On perfect isometries and isotypies in alternating groups
On perfect isometries and isotypies in alternating groups
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关于交替组中的完美等距和同型
DOI：
10.1090/s0002-9947-97-01793-5
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
M. E. Harris
中科院分区：
文献类型：
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作者：
P. Fong;M. E. Harris
在具有阿贝尔缺陷群和这些块的Brauer对应体的块之间的交替组中构造了完美等距和等型。这些完全等距和等型满足附加的相容条件，这意味着对于一个与交替群同构的有一个阿贝尔Sylow psubgroup和一个广义拟合子群的几乎简单群的主块，扩展Broue猜想成立。设G是一个有限群，设0是一个具有特征为0的商域K和特征为p >的残类域K的完全离散赋值环。我们假设K包含一个原始的单位的i次方根。在[4，（6.1）]中，Michel Broue提出了如下同型猜想。设e为具有阿贝尔缺陷群D的OG块，设f为e在(9NG (D)中的Brauer对应体。然后e和f是同型块。如果G有一个阿贝尔Sylow p-子群，则可以对0G的主块提出该猜想。在这种情况下，作者证明了猜想保持，为具有阿贝尔Sylow p-子群的几乎单群的主块提供了一个扩展的猜想保持（见[10，（5E）]）。本文证明了具有阿贝尔缺陷群的任意块的同型猜想。此外，证明了具有与交变群同构的阿贝尔Sylow p-子群和广义拟合子群的概简群的主p块的扩展猜想。我们回顾一下基本定义。设-:0 k为正则商映射，设-:OG kG为群代数的诱导0代数同态。特别地，-:e -*e在OG和kG的中心幂等之间引起双射。若e是OG的块幂等，设KGeMod为有限型左kge模的范畴，设ZK （G, e）为KGeMod的Grothendieck群。设Gv是G / k上不可约字符的集合，我们在（G, e）v = {X e Gv X(ge) = X(G)上对所有G e G}用自由阿贝尔群识别R.K （G, e）。设CF（G， K）是G上的K值类函数的K空间，设CF（G, e， K）是CF（G， K）中类函数ae的K子空间，使得ao(ge) = a(G)。编辑收到，1996年3月26日。1991数学学科分类。初级201c15、201c20；二次20 c30。第一作者部分由美国国家科学基金会资助DMS 9100310。第二作者得到了NSA拨款MDA 904 92-H-3027的部分支持。？1997年美国数学学会
Perfect isometries and isotypies are constructed for alternating groups between blocks with abelian defect groups and the Brauer correspondents of these blocks. These perfect isometries and isotypies satisfy additional compatibility conditions which imply that an extended Broue conjecture holds for the principal block of an almost simple group with an abelian Sylow psubgroup and a generalized Fitting subgroup isomorphic to an alternating group. Let G be a finite group and let 0 be a complete discrete valuation ring with field of quotients K of characteristic 0 and residue class field k of characteristic p > 0. We suppose that K contains a primitive IGI-th root of unity. In [4, (6.1)] Michel Broue posed the following Isotypy Conjecture. Let e be a block of OG with abelian defect group D and let f be the Brauer correspondent of e in (9NG (D). Then e and f are isotypic blocks. If G has an abelian Sylow p-subgroup, then the conjecture can be posed for the principal block of 0G. In this case the authors have shown that the conjecture holds provided an extended conjecture holds for the principal block of almost simple groups with an abelian Sylow p-subgroup (see [10, (5E)]). In this paper the isotypy conjecture for an arbitrary block with abelian defect group is proved for alternating groups. In addition, the extended conjecture is proved for the principal p-block of almost simple groups with abelian Sylow p-subgroups and generalized Fitting subgroup isomorphic to an alternating group. We recall the basic definitions. Let -: 0 k be the canonical quotient mapping and let -: OG kG be the induced 0-algebra homomorphism of the group algebras. In particular, -: e -*e induces a bijection between central idempotents of OG and kG. If e is a block idempotent of OG, let KGeMod be the category of left KGe-modules of finite type and let ZK (G, e) be the Grothendieck group of KGeMod. Let Gv be the set of irreducible characters of G over K. We identify R.K (G, e) with the free abelian group on (G, e)v = {X E Gv X(ge) = X(g) for all g E G}. Let CF(G, K) be the K-space of K-valued class functions on G, and let CF(G, e, K) be the K-subspace of class functions ae in CF(G, K) such that ao(ge) = a(g). The Received by the editors March 26, 1996. 1991 Mathematics Subject Classification. Primary 20C15, 20C20; Secondary 20C30. The first author was supported in part by NSF grant DMS 9100310. The second author was supported in part by NSA grant MDA 904 92-H-3027. ?1997 American Mathematical Society