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REPRESENTATION TYPE of Q-SCHUR ALGEBRAS 1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 12, Pages 4729{4756 S 0002-9947(01)02849-5 Article electronically published on July 11, 2001 REPRESENTATION TYPE OF q-SCHUR ALGEBRAS KARIN ERDMANN AND DANIEL K. NAKANO Abstract. In this paper we classify the q-Schur algebras having finite, tame or wild representation type and also the ones which are semisimple. 1. Introduction 1.1. Let A be a finite-dimensional algebra over a field k. Furthermore, let mod(A) denote the category of finite-dimensional A-modules. The algebra A has finite rep- resentation type if and only if there are only finitely many indecomposable modules in mod(A). Otherwise, A has infinite representation type. Algebras of infinite rep- resentation type can be divided into two mutually exclusive categories|those that have either tame representation type or wild representation type.AnalgebraA has tame representation type if for each dimension d there are finitely many 1-parameter families of modules which contain all d-dimensional indecomposable modules. In this situation there is a good chance one may be able to describe all indecomposable objects in mod(A). On the other hand, algebras of wild representation type are those algebras whose representation theory is as complicated as modules over the free associative algebra khx; yi. Given a class of algebras a fundamental problem is to classify the representation type of the algebras and to gain as much insight into those of finite or tame type as possible. Let GLn(k) be the general linear group over k and V be the n-dimensional ⊗d natural representation. The Schur algebra S(n; d) is defined as EndΣd (V )where Σd is the symmetric group on d letters. These algebras arise naturally in the study of the polynomial representation theory of GLn(k). In [Erd2], [DN], and [DEMN], a complete classification of the representation type for the Schur algebra was obtained. The purpose of this paper is to provide a complete classification of the representation type of the quantum (or q) Schur algebras. 1.2. Definitions and Notation. Let R be an integral domain and q be a unit in R, we assume q =6 1. Let us regard the symmetric group Σr as the set of bijections of f1;:::;rg where r is a natural number which we assume to be ≥ 2. The Hecke algebra Hq(r)=HR;q(Σr)isthefreeR-module with basis fTw : w 2 Σrg.The multiplication in the algebra is defined by the rule ( Tws if `(ws) >`(w); TwTs = qTws +(q − 1)Tw otherwise; Received by the editors September 24, 1999 and, in revised form, September 13, 2000. 2000 Mathematics Subject Classification. Primary 16G60, 20G42. Research of the second author partially supported by NSF grant DMS-9800960. c 2001 American Mathematical Society 4729 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4730 KARIN ERDMANN AND DANIEL K. NAKANO where s =(i; i +1) 2 Σr is a basic transposition and w 2 Σr. The function ` :Σr ! N is the usual length function. For each partition λ of r there is a q-Specht module of the Hecke algebra Hq(r), denoted by Sλ. Assume that R = k is a field and that q 2 k with q =06 ; 1. Let λ be a composition of r and let Σλ be the corresponding Young subgroup of λ ∼ Σd,soΣ = Σλ × Σλ ×···.Let 1 2 X xλ = Tw: w2Σλ λ λ Define M := xλHq(r). The module M has a unique submodule isomorphic to the Specht module Sλ, and there is a unique indecomposable direct summand of ∼ M λ containing Sλ; this is the Young module Y λ.WehaveY λ = Y µ if and only if λ = µ.EachM α (for α 2 Λ(n; r)) is a direct sum of such Young modules (see [Mar]). The q-Schur algebra was introduced by Dipper and James as follows. Let Λ(n; r) be the set of compositions of r into n parts. The q-Schur algebra is defined as 0 1 M @ H A Sq(n; r)=SR;q(n; r)=EndHq (r) xλ q(r) : λ2Λ(n;r) Note that if q =1withR = k where k is a field, then this is the classical Schur algebra S(n; r)=Sk(n; r)[Gr]. The q-Schur algebra over a field can also be constructed via quantum GLn;here we use the version introduced in [DD], and we take advantage of the work done in [Don]. Let Aq(n) be the bialgebra defined there; for each r ≥ 0 its homogeneous part Aq(n; r) is a finite-dimensional subcoalgebra, and its dual is therefore a finite- dimensional algebra which can be identified with the q-Schur algebra. We note that the representation theory works well for arbitrary nonzero q. 1.3. Main results. For the remainder of the paper we will assume that R = k is an algebraically closed field of characteristic p ≥ 0(p =0oraprimenumber). We make the convention that p is arbitrary unless it is explicitly specified. We also assume q 2 k and q =06 ; 1andn ≥ 2. Let l be the multiplicative order of q in k∗, that is, either q is a primitive lth root of 1, or l = 1.Notethatifq is a primitive lth root of 1 and p>0, then l and p must be coprime. The main results of this paper are listed below. Theorem (A). The algebra Sq(n; r) is semisimple if and only if one of the follow- ing holds: (i) q is not a root of unity; (ii) q is a primitive lth root of unity and r<l; (iii) n =2, p =0, l =2and r is odd; (iv) n =2, p ≥ 3, l =2and r is odd with r<2p +1. Theorem (B). The algebra Sq(n; r) has finite representation type but is not semi- simple if and only if q is a primitive lth root of unity with l ≤ r, and one of the following holds: (i) n ≥ 3 and r<2l; (ii) n =2, p =06 , l ≥ 3 and r<lp; (iii) n =2, p =0and either l ≥ 3,orl =2and r is even; License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use REPRESENTATION TYPE 4731 (iv) n =2, p ≥ 3, l =2and r even with r<2p,orr is odd with 2p +1≤ r< 2p2 +1. Theorem (C). The algebra Sq(n; r) has tame representation type if and only if q is a primitive lth root of unity and one of the following holds: (i) n =3, l =3, p =26 and r =7; 8; (ii) n =3, l =2and r =4; 5; (iii) n =4, l =2and r =5; (iv) n =2, l ≥ 3, p =2or p =3and pl ≤ r<(p +1)l; (v) n =2, l =2, p =3and r 2f6; 19; 21; 23g; We remark that Theorem 1.3(A) corrects several misconceptions in the literature on the semisimplicity of quantum Schur algebras (see [PW, (11.4.1) Lem.], [Mar, p. 191 Rem. (1)], [Don, p. 96 (7)]). It is also interesting to note that when p =0 the algebra Sq(2;r) is always semisimple, or of finite representation type. 1.4. For the sake of convenience it will be useful to reformulate the results given in the preceding section. Theorem (A). The algebra Sq(n; r) is not semi-simple if and only if q is a prim- itive lth root of unity, such that r ≥ l and one of the following holds: (a) n ≥ 3; (b) n =2and l ≥ 3; (c) n =2, l =2and r even; (d) n =2, l =2, p ≥ 3 and r odd with r ≥ 2p +1. Theorem (B). The algebra Sq(n; r) is of infinite type if and only if q is a primitive lth root of unity and one of the following holds: (a) n ≥ 3 and r ≥ 2l; (b) n =2, p =06 , l ≥ 3 and r ≥ lp; (c) n =2, p ≥ 3, l =2and r is even, r ≥ 2p,orr is odd, r ≥ 2p2 +1. Theorem (C). The algebra Sq(n; r) has wild representation type if and only if q is a primitive lth root of unity and one of the following holds: (a) n ≥ 3, l ≥ 4 and r ≥ 2l; (b) n ≥ 3, l =3and r ≥ 2l except for the algebras in Theorem 1.3(C)(i); (c) n ≥ 3, l =2and r ≥ 2l, except for the algebras in Theorem 1.3(C)(ii) and (iii); (d) n =2, l ≥ 3, p =06 and r ≥ lp except for the algebras in Theorem 1.3(C)(iv); (e) n =2, l =2, p ≥ 3 and r is odd, r ≥ 2p2 +1,orr is even and r ≥ 2p,except for the algebras in Theorem 1.3(C)(v). It may be surprising that, for example, when n =3,l =3andp =2thealgebra6 is wild for r = 6 but tame when r =7; 8. But algebras Sq(n; r)andSq(n; r + t) for 1 ≤ t<nand r ≥ n can be quite different; one can see this directly in small 2 cases. Let l = 2, then the algebra Sq(2; 2) is isomorphic to k[T ]=((T +1) )and hence is not semisimple. On the other hand, Sq(2; 3) is semisimple, there are just two simple modules and they lie in different blocks (and simple modules for q-Schur algebras have no non-split self-extensions). We start with general reductions (in x2).
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