On Modular Homology in the Boolean Algebra, II

Transcription

1 JOURNAL OF ALEBRA 199, ARTICLE NO JA O Modula Homology i the Boolea Algeba, II Steve Bell, Philip Joes,* ad Johaes Siemos School of Mathematics, Uiesity of East Aglia, Nowich, NR4 7TJ, Uited Kigdom Commuicated by Walte Feit Received Decembe 10, 1996 Let R be a associative ig with idetity ad a -elemet set Fo k coside the R-module Mk with k-elemet subsets of as basis The -step iclusio map : Mk Mk is the liea map defied o this basis though whee the ae the Ž k -elemet subsets of Ž 1 2 k i Fo m oe obtais chais M :0M M M M 0 m m m2 m3 of iclusio maps which have iteestig homological popeties if R has chaacteistic p 0 V B Mukhi ad J Siemos Ž J Combi Theoy 74, ; J Algeba 179, 1995, itoduced the otio of p-homology to examie such sequeces whe 1 ad hee we cotiue this ivestigatio whe is a powe of p We show that ay sectio of M ot cotaiig cetai middle tems is p-exact ad we detemie the homology modules fo such middle tems Numeous ifiite families of ieducible modules fo the symmetic goups aise i this fashio Amog these the semi-simple iductie systems discussed by A Kleshchev Ž J Algeba 181, 1996, appea ad i the special case p 5we obtai the Fiboacci epesetatios of A J E Ryba ŽJ Algeba 170, 1994, Thee ae also applicatios to pemutatio goups of ode co-pime to p, esultig i EulePoicae equatios fo the umbe of obits o subsets of such goups 1998 Academic Pess 1 INTRODUCTION Let be a set of fiite size ad R a associative ig with idetity Fo k coside the R-module Mk which has k-elemet subsets of as basis So M cosists of all fomal sums f Ý f with ad k k * Suppot fom the EPSRC is ackowledged addess: philipjoes@ueaacuk addess: jsiemos@ueaacuk $2500 Copyight 1998 by Academic Pess All ights of epoductio i ay fom eseved 556

2 ON MODULAR HOMOLOY 557 f R Whe 0 is a itege the -step iclusio map : Mk Mk is the liea map defied though k 1 2 Ž whee i ae the Ž k -elemet subsets of So fixig some m we obtai a chai of iclusio maps M :0M M M M m m m2 m3 Whe R has chaacteistic p 0 the p is the zeo map ad oe is iteested i the homological popeties of M I the papes 10, 11 with Valey Mukhi we have defied the otio of a p-homological ad p-exact sequecesee the defiitio i Sectio 3ad this pape is a cotiuatio of that wok Fo the pupose of p-homology it is ecessay to ivestigate cetai subchais of M We assume ow that R is a ig of pime chaacteistic p 0 ad that is a powe of p Fo iteges 0 i p ad 0 k mmodž with k i p we coside the subsequece M :0M M M M M k k i k p k Ži p k 2i Hee each aow epesets the elevat powe of ad as p 0 we see that M is homological Ž M would be p-exact by defiitio if all subsequeces M of the kid just descibed wee exact Let Ž : Mki Mk MkŽ pi be ay thee cosecutive tems of M, thus eithe k k modž p with i p i o k k i* modž p with i i We say that o Ž k, i is a middle tem fo M if 2k Ž p ip Note that M may have o middle tems Ž take odd, p 2, ad 1, fo example, but if thee is oe the it is uique ad we ca speak of the middle tem fo M I Theoem 32 we pove that ay sectio of M ot cotaiig its middle tem is exact This esult is aleady cotaied i Bie s pape 2 ad it is poved thee via ak agumets based o Wilso s wok 15 Ou poof is moe diect I Sectio 2 we develop the geeal calculus fo iclusio maps i k 0 M k This leads to a itegatio theoem which pi allows us to wite dow a pe-image F with Ž f f fo ay f with i Ž F 0 uless Ž k, i is a middle tem If Mki Mk kž pi is a middle tem deote the keel of i by K k, i ke i Mk ad so let pi Ž H K M k, i k, i kž pi be the oly o-tivial homology module of M I Sectio 5 we detemie explicit geeatos fo all Hk, i ad i Sectio 6 we classify completely the homologies of the 1-step map I Theoem it is show that H is isomophic to H H as SymŽ k, i k, i1 k1, i1 - modules This is used to costuct ifiite families of ieducible Sym -

3 558 BELL, JONES, AND SIEMONS modules fo abitay pimes p 2 occuig as the homology modules of the 1-step iclusio map Fo a abitay pime p 2 the family H k,i :0ip, k, 2ki14 is a example of a semi-simple iductie system as cosideed by Kleshchev 9 ad i the case p 5 we obtai Ryba s Fiboacci epesetatios of 13 I 3 Bie has show how cetai spi modules ove FŽ 2 aise as the homology modules of the 2-step map whe R has chaacteistic 2 These examples illustate that the otio of p-homology leads to wothwhile esults o the modula epesetatio theoy of the symmetic goups Fo the modules of the -step maps with 1 i geeal ot much appeas i the liteatue Ou desciptio of the Hk, i i Sectio 5 is explicit eough to allow a full aalysis of these modules which may be peseted i a subsequet pape Othe applicatios coce pemutatio goups Pemutatios act atually o M ad fo ay goup Sym k 0 k we ca coside the g obit module i M defied as M f M : f f g 4 k k k Its atu- al basis ae the obit sums whee as usual deotes g 4 : g I paticula, the dimesio of Mk is the umbe of -obits o the k-elemet subsets of AsŽ M ŽM k k we obtai sequeces of the kid M :0M M M M M, k k i k p k Ži p k 2p whee each tem is a submodule of the coespodig tem i M Such a sequece is automatically homological but may fail to be exact at tems whee M is exact Howeve, i Theoem 41 we show that if the ode of is ot divisible by p, the ay sectio of M ot cotaiig its middle tem is exact If k, i is a middle tem of M we let Hk, i K k, i Mk pi Ž MkŽ pi deote the oly o-tivial homology module i M k, i The esults i Sectio 5 give geeatos fo Hk, i ad i Theoem 45 we obtai EulePoicae equatios fo the umbe of -obits o k-elemet subsets of whe p does ot divide the ode of Fo p 2 we have made use of such equatios befoe i 10 fo goups of odd ode Modula p -homology ove igs R of chaacteistic p 0 fo sequeces such as M above ca be cosideed fo moe geeal classes of patially odeed sets Note that it will be ecessay to distiguish betwee the chaacteistic of the ig ad the iteval legth p appeaig i the defiitio of p -homology; fo the Boolea lattice these happe to coicide If Ž P,, is a aked poset ad if Mk deotes the R-module with x P: xk4 as basis the oe ca defie ode maps : Mk Mk1 aalogous to ad theefoe the questio of p -homology ca be studied also fo such aked posets I 12 we have doe this fo pojective spaces

4 ON MODULAR HOMOLOY 559 ove FŽ q whe the coefficiet ig R has chaacteistic p q Žfo p q o homologies occu The esults ae iteestig: Oe picipal diffeece to the Boolea case is that p ideed is diffeet fom the chaacteistic of the coefficiet ig O the othe had, also i pojective spaces evey chai such as M is iexact i at most oe positio ad so gives ise to just oe o-tivial homology We hope that these facts suppot the suggestio that the modula homology cosideed hee ucoves some deepe popeties of patially odeed sets 2 THE -STEP INCLUSION MAP Thoughout R will be a associative ig with idetity ad a fiite set of cadiality We let 2 deote the collectio of subsets of ad R2 the R-module with 2 as a basis Fo a itege k the collectio of all k-elemet subsets of is deoted by k4 Futhemoe, we let R k4 R2 deote the submodule with k-elemet subsets as basis We will abbeviate R k4 by Mk o simply Mk if the cotext is clea We idetify 14 with ad 1 with 1 R so that i paticula M : R4 0 R Also, we put Mk 0 wheeve k 0okad efe to R as the coefficiet ig of M k Fo f Ý f R2 the suppot of f is the uio of all fo which f 0; we will deote it by supp f The suppot size of f is f supp f Two elemets f ad h of R2 ae said to be disjoit if supp f ad supp h ae disjoit sets The Boolea opeatio of set uio is easily exteded to a poduct o R2 :if fýf ad h Ýh ae elemets of R2 we defie fh Ý f h Ž, It is a simple matte to check that R2 with this poduct is a associative algeba with the empty set as idetity Fo a itege 0 the -step iclusio map o 2 is the liea map : R2 R2 give by, 1 2 k whee is a k-elemet subset of ad whee the ae the Ž k i - elemet subsets of I paticula, is the idetity map, 0 0if, ad 1 R if Clealy, this map esticts to homo

5 560 BELL, JONES, AND SIEMONS mophisms : M M k k Futhemoe, it is easily veified that fo ay positive itege s we have ž / s s s Let f be a elemet of R2 The fo ay i we ca wite f uiquely as f f l i such a way that does ot belog to supp f supp l Also vey impotat is the followig LEMMA 21 If f ad h ae disjoit elemets i R2 the Ž f h Ý Ž f Ž h j0 j j Poof The idetity is obvious whe f ad h ae two disjoit sets ad follows fom lieaity i geeal We shall use these basic facts without futhe efeece The ext theoem geealizes Theoem 21 of 10 whee the esult was poved fo the oe-step map THEOREM 22 Fo ay coefficiet ig with idetity the keel of : Mk M is geeated by elemets of suppot size at most max2k 1, k 4 k Remak Whe the coefficiet ig is a itegal domai of chaacteistic 0 the the miimum suppot size of elemets i the keel is max2k 1, k4 exactly This ca be see fom simple ak agumets Howeve, i o-zeo chaacteistic thee may be elemets of smalle suppot size Coollay 34 late gives moe ifomatio Poof The poof follows closely that of Theoem 21 i 10 ad poceeds by iductio o Fo 1, 2 the esult is easily veified Theefoe suppose that 2 ad that f ke M with f k max2k 1, k 4 Futhemoe, we may assume that k, fo if k the the stadad basis of Mk is cotaied i the keel of ad the esult clealy holds We pick supp f ad wite f f l whee supp f supp l The by Lemma 21 0 Ž f Ž f 1Ž f Ž l Theefoe, f ke Mk1 1 ad by the iductive hypothesis we may s 1 wite f Ý w with w ke M ad w max2k 1, k i1 i i k1 i

6 ON MODULAR HOMOLOY As f2k1k1ž k k1, we have f w i 2 fo i 1,,s Hece, wheeve 1 i s, we may choose i i with ad supp f supp w The Ž w i i i i i 2 w max2k 1, k 14 max2k 1, k4 ad ŽŽ w i i i s s 0 by Lemma 21 Futhemoe, f Ýi1 i wi Ýi1i wi l ad as Ýi1 s i wi l ke Mk 1 we may ivoke the iductio hypothesis to complete the poof LEMMA 23 Suppose that R is a associatie ig with idetity ad has pime chaacteistic p 0 Let 1 be a powe of p ad let f be a elemet of ke M k If 0 s p satisfies 2k s the thee exists F i M with s Ž F f ks Poof By the theoem above we may assume that f max2k 1, k 4 Suppose fistly that f k ad that k 2k 1 Theefoe we also have k ad 1 As k s k thee exists with supp f ad s k We defie F Ž s! 1 k f M ad show by iductio o t s that t Ž F t! Ž s! 1 ks tk f Takig t s will the complete the poof i this case Fo t 0 the esult is cetaily tue ad supposig the esult holds fo t s 1 we calculate, usig Lemma 21, Ž t1 1 Ž F t! Ž s! tk f 1 t! Ž s! Ž f ž Ý j Ž tk j / j0 k 1 Ž t1 kj Ýž / Žt1kj j ž t k / j0 t! s! f To deduce that Ž Ž t 1 k j t k 0 mod p if 0 j k we use Fact 1 Žsee 1, p 8 Fo a positive itege m ad a pime p the l lagest itege l such that p divides m! isý mp Žwhee x 1 deotes the itegal pat of the eal umbe x Ad Fact 2 Fo ay eal umbes x ad y we have x y x y

7 562 BELL, JONES, AND SIEMONS Theefoe, if p d with d 1, the the lagest itege l such that p l divides Ž Ž t 1 k j is t k ž / d1 Ž t1 kj tk j Ž t1 t Ý ž p p p / 1 Ž t 1 k j tk j p p p Ý 1 which is stictly geate tha zeo ŽNote that each tem of the sum is oegative by Fact 2 So we have ideed show that Ž Ž t 1 k j t k 0 modž p if 0 j k Futhemoe, we ca calculate Ž t1 k Ž t 1 l ž Ł t k / ž t l / tlk Ł ž 1 t l / tlk ž /ž / ž /ž / t t1 2 1 t1 which completes the iductio ad gives the esult i this case We may ow suppose that f 2k 1 Ž s 1 1 Theefoe thee exists with supp f ad Ž s 1 1 We the defie F Ž s! 1 f ad show iductively that t Ž F 1 t! Ž s! 1 Žt11 f wheeve t s Takig t s will the complete the poof i this case Fo t 0 the esult is cetaily tue ad if we suppose the esult holds fo t s 1 the ž / t1 1 Žt11 F t! s! f 1 Ý j Ž Žt11 j j1 t! Ž s! Ž f ž / t! Ž s! Ž f 1 t1 j1 Ý Žt11j j j j1

8 ON MODULAR HOMOLOY 563 We ote that we ae doe if 1 Theefoe we suppose that 1 But the ž / Ž t1 j1 Ž Ž t 1 j 1 Ž Ž t 1 j 2 j Ž j1 Ž j2 Ž t1 1 Ž t1 1 j ad we see that all tems i the poduct will be 1 modž p except the last This will be 0 modž p uless j whe it will be t 1 This completes the iductio ad hece also the poof The mai esult of this sectio is the ext theoem which shows that Coollay 23 of 10 ca be exteded to -step maps THEOREM 24 Ž The Itegatio Theoem Suppose that R is a associatie ig with idetity of pime chaacteistic p Let 1 be a powe of p ad let f be a elemet of M Suppose futhe that i Ž f k 0 with 0 i p ad that j 1,, pi4 satisfies 2k j The thee exists F i Mkj with j Ž F f Poof The poof is by iductio o i Fo i 1 the esult holds by the pecedig lemma Suppose the esult holds fo i p 2 ad that f i1 ke M with j 1,, pž i14 k satisfyig 2k j The i Ž Ž f 0 Also Ž f M ad 2Ž k j 2 Ž k j 1 whee j 1 2,, pi 4 Theefoe by the iductive hypothesis j1 thee exists H M with Ž H Ž f But the Ž j Ž H f kj 0 ad 2k j So by the pecedig lemma, thee exists J Mkj with j Ž J j Ž H f Hece f j Ž H J ad the iductio is complete 3 HOMOLOICAL SEQUENCES Thoughout this chapte the coefficiet ig R has pime chaacteistic p 0, 1 is a powe of p, ad is some fiite of cadiality We obseve that p : R2 R2 is the zeo map To see this ecall the Ž s fomula fom Sectio 2 By iductio it follows that s s ž / ž / ž / j j 2 j Ž p Ž p 1 1 Now otice that p 0 mod p

9 564 BELL, JONES, AND SIEMONS The esults i Sectio 2 lead us to ivestigate homology We ecall the defiitios: if : A B ad : B C ae homomophisms the the sequece A B C is homological at B if kež Ž A I this case HkeŽ Ž A is the homology module at B ad the sequece is exact if H 0, that is, if kež Ž A A loge sequece A: A A A A A j2 j1 j j1 j2 is homological Ž exact if it has that popety at evey A I 10, 11 i we have itoduced the followig DEFINITION A is p-exact Ž p-homological if all subsequeces of the kid A : A A A A A k k i k p k i p k 2p A ae exact Ž homological k i 2 p fo evey k ad 0 i p Ž The aows ae the atual compositios of the maps i A As is poited out i Bie s pape 2, this kid of homology was fist cosideed i the woks 8 of Maye i 1947, see also 14 Now select some m ad coside the sequece M :0M M M M m m m2 m3 I ode to ivestigate its p-homological popeties we fix iteges 0 i p ad 0 k m mod with k i p to obtai the subsequece M :0M M M M M k k i k p k Ži p k 2p i which each aow epesets the elevat powe of Sice p : R2 R2 is the zeo map this sequece is homological Fo geeal paametes, 0ip, ad k we let K k, i deote ke i Mk ad let Hk, i Kk, i pi Ž MkŽ pi be the coespodig homology module If f K coset i H by k, i pi f f M kž pi k, i the we deote its As befoe the supescipt ca be dopped if the cotext is clea We begi by statig a cosequece of the Itegatio Theoem of Sectio 2: LEMMA 31 Suppose that R is a associatie ig with idetity ad has pime chaacteistic p Let 1 be a powe of p ad suppose that 0 i p satisfies 2k p ithe H 0 k, i

10 ON MODULAR HOMOLOY 565 To exted this esult let ow Mki Mk MkŽ pi be ay thee cosecutive tems of M ŽSo eithe k k modž p ad i p i o kk i modž p ad i i We say that Ž k, i is a middle tem fo M if k Žk Ž p i p, idicatig that Mki Mk M is eaest to the middle of M kž pi Note that thee may be o middle tems fo M Ž take odd, p 2, ad 1, fo example Howeve, if thee is a middle tem, the it is easy to see that thee is at most oe so that we ca talk of the middle tem fo M We exted the use of this tem slightly ad efe also to Mki Mk MkŽ pi as the middle tem of M Futhe, Mki Mk MkŽ pi will be called a middle tem, o a middle tem of M, if it is the middle tem fo some M The followig esult appeas aleady i Bie s pape 2, Satz 2 The poof thee is based o Wilso s ak fomula 15 which yields the p-ak of the icidece matix of k-subsets vesus Ž k i -subsets of THEOREM 32 Suppose that R is a associatie ig with idetity ad has pime chaacteistic p 0 Let 1 be a powe of p The Hk, i 0 uless Ž k,i is a middle tem COROLLARY 33 A sectio of M cotaiig o middle tems is p-exact COROLLARY 34 If the coefficiet ig has pime chaacteistic p 0 ad i if k, i is ot a middle tem the the keel of : Mk Mki is geeated by elemets of suppot size at most k Ž p i Poofs The coollaies ae clea To pove the theoem we itoduce a ew liea map U : R2 R2 defied by U k 1 2 Ž whee is a k-elemet subset of ad whee the ae the Ž k i - elemet subsets of cotaiig Note that fo 0 i p the matix epesetig U i : Mk Mki is the taspose of the matix epesetig i : Mki M k I paticula, U i : Mk Mki ad i : Mki Mk have the same ak Futhemoe, the liea map c : R2 R2 defied by c is a module isomophism which satisfies cuc Let M M M be cosecutive tems of M a b ap whee without loss of geeality b a i If abp the this sequece will be exact by Lemma 31 Hece we may assume that a b p p We coside the sequece of modules MŽ ap Mb M a Sice 2 Ž ab, Lemma 31 implies that this sequece is exact But the we

11 566 BELL, JONES, AND SIEMONS calculate dim Hb, i dim K b, i dim pi Ž Map dim Mb dim i Ž Mb dim pi Ž Map dim Mb dim U i Ž Ma dim U pi Ž Mb dim Mb dim i Ž Ma dim pi Ž Mb dim K b,ž pi dim i Ž Ma dim H b, Ž pi 0 This completes the poof 4 ROUP ACTIONS AND THE EULERPOINCARE EQUATION We shall show that thee is a caoical way to attach submodules of R2 to ay pemutatio goup o These give ise to homological sequeces to which we ca apply the esult of the last sectio i ode to establish exactess As befoe, the coefficiet ig R has pime chaacteistic p 0, 1 is a powe of p, is some fiite set of cadiality, ad Sym is a pemutatio goup o g g Let g be a pemutatio of The g acts o 2 by : 4 which ca be exteded liealy to the whole of R2 It is ot difficult to see that g commutes with ad so images ad keels of i ae left ivaiat by pemutatios This also implies that pemutatios act as liea maps o the homology modules H k, i We defie the obit module of i M as 4 M f M : f g f, g k k The atual basis fo M ae the obit sums Ý whee as k g 4 usual deotes : g I paticula k dim Mk k4 is the umbe of -obits o As ŽM ŽM k k we obtai sequeces of obit modules Theefoe select as befoe some m ad coside the sequece m m m2 m3 M :0M M M M k

12 ON MODULAR HOMOLOY 567 which is cetaily p-homological I ode to ivestigate p-exactess we fix iteges 0 i p ad 0 k m mod with k i p to obtai the subsequece M :0M M M M M k k i k p k Ži p k 2p of M i which aows ae appopiate powes of Fo abitay paametes, 0ip, ad k we let K ke i Mk ad let Hk, i Kk, i pi Ž MkŽ pi k, i deote be the coespodig homology module If f K H by k, i k, i Ž kž pi pi f f M we deote its coset i The dimesio of H k, i is the Betti umbe k, i dim Hk, i I paticula, if is the idetity goup the H H ad we put k, i k, i k, i dim Hk, i By Theoem 32 we have 0 uless Ž k, i k, i is the middle tem of M i which case we efe to as the Betti umbe of M k, i Middle tems fo M ad M ae defied as befoe We ow examie M fo exactess THEOREM 41 Suppose that R is a ig of pime chaacteistic p 0 Let 1 be a powe of p ad a pemutatio goup o whose ode is ot diisible by p The H 0 uless Ž k, i is a middle tem k, i COROLLARY 42 If p does ot diide the ode of the ay sectio of M cotaiig o middle tems is p-exact Remak Theoem 32 is the special case of Theoem 41 whe is the idetity goup o Theoem 41 states that all but oe of the Betti umbes of M ae tivial Theefoe, if k, i is the middle tem of M, we call dim H the Betti umbe of M k, i k, i Poof Let M M M be cosecutive tems of M whee a b ap without loss of geeality b a i Suppose that b a p o that b a p p If fk K the by Theoem 32 thee exists b, i b, i

13 568 BELL, JONES, AND SIEMONS pi 1 g FMap with F f But the ÝgF Map ad pi Ž 1 g 1 pi Ý F Ý Ž F g f This completes the poof g g Befoe we cotiue we ote that Theoem 41 ca be used to compute the modula ak of cetai obit iclusio matices of : Fo s t let Ws, t be the matix whose colums ae idexed by -obits o t4, ows by s4 th -obits o, with i, j -ety, fo a fixed t-set i the j obit, coutig the umbe of s-elemet subsets belogig to the i th obit It is easy to see that Wk, k, viewed as a matix ove R, is the matix of : M M The followig exteds Theoem 42 of 10 k k COROLLARY 43 If p does ot diide the ode of, if is a powe of p, ad if k, 0ip satisfy 2k i the the p-ak of Wki, k is ki kp kž pi k2p Poof 0 Mkpi Mkp Mki Mk is exact accod- ig to the pecedig coollay The EulePoicae Equatio fo a homological sequece states that its chaacteistic Ž ie, the alteatig sum of the dimesios is equal to the alteatig sum of its Betti umbes, see fo istace Chapte IX4 i 7 o Chapte XX3 i 6 As Mk, i has paticulaly simple homologies whe has ode co-pime to p this becomes a stog esult We deote by g H k, i 4 k, i C f H : f f g the cetalize of i H k, i, o i othe wods, the fixed-module of o H k, i We give a alteative chaacteizatio of Hk, i PROPOSITION 44 If the coefficiet ig has pime chaacteistic p 0 ad if Sym has ode co-pime to p the H C Ž Poof Fist we ote that k, i H k, i g H k, i 4 k, i C f H : f f g K pi M pi M k, i kž pi kž pi sice if f f g fo all g the f 1 Ý f g g ad 1 g pi Ž Ý f is fixed by the goup We clealy have M g kž pi pi pi M K Moeove, if F M with Ž F fixed kž pi k, i kž pi

14 ON MODULAR HOMOLOY 569 pi pi by the goup the F Ž 1 Ý F g g showig that pi Ž pi M M K But the kž pi kž pi k, i Hk, i Kk, i pi Ž MkŽ pi Kk, i K pi M pi M k, i kž pi kž pi CH k, i Ž As usual, we put the biomial coefficiet equal to zeo if k 0oif k: THEOREM 45 Ž The EulePoicae Equatio If the coefficiet ig has pime chaacteistic p 0 ad if 1 is a powe of p, let Ž k, i with 0 i p be the middle tem of M ad k, i dim Hk, i its Betti umbe Suppose that Sym has ode ot diisible by p ad let k, i dim H be the Betti umbe of M k, i The Ž k k, i Ýž / kpt ž k i pt/ tz Ý kpt kipt tz ad iduces a fixed-poit-fee epesetatio of degee o k, i k, i H C whee C H is the fixed module of o H k, i k, i k, i Poof By Theoem 41, M has at most oe o-tivial homology ad so the EulePoicae fomula gives k, i ÝtZ kpt kipt as the ae the dimesios of the modules i M j The equatio fo k, i is the special case whe 1 ad the iequality follows fom Popositio 44 Fially, the cetalize of i Hk, ic is tivial as p does ot divide Ž Ž Ž k, i tz k pt k i pt Remaks 1 Coside the fuctio Ý fo geeal, k, i, It is clealy peiodic i k ad i ad Theoem 45 states that agees with whe Ž k, i k, i k, i is a middle tem while k, i 0 othewise Some fasciatig obsevatios ca be made: Fo p2, 1 we have 0; fo p 3 ad 1 we get 0, 14 k, i k, i st th while fo p 5, 1 we fid that k, i is 0 o the 1,, o Ž 1 st Fiboacci umbe See also Remak 2 followig Theoem 65 Ž k, i k, i 2 The iequality may ot hold fo goups of ode divisible by p Fo istace, whe p 3 ad is C6 actig o six poits, we have 3, but 3, 1 2

15 570 BELL, JONES, AND SIEMONS Ž k, i k, i 3 Fo middle tems the iequality gives iteestig esults about the obits o subsets of pemutatio goups of ode co-pime to p, i paticula if k, i is small This was fist used i Theoem 61 i 10 Ž 4 Ay fuctioal elatio fo k, i will give ifomatio about k, i Fo istace, it is clea that k, i k, 1 i k1, 1 i as this holds fo biomial coefficiets This leads to the coollay below But thee ae less obvious elatios ad some of these will be made moe explicit i Sectio 6 COROLLARY 46 Ž i If 0 i p ad 1 2k Ž p ip the Ž ii k, i k, i k1, i If 0 i p ad 2k p i p, the k, i k, 1 i1 k1, 1 i1 Poof I Ž i the coditios o the paametes mea that Ž k, i is a middle tem fo a set of size ad that Ž k, i ad Ž k 1, i ae middle tems fo a set of size 1 Hece k, i k, i, k, 1 i k, 1, ad k1,i 1 k1, 1 i The esult follows fom k, i k, 1 i k1, 1 i Simi- laly, fo ii wite out the tems of k, i ad use the elatio fo the biomial coefficiets 5 ENERATORS OF THE KERNELS I this sectio we costuct geeatos fo K k, i fo geeal 0 k ad 0 i p This the also povides geeatos fo the homology modules Hk, i ad Hk, i fo goups of ode co-pime to p If 2k Ž p io 2k Ž p ip the Theoem 32 impi plies that K Ž M k, i kž pi which povides a efficiet set of gee- atos Theefoe we estict ou attetio to fidig a geeatig set fo K whe 2k Ž p ip, that is, whe Ž k, i k, i is a middle tem Moeove, if k i the K k, i Mk ad so we ca assume that k i Whe i k ad 2k i 1 we defie Ck, i Ž 1 1 Ž 2 2 Ž t t :,,, j j j j fo 1 j, j t, t k i 1, i 1 4 LEMMA 51 Let R be ay coefficiet ig with idetity ad let ad i be positie iteges If k i ad k 1 the K ² C : k, i k, i

16 ON MODULAR HOMOLOY 571 i Poof Cetaily ke M is spaed by :, ad k k 4 We show that ² C : by iductio o k, i Fo 0 o 1 this cetaily holds Theefoe suppose that 2 ad that,,,,,, j j1 k ad 1, 2,, j, j1,, k4, whee j k The we let,,,,,,, j j1 j2 k ad ote that whee 1 ad 1 Ivokig the iductio hypothesis completes the poof LEMMA 52 Let R be a ig of pime chaacteistic p ad 1 a powe of p If 0 i p ad i k let k, i be a middle tem The K k, i ² pi Ž M,C : kž pi k, i Poof We poceed by iductio o Fo small values of the esult is easily veified So suppose that the lemma is tue fo all values By the above lemma we may assume that k i Let f K k, i be give We assume that supp f othewise we may use iductio o Theoem 32 to complete the poof We wite f f l whee supp f supp l The i Ž f 0 implies that i Ž f 0, that is, f Kk1, 1 i The eithe by iductio, the above lemma, o Theoem 32 we ² pi Ž 1 1 : pi see that f M, C We wite f Ž F k1ž pi k1, i Ýjc j j with j R, cj Ck1, 1 i, ad F Mk1Ž 1 pi Sice 2k i ad c j 2k i 1 we may select j supp c j We let pi 1 h f Ýjj j cj F ad ote that h K k, i, pi pi Ý c C ad F Ž M j j j j k, i kž pi Theefoe by iductio o Theoem 32 the poof is complete We collect the esults of this sectio so fa togethe i the followig THEOREM 53 Let R be a ig of pime chaacteistic p, 1 a powe of the pime p, ad let 0 i p Ž pi i If k, i is ot a middle tem the K Ž M k, i kž pi, ad Ž ii If Ž k, i is a middle tem the K k, i Mk fo k i ad K k, i ² pi Ž M,C : fo i k kž pi k, i

17 572 BELL, JONES, AND SIEMONS Fom this we obtai immediately expessios fo the homology modules: COROLLARY 54 If Ž k, i is a middle tem, the H k, i ½ ² : ² : M pi M if ki k kž pi pi Ck, i Ck, i MkŽ pi if ik Futhe, if Sym has ode co-pime to p, the ½ k k Ž kž pi ² : ² : M M pi M if ki H k, i pi k k, i k k, i kž pi M C M C M if ik Poof The fist pat is clea ad the secod follows fom Popositio 44 It is clea that the module geeated by Ck, i is of special impotace ad we will examie it i tems of the stadad epesetatio theoy of the symmetic goups; as a efeece we suggest Chapte 7 of 5 Suppose ow that R is a field of chaacteistic p 0 Let c Ž 1 1 Ž 2 2 Ž t t be a elemet i C with t k i 1 ad,,, 4 k, i 1 2 i1 ad defie,,,,,,,, 4 1 t 1 1 i1 We otice that c coespods to the polytabloid o whee 1 2 t 1 2 i1 ž 1 2 t / ad whee Ž1 Ž, Ž1 Ž, Ž1 Ž, t t Ž RSym is the siged colum stabilize of So if we let Mk deote the R-module with k-elemet subsets of as basis we have obtaied LEMMA 55 S ² M C : k k, i is isomophic to the Specht module fo the patitio of ito 2 pats of size k ad k i 1 Futhe, ² C : k, i S SymŽ is the module iduced fom S With the use of this lemma ad ecipocity agumets oe ca detemie the stuctue of ² C : This is the case i paticula whe k, i is close to ad we will use this lemma i the ext sectio to detemie the stuctue of some homology modules i tems of Specht modules

18 ON MODULAR HOMOLOY THE HOMOLOIES OF THE 1-STEP MAP I this sectio we estict ou attetio to the case 1 ad fo simplicity the 1-step map 1 is deoted by Thoughout this sectio R is a ig of pime chaacteistic p I 10 we have show that i chaacteistic p 2 all homologies of the 1-step map ae tivial So hee thoughout p 2 If Sym is a pemutatio goup o ad 0 i p the Hk, i is the homology module elative to as defied i Sectio 4 If the we egad the stabilize of as a pemutatio goup o ad so H k, i deotes the homology module elative to To avoid upleasat case distictios we will put H H H 0 whe i 0oip k, i THEOREM 61 Let R be a ig of pime chaacteistic p, 0ip,ad let be a pemutatio goup o Suppose that fo some the size of the obit is co-pime to p ad let N be the omalize of i SymŽ The thee exists a moomophism : H H H k, i k1, i1 which k, i1 commutes with N A special case of this theoem is woth metioig sepaately Note that i both theoems we do ot equie that Ž k, i be a middle tem: THEOREM 62 Let R be a ig of pime chaacteistic p, 0ip,ad let be a abitay elemet of The Hk, i Hk, 1 i1 Hk1, 1 i1 as Sym -modules I paticula, if p 2 ad 0 k the Hk, i 0 if ad oly if Ž k, i is a middle tem fo Remak Note that Ž k, i is a middle tem with espect to if ad oly if Ž k, i 1 ad Ž k 1, i 1 ae middle tems with espect to Hece Theoem 52 ad iductio o ca be used to give the shotest self-cotaied poof that H is tivial if ad oly if Ž k, i is a middle tem k, i Poof of Theoem 61 Let f f l be a elemet of K k, i whee i i i1 supp f supp l The 0 f f i Ž f i Ž l i Ž ad so f 0, l K, ad if l K k, i1 k1, i1 Now defie the map : H H H k, i k1, i1 by puttig k, i1 k,i k, i : f Ž l, if Ž l To show that this is well defied suppose that f h with h h m ad suppž h suppž m So thee exists some F F L pi M with supp F supp L ad Ž F f h Žf kpi

19 574 BELL, JONES, AND SIEMONS g g h lm Note that F F ad L L fo all g ad we calculate Ž f h l m pi Ž F L pi Ž F Ž pi pi1 Ž F pi Ž L pi1 Ž pi implyig that l m M ad that Ž F kpi1 f h Applyig to the equatio gives Ž f h f h Ž l m pi1 Ž F L pi1 Ž F Ž pi1 pi Ž F pi1 Ž L pi1 Ž so that i f g lm M Theefoe is well kpi defied, clealy liea, ad it is a simple matte to check that it commutes with N Ž Suppose ow that f 0, 0 The thee exists F M kpi1 pi1 pi1 with F l ad thee exists H M with Ž H kpi if Ž l The pi Ž F Ž l Ž p i l, pi1 Ž H if Ž l Ž p i 1 pi Ž H pi1 pi pi ad hece H pi1 H Ž F if Let J Ž H Ž pi1 HF The J is fixed by ad we may defie Ý 1 g J J gcosž : piž The J is fixed by ad J if Hece is ijective Poof of Theoem 62 Hee we suppose 14 ad let Ž l, m 1 1 H H The i ŽŽi 1 ŽmŽ l l 0 ad Ž k, i1 k1, i1 1 i ŽmŽ l l Ž l, m showig that is sujective Alteatively, 1 1 use Coollay 46 ii to show that Hk, i1 Hk1, i1 has dimesio k, i As Ž k, i is a middle tem fo if ad oly if Ž k, i 1 ad Ž k 1, i 1 ae middle tems fo 1 the statemet about the o-tiviality of Hk, i is poved by iductio o This completes the poof Theoem 62 is useful fo ivestigatig the ieducibility of the homology modules which we deal with i the ext two esults

20 ON MODULAR HOMOLOY 575 THEOREM 63 Let R be a ig of pime chaacteistic p 2 Fo assume that Hk, i Hk, 1 i1 Hk1, 1 i1 is o-zeo ad suppose futhe that 1 1 H ad H ae zeo o ieducible R SymŽ k, i1 k1, i1 -modules ad that they ae o-isomophic if they ae both o-zeo The Hk, i is a ieducible RSym -module Poof Fo a cotadictio we will suppose that U is a o-tivial RSym -submodule of Hk, i ad so if is the map of Theoem 62 the 1 1 U is a o-tivial R Sym -submodule of Hk, i1 H k1, i1 Theefoe we ae doe if eithe Hk, 1 i1 o Hk1, 1 i1 is zeo Hece we may assume that Hk, 1 i1 ad Hk1, 1 i1 ae ieducible o-isomophic RSymŽ -modules ad futhe that 3, k, ad 1 i p Theefoe U is eithe H o H k1, i1 k, i1 1 1 Case 1 U H k1, i1 Let f be a geeato of Hk1, i1 as give 1 i Theoem 53 So eithe f C if k i o f is a Ž k 1 k1, i1 -subset of 1 1 if k i As f U we have i f Žf U But ² g : agai by Theoem 53 we have f : g Sym H k, i, a cota- dictio 1 1 Case 2 U H k, i1 Fist assume that i k ad let f C k, i1 4 i1 The we may wite f A l whee A 1,,i ad l C ki,1 1 1 Hece puttig f i f f we see that Žf f, ad as 1 Ž, 1 f Hk, i1 it follows that f belogs to U Let f be the esult of applyig the taspositio Ž, to f The 1 Ž Ž, 1 1 f i 1 A1 l A1 l Ž Ž i 1 A1i A l i Al Ž, 1 2 As i 1 coside the taspositio, ad compute f Ž, 2 Ž 1 f i 1 Ž A, 4 l By Theoem 53 4 the Sym -images of 2 1 A 1, 2 l geeate Hk, i which is a cotadictio Secodly we assume that 2 k i ad hee we let A,, 4 1 k so 1 1 that A H Puttig f i Ž A A, we see that f k, i1 1 ŽA so that f U Futhe, Ž, 1 1 f i 1 A1 A1 Ž i 1 A1i A i A Ž, 1 Ž, ad hece f f 2 Ži 1 1 Ž A 2 1 1, 4If ikthe Theoem 53 implies as befoe that the Sym 2 -images 4 of 2 1 A 1, 2 geeate Hk, i which is a cotadictio

21 576 BELL, JONES, AND SIEMONS I ay case expessios of the fom Ž A, ae diffeeces of two k-elemet sets ad so U has co-dimesio as most oe Ž i H We suppose theefoe that k i ad that 1 dim H U k, i k, i dim Hk, i dim Hk, 1 i1 dim Hk1, 1 i1 by Theoem 62 As 2 k we have 1 i 1 i 1 p so that both Ž k 1, i ad Ž k 2, i 2 ae middle tems with espect to 2 ad 0 k 2 k 1 2 Hece by Theoem 62 we have k1, 2 i 0 ad k2, 2 i2 0 which cotadicts 1 k1, 1 i1 k1, 2 i k2, 2 i2 Theefoe fially assume that k 1 ad as Hk1, 1 i1 0 it follows fom Theoem 62 that Ž 0, i 1 is a middle tem so that p i 1 1 As pži1 Ž 1 K M fo i 1 1 we have H M M 1, i1 1 k, i1 1 pi As Hk, 1 i1 is ieducible by assumptio while M1 1 is ot, we caot have pži1 Ž 1 M pi 0 Togethe with p i 1 1 this implies that 1 1pi which meas that H M ŽM k, i 1 ad this module is kow to be ieducible fo p, see 4, p 18 This completes the poof THEOREM 64 Let R be a ig of pime chaacteistic p 2 ad suppose that 0 k ad 0 i p satisfy 2k p i p 1 The Hk, i is ieducible Futhemoe, if Žk, i is aothe pai of positie iteges satisfyig the aboe coditios the H H k, i Poof We pove this esult by iductio o Fo small values of the esult is easily veified Theefoe suppose the esult holds fo 1 Sice 2k p Ž i 1 Ž 1 p 1 ad 2Ž k 1 p Ž i p1 we see iductively that Hk, i1 ad Hk1, i1 ae eithe zeo o ieducible, ad that if they ae both ieducible the they ae o-isomophic Howeve, as p 2 it is easy to see that Hk, 1 i1 ad Hk1, 1 i1 caot both be zeo So Theoem 63 implies that Hk, i is ieducible Now suppose that Žk, i is aothe pai of positive iteges satisfyig 2k p i p 1 ad 0 i p We assume fo a cotadictio that H H k, i ad so by Theoem 62 we have Hk, 1 i1 Hk1, 1 i1 H 1 The iductio hypothesis the implies that H 1 Hk 1, i 1 k 1, i 1 k1, i1 H 1, hece H 1 H 1 ad H 1 H 1 k 1, i 1 k, i1 k 1, i 1 k1, i1 1 Moeove, if H 1 ad H 1 ae both o-zeo the the iductio k, i1 k1, i1 hypothesis implies that k k 1 ad k 1 k, givig us a cotadictio Suppose theefoe that Hk, 1 i1 is o-zeo ad Hk1, 1 i1 is zeo But the we see that i 1 i 1 0 modž p ad by the iductio hypothe- sis that i 1 i 1 Hece 2 i 1 i 1 2 modž p, a cotadictio A simila agumet woks i the case whe Hk, 1 i1 is zeo ad 1 H is o-zeo k1, i1

22 ON MODULAR HOMOLOY 577 Fially we ae i a positio to idetify cetai of the homology modules i tems of Specht modules ad patitios of : THEOREM 65 Let R be a ig of pime chaacteistic p 2 ad suppose that i k ad 0 i p satisfy 2k p i p 1 The Hk, i is isomophic to S S S whee S is the Specht module coespodig to a patitio of ito 2 pats of legth k ad k i 1 Poof By Coollay 54 we have H ² C : ² C : k, i k, i k, i pi M ad as i Lemma 55 we have S ² C : kžpi k,i As S S is the uique maximal submodule of S, the esult follows fom Theoem 64 Remaks Ž 1 Povided that 2 p thee ae Ž p 1 2 distict pais of positive iteges k ad 0 i p with 2k p i p 1 So Theoem 64 povides Ž p 1 2 o-isomophic ieducible Sym - modules ad thei dimesios ae give by the fuctio k, i of Sectio 4 Ž 2 Whe p 5 these modules ae pecisely the Fiboacci epesetatios of the symmetic goups descibed i Ryba s pape 13 Such systems of epesetatios have bee geealized i Kleshchev s wok 9 Fo geeal pime p 2 the collectio H H k, i: k, 0ip, 2ki14 is a example of the semi-simple iductie systems discussed i 9 I fact, H cosists pecisely of the modules aisig fom 2-pat patitios which satisfy Kleshchev s coditio of Theoem 21 i 9 We cojectue that such semi-simple iductive systems fo patitios with moe tha 2 pats aise also as homologies fo suitable posets I the emaide of this sectio we give the complete decompositio of the Hk, i Let a be a itege satisfyig 0 a p Fo 0 i p we defie module homomophisms ad : H : H H H k, i1 k1, i1 by f f ad f f, espectively It is a simple matte to check that these maps ae well-defied We ecod some popeties of these homomophisms i the followig: LEMMA 66 Ž k, i k, i1 If 2k p i a the a : H H is sujectie ifia1ad Ž b : H H is sujectie ifipž a1 k,i k1, i1 Ž i Poof a Let f be i K The f K ad 2Ž k i k, i1 ki,1 p1ž a1 i By the Itegatio Theoem Žo ideed, by

23 578 BELL, JONES, AND SIEMONS p1 Lemma 23 thee exists F i M with Ž F i Ž f kpži1 But the i Ž pži1 Ž pži1 f F 0 ad f Ž F f b Let f be i K The 2Ž k 1 1 Žp Ž a 1 k1, i1 i ad by the Itegatio Theoem Ž o ideed, by Lemma 23 thee exists F i K with Ž F f But the ŽF f k, i We ow peset two futhe esults which will help us detemie the compositio factos of the homology modules LEMMA 67 If 2k p i a the H H k, i k, a Poof We otice that k p i k p a mod p so that H k, i ad Hk, a will have the same dimesio Without loss of geeality we may suppose that i a ad the we look at the map ai : Hk,i Hk, aif i fk the f K ad 2Ž k i p Ž a i k, a ki, ai so that, pžai by the Itegatio Theoem, thee exists F i M with Ž F kpa i i Ž pa ai Ž pa f But the f F 0 ad f Ž F f LEMMA 68 H H k, i k, pi Poof Suppose 2k p i a ad, without loss of geeality, that k k But the 2k p Ž i a ad we ca look at the map pžia : Hk, pi Hk, a Suppose that f K k, a The 2k p ia ad by the Itegatio Theoem thee exists F K k, pi pžia pžia with F f But the F f Howeve, H k, pi ad Hk, i have the same dimesio ad hece applyig the pevious esult completes the poof We ae ow i a positio to detemie the compositio factos of all homology modules Sice Hk, i Hk, pi it suffices to coside the case whe 2k p i a ad 0 a p2 THEOREM 69 Let 2k p i a ad 0 a p2 The the compositio factos of Hk, i each hae multiplicity oe ad ae gie as follows: Ž a H :j0,,a14 kj, ia12 j if a i p a b H : j0,,i14 kj, ia12 j if i a ad Ž c H : již pa,,a14 if i p a kj, ia12 j Poof The poof is by iductio o a Suppose fistly that i a The Hk, i Hk, a ad 2k p a i with i a p i so that, by iduc- tio, the compositio factos of H ae H : j 0,,i1 4 k,i kj, ia12j Secodly suppose that i p a The H H ad 2Ž k k, i k, pa pž pa pia with p i p a i By iduc-

24 ON MODULAR HOMOLOY 579 tio the compositio factos of H k, i ae theefoe Hkj,2pŽai12j : j 0,, pi14 4 H : liž pa,,a1 kl, ia12l Fially suppose that a i p a By Lemma 66 all compositio factos of Hk, i1 ad Hk1, i1 will be compositio factos of Hk, i Sice 2Ž k1 pž i1 2kpŽ i1 Ž a1 ad a 1 i 1i1pŽ a1 we ca assume iductively that Hk1j, i1a22j : j 0,,a24 H : l0,,a24 kl, i1a22l H : j0,,a14 kj, ia12j ae all compositio factos of H This set cosists pecisely of the k, i compositio factos of Hki, i1 togethe with Hk, ia1 Futhemoe, we otice that dim Hk, i dim Hk1, i1 dim Hk, ia1 sice k 1 k ia1 mod p Sice all the modules i H : j kj, ia12 j 0,,a14 ae ieducible ad o-isomophic we ae doe REFERENCES 1 A Bake, A Cocise Itoductio to the Theoy of Numbes, Cambidge Uiv Pess, Cambidge, UK, T Bie, Eie homologische Itepetatio gewisse Izidezmatize mod p, Math A 297 Ž 1993, T Bie, Zweischitthomologie des Simplex ud Biae Spioe, i Egazugseihe , SFB Diskete Stuktue i de Mathematik, Uivesitat Bielefeld, D James, The Repesetatio Theoy of the Symmetic oups, Spige-Velag, New YokBeli, D James ad A Kebe, The Repesetatio Theoy of the Symmetic oups, Ecyclopedia of Mathematics ad Its Applicatios, Cambidge Uiv Pess, Cambidge, UK, S Lag, Algeba, 3d ed, AddisoWesley, Readig, MA, W S Massey, A Basic Couse i Algebaic Topology, Spige-Velag, New YokBeli, W Maye, A ew homology theoy, A Math 48 Ž 1947, , A Kleshchev, Completely splittable epesetatios of symmetic goups, J Algeba 181 Ž 1996, V B Mukhi ad J Siemos, O the modula theoy of iclusio maps ad goup actios, J Combi Theoy, 74 Ž 1996, V B Mukhi ad J Siemos, O modula homology i the Boolea algeba, J Algeba 179 Ž 1995,

25 580 BELL, JONES, AND SIEMONS 12 V B Mukhi ad J Siemos, O modula homology i pojective spaces, i pepaatio 13 A J E Ryba, Fiboacci epesetatios of the symmetic goups, J Algeba 170 Ž 1994, E H Spaie, The Maye homology theoy, Bull Ame Math Soc 55 Ž 1949, R M Wilso, A diagoal fom fo the icidece matix of t-subsets vesus k-subsets, Euopea J Combi 11 Ž 1990,