = GL(n)overthefieldK, that has the property that the category M(n, r) of polynomial G-modules that are homogeneous of degree r is equivalent to the category of S(n, r)-modules. In the first paper [25] Schur determined the polyno- mial representations of GL(n) over the field of complex numbers. This was done using what is now called “Schur functor” that provides an equivalence between the category M(n, r) and the category of modules over the group algebra KSr of the symmetric group Sr.
    [Show full text]
  • Schur Algebras of Classical Groups II 1 Introduction
    Schur Algebras of Classical Groups II Qunhua Liu1 Mathematics Institute, University of Koeln, Weyertal 86-90, 50931 Koeln, Germany. Email address: [email protected] Abstract We study Schur algebras of classical groups over an algebraically closed ¯eld of characteristic not 2. We prove that Schur algebras are generalized Schur al- gebras (in Donkin's sense) in types A, C and D, while are not in type B. Consequently Schur algebras of types A, C and D are integral quasi-hereditary by Donkin [8, 10]. By using the coalgebra approach we put Schur algebras of a ¯xed classical group into a certain inverse system. We ¯nd that the correspond- ing hyperalgebra is contained in the inverse limit as a subalgebra. Moreover in types A, C and D, the surjections in the inverse systems are compatible with the integral quasi-hereditary structure of Schur algebras. Finally we calculate some Schur algebras with small parameters and prove Schur{Weyl duality in types B, C and D in a special case. Key words: Schur algebra, Classical group, Generalized Schur algebra, Inverse system, Hyperalgebra. 1 Introduction Throughout the paper K is an algebraically closed ¯eld of characteristic not 2, G a classical group and G0 the corresponding group of similitudes de¯ned over K. Let E be the standard representation of G and G0, and E­r (r > 1) the r-fold tensor space on which G and G0 act diagonally. Doty [11] has de¯ned the corresponding Schur algebra to be the image of the ­r 0 ­r representation map KG ¡! EndK (E ), which equals that of KG ¡! EndK (E ).
    [Show full text]
  • On Comparing the Cohomology of General Linear and Symmetric Groups
    Pacific Journal of Mathematics ON COMPARING THE COHOMOLOGY OF GENERAL LINEAR AND SYMMETRIC GROUPS Alexander S. Kleshchev and Daniel K. Nakano Volume 201 No. 2 December 2001 PACIFIC JOURNAL OF MATHEMATICS Vol. 201, No. 2, 2001 ON COMPARING THE COHOMOLOGY OF GENERAL LINEAR AND SYMMETRIC GROUPS Alexander S. Kleshchev and Daniel K. Nakano In this paper the authors explore relationships between the cohomology of the general linear group and the symmetric group. Stability results are given which show that the coho- mology of these groups agree in a certain range of degrees. 1. Introduction. 1.1. Let k be an algebraically closed field, GLn(k) be the general linear group over k, and Σd be the symmetric group on d letters. For k = C, Frobenius and Schur discovered that the commuting actions of GLn(k) and ⊗d Σd on V can be used to relate the character theory of these two groups. For modular representations of these groups the relationship between their representation theories is not as direct. However, James [12] showed that the decomposition matrix of the symmetric group is a submatrix of the de- composition matrix of the general linear group, and Erdmann [9] used tilting modules to prove that one can also recover the decomposition numbers of the general linear groups from those of the symmetric groups. Let M(n, d) be the category of polynomial representations of GLn(k) of a fixed degree d ≤ n. The Schur functor F is a certain covariant exact functor from M(n, d) to modules for the group algebra kΣd ([10]).
    [Show full text]
  • ON COMPLEX REFLECTION GROUPS G(M,1,R)
    ON COMPLEX REFLECTION GROUPS G(m; 1; r) AND THEIR HECKE ALGEBRAS CHI KIN MAK A thesis submitted in fulfilment Of the requirements for the degree of Doctor of Philosophy School of Mathematics University of New South Wales AUGUST, 2003 Abstract We construct an algorithm for getting a reduced expression for any element in a complex reflection group G(m; 1; r) by sorting the element, which is in the form of a sequence of complex numbers, to the identity. Thus, the algorithm provides us a set of reduced expressions, one for each element. We establish a one-one correspondence between the set of all reduced expressions for an element and a set of certain sorting sequences which turn the element to the identity. In particular, this provides us with a combinatorial method to check whether an expression is reduced. We also prove analogues of the exchange condition and the strong exchange condition for elements in a G(m; 1; r). A Bruhat order on the groups is also defined and investigated. We generalize the Geck-Pfeiffer reducibility theorem for finite Coxeter groups to the groups G(m; 1; r). Based on this, we prove that a character value of any element in an Ariki-Koike algebra (the Hecke algebra of a G(m; 1; r)) can be determined by the character values of some special elements in the algebra. These special elements correspond to the reduced expressions, which are constructed by the algorithm, for some special conjugacy class representatives of minimal length, one in each class. Quasi-parabolic subgroups are introduced for investigating representations of Ariki- Koike algebras.
    [Show full text]
  • G-SCHUR ALGEBRAS CORRESPONDING to HECKE
    Transformation Groups ©Springer Science+Business Media New York (??) q-SCHUR ALGEBRAS CORRESPONDING TO HECKE ALGEBRAS OF TYPE B C.-J. LAI D. K. NAKANO˚ Institute of Mathematics Department of Mathematics Academia Sinica University of Georgia Taipei 10617, Taiwan Athens, GA 30602, USA [email protected] [email protected] Z. XIANG˚˚ Institute of Mathematics Academia Sinica Taipei 10617, Taiwan [email protected] Abstract. In this paper the authors investigate the q-Schur algebras of type B that were constructed earlier using coideal subalgebras for the quantum group of type A. The authors present a coordinate algebra type construction that allows us to realize these q-Schur algebras as the duals of the dth graded components of certain graded coalgebras. Under suitable conditions an isomorphism theorem is proved that demonstrates that the representation theory reduces to the q-Schur algebra of type A. This enables the authors to address the questions of cellularity, quasi-hereditariness and representation type of these algebras. Later it is shown that these algebras realize the 1-faithful quasi hereditary covers of the Hecke algebras of type B. As a further consequence, the authors demonstrate that these algebras are Morita equivalent to the category O for rational Cherednik algebras for the Weyl group of type B. In particular, we have introduced a Schur-type functor that identifies the type B Knizhnik-Zamolodchikov functor. 1. Introduction 1.1. Schur-Weyl duality has played a prominent role in the representation theory of groups and algebras. The duality first appeared as method to connect the rep- resentation theory of the general linear group GLn and the symmetric group Σd.
    [Show full text]