The Pseudoblocks of Endomorphism Algebras

Transcription

1 Internatonal Mathematcal Forum, 4, 009, no. 48, The Pseudoblocks of Endomorphsm Algebras Ahmed A. Khammash Department of Mathematcal Scences, Umm Al-Qura Unversty P.O.Box 796, Makkah, Saud Araba Abstract. In [5], t s shown that the block dstrbuton of an endomorphsm algebra of a module Y over a fnte dmensonal algebra mples the -block dstrbuton of the ndecomposable drect summands of Y n the lght of the Brauer-Fttng correspondence. Towards the end of that paper we gave an example whch shows that the converse s not true even when the endomorphsm algebra of Y s Frobenous and conectured that the converse s true when the endomorphsm algebra s symmetrc. However the author thankfully receved a counterexample to that conecture from S. Donkn [4] whch we present n ths note. We also ntroduce the noton of the pseudoblock of endomorphsm algebras and show that the - pseudoblock dstrbuton of ndecomposable summand of Y s compatble wth the block dstbuton of the smple modules for the endomorphsm algebra E (Y ). Mathematcs Subect Classfcaton: 0C30, 0C33, 0C05 Keywords: Blocks, Endomorphsm algebra, Symmetrc algebras, Extensons, pseudoblocks INTRODUCTION Let be a fnte dmensonal k-algebra where k s an algebracally closed feld of characterstc p f 0. Let mod denote the category of fnte dmensonal left -modules and M mod denote the category whose obects are all the covarant functors F : mod modk, where modk s the category of vector spaces over k. Let Y be a fnte dmensonal -module and wrte E( Y ) = End ( Y ) ; the endomorphsm algebra of Y. The nterplay between the representatons of E(Y) and representatons of s an obect of study n the

2 364 A. A. Khammash representaton theory, motvated by many concrete examples n representaton of fnte groups. One of the early and drect connectons between the representaton theory of the endomorphsm algebra E(Y) and representaton theory of s provded by the Brauer-Fttng theorem (see [6], 1.4) whch gves a becton (Brauer-Fttng correspondence) between the somorphsm classes of smple E(Y)- modules and the somorphsm classes of ndecomposable drect -summands of Y. In fact, f Y s an ndecomposable drect summand of Y, then t s known that P : = ( Y, Y ) = Hom ( Y, Y ) s a proectve ndecomposable E(Y)-module and the Brauer-Fttng correspondence mentoned earler s gven by the followng becton ( 1) S := P / rp Y In [5, Theorem1] t s shown that f two smple E(Y)-modules are lnked (blockwse) as E(Y)-modules, then the correspondng summands under the Brauer-Fttng correspondence (1) are also lnked as -modules. In fact f we use the notaton to mean " le n the same block of " or "lnked" then we have THEOREM1 ( [5], Theorem1): If S S Y Y. then E( Y ) The proof of theorem1 depends on the descrpton of the lnkage prncple n terms of extenson bfunctor Ext 1 (, ) as well as the structure of the morphsm space n the category of functors M mod. Theorem1 was appled to study the block dstrbuton of the smple modules for the fnte groups of Le type G n the defnng charactestc n the lght of the Cabanes work [1] n whch he determned the block dstrbuton for the smple modules of the endomorphsm algebra E (Y ) where Y s the permutaton module on the Sylow p-subgroup of G where p s the defnng characterstc of G. It was shown also that the converse of theorem1 s not true n general by provdng a counterexample for whch the endomorphsm algebra E(Y) s a Frobenus algebra accordng to the frst part of the followng defnton DEFINITION: (1) A fnte dmensonal algebra A over k s sad to be Frobenous f there s a lnear map λ : A k such that Ker λ contans no non-zero left or rght deal. () A s symmetrc f t satsfes (1) together wth the condton λ ( ab) = λ( ba) for all a, b A. The concepts of Frobenus and symmetrc algebras proved to be vtal n studyng dualty and selfenectve modules as well as the structure and propertes of quotent algebras. If we take A = kg ; the group algebra of a fnte group G over k and defne λ : A k as λ ( cg g) = c1 ; the coeffcent of the dentty g G element of G, then λ satsfes condton () n the prevous defnton (for the proof see [3], 9.6) and hence A = kg s symmetrc algebra. It s also not dffcult

3 The pseudoblocks of endomorphsm algebras 365 (see [1], exercse1 p.10) to show that every fnte dmensonal semsmple algebra over a feld s Frobenous. Motvated by two examples for whch E(Y) s symmetrc algebra (one s when E (Y ) s the group algebra of a fnte group and the other s when t s semsmple algebra), t was conectured n [5] that the converse of theorem1 s true for the case when the endomorphsm algebra E(Y) s symmetrc. However the author thankfully receved a counterexamople for that conecture from S. Donkn. To present hs example, we need to ntroduce some prelmnary facts and results concernng the endomorphsm algebras of certan extenson modules. In order to adapt the block dstrbuton of smple E (Y ) -modules wth the Brauer-Fttng correspondents, we ntroduce the concept of the pseudo-blocks of E (Y ) and show that the ndecomposable summands of Y dstrbute n pseudo-blocks n a compatble way wth the block dstrbuton of the correspondng smple (Y ) -modules. E EXTENSIONS AND ITS ENDOMORPHISM ALGEBRS The structure for the endomorphsm algebra of smple obects n mod s descrbed by the followng well known lemma whch we state here for latter reference SHUR'S LEMMA: If U and V are two smple -modules then for U / V, Hom ( U, V ) = 0, whle the endomorphsm algebra E ( U ) Hom ( U, U k. = ) If X, Y, E mod such that there s a short exact sequence 0 X E Y 0 n mod then we say that the module E s an extenson of the module X by Y and t s called non-splt extenson f E s ndecomposable. Schur's lemma can be extended to descrbe the endomorphsm algebra for an extenson of a smple module by another non-somorphc smple module LEMMA1: If 0 X E Y 0 s a non-splt extenson n mod where X, Y are two nonsomorphc smple modules then End ( E) k. PROOF: Each morphsm 0 f End ( E) s determned by a trple ( f 1, f, f ) where 0 f1 End ( X ), 0 f End ( Y ) whch makes the followng dagram 0 X E Y 0 f1 f f 0 X E Y 0

4 366 A. A. Khammash commutes. But snce X and Y are smple, Shur's lemma mples that both f 1, f are somorphsms. It follows then that f s an somorphsm and therefore ; as the feld k s algebrac closed, End ( E k. ) Incdently lemma1 provdes a counterexample for the converse of Schur's lemma snce extensons of a smple module by a dfferent smple module exst for fnte-dmensonal algebras; e.g. over a feld of characterstc 3, the group algebra for the symmetrc group S 3 has two ndecomposable modules of dmenson each wth two nonsomorphc composton factors. We are now ready to present S. Donkn's example EXAMPLE1: (S. Donkn [4]): Take any algebra whch has a block wth at least 3 smples n whch there s an extenson of a smple by a dfferent smple (e.g. example below for the group algebra of G = SL(,4) over algebracally closed feld of characterstc provdes such an algebra). Take such an extenson E and a smple L, say, n the same block whch s not a composton factor of E. Then, as k s an algebracally closed feld, Schur's lemma and lemma1 mples that End kg ( E L) EndkG ( E) EndkG ( L) k k whch s semsmple algebra havng two blocks although E and L are lnked and ths gves a counterexample for the conecture. The PSEUDOBLOCKS In ths secton we ntroduce the noton of pseudo-block of the endomorphsm algebra E (Y ) and show the compatblty of the pseudo-block dstrbuton of the ndecomposable summands of Y wth the the block dstrbuton of the (Brauer-Fttng correspondents) smple (Y )-modules. E DEFINITION: Defne the relaton PSE (Y ) on the set of ndecomposable summands of Y Inds ( Y ) = { Y, Y,..., Y n } as follows: If 1 Y, Y Inds( Y ) then Y Y f = 1,..., f = such that The relaton n, Hom( Y, Y ) 0 or Hom( Y, Y ) 0. PSE (Y ) n n + 1 n + 1 n clearly defnes an equvalence relaton on Inds (Y ) whose equvaence classes are called the pseudo-blocks of the endomorphsm algebras E (Y ). Note that, although Y Y mples that Y Y, the PSE (Y ) classes of Inds (Y ) n general dffer from ts followng example. classes as shown by the

5 The pseudoblocks of endomorphsm algebras 367 EXAMPLE: Take G = SL(,4). Then t s known that n characterstc,g has blocks and 4 rreducble kg -modules : 1, 1,, 4, where upper case number refers to dmensons, the frst three rreducble kg-modules belong to the G same -block. Let Y = IndV ( k), where V s the Sylow - subgroup of G whch somorphc to C C. Then Y s of dmenson 15 and has the followng decomposton Hence ( Y = Inds ( Y ) = Y1 = 1, Y = 1, Y3 = 1, Y4 = 4 and the pseudo-blocks 1 Y, Y, Y Y whle there are only two classes classes) are PSE (Y ) { 1 } { 3 }, { 4 } Y 1, Y, Y 3, { Y 4 } namely { }. We note that the pseudo-blocks dstrbuton above s compatble wth the block dstrbuton for the smple representatons of E = End kg ( Y ) (see the example n [5]) and the followng theorem shows that ths n fact s the case n general. THEOREM: S S E ( Y ) f and only f 1 Y Y. PROOF: It s clear from the proof of theorem1 n [5] that f P P E ( Y ) where P : = ( Y, Y ) kg Y Y f and only s the proectve cover of the smple E(Y ) module S whch s equvalent to the fact that S S. E ( Y ) Acknowledgments. The author thanks S. Donkn for hs ntatve of sendng hm the counterexample to the conecture. I also thanks A. Alarad for a useful dscusson on the subect of ths paper. REFERENCES [1] D. Benson, Modular representaton theory: New trends and Methods, LNM #1081, Sprnger, Berln 1984 [] M. Cabanes, Extenson groups for modular Hecke algebras, J. Fac. Sc. Unv. Tokyo Math. Sect. IA Math. 36 (1989 ),

6 368 A. A. Khammash [3] C. Curts and I. Rener, Methods n representaton theory I, John Wley, New York 1981 [4] S. Donkn, prvate communcatons [5] Ahmed Khammash, On The Blocks Of Endomorphsm Algebras, Internatonal Math. Journal Vol. No.6 00 [6] P. Landrock, Fnte group algebras and ther modules, Cambrdge Unversty Press, 1984 Receved: November, 008