DSpace at VNU: Splitting B(Z p)n+ from the multiplicative group of the subfields of Finite Fields

" - 1) Fi Fp = Fp{uj) and F*,: = < >, we have K = where r = and i r = < tì'- >, N o t a t i o n 2.2 We let c,„ =< c > act on z / ( j f - 1) by d l l ) = (/;)” /"' (mod p" - 1) L be the orbit containing 11 tinder the action of c,,, On the S ( ' t [ Z / i j f — 1))/C„1, the action u c , = < rt { Z / i p " - l ) ) / ( Z / ( / / " - 1)) is given by a{ Foi cacli v/o choor.o If to Uo tho oloi^iont i>f Thi*ii v,,(ííí) it; OY‘A /'t]\ thí' ^ consisting of the elements of z / (;>" —1) in form //.// —1) with < / < 77/ “ , < // < r —1 the set consisting of such representatives Ỉ/ is I(n i) Hero, /„(ni.) and z„(in) are (letonnim as follows: We let c, n > act on z / ( j / " 1) by ộ { u ) - up Let Jui^n) be the orl)it roiiTaiiti] u, and let / ( m ) be a set consisting of one ('leineiit from each orbit The caulinality i)i -/„(///) wiiere z,^{ni) is the smallest positive exponent k with = u (mod p ”' - 1) Harris has proved the elements ([7],3.5,4.4) p"’-2 fu{rn) - ^ e„(;n) , V e I { m ) , where e„(m ) = i’€ Ju{rn) form a primitive orthogonal set of idernpotent/s v/ith splitting of B { Z / p ) from - k'Vi ^kr k~ — in Fp[K*\ In order to give each /u (n ỉ) have to be expressed as a sum of primitive orthogor idenip)Otents , J e z / i p ” - 1) , in Fp-[F*n] (see [4]) We have the following m ain lemma: p u t t i n g B ( Z /p ) “ f r o m t h e m u l t i p l i c a t i v e g r o u p e m m a 2.3 For m I n aiid 11 we have V, G / ( m ) z,.(m)- fuim ) = ^ e, = 5^ jeV.Arn) l=zQ 7= up^(mod ^roơ/ We have u(m)~l p"-2 r-1 76 Vu(m) Vu(ni) /=0 i;=0 /i=0 [here r =: ^nZi\ ^ ^ — l(niocl p), and r-l + ^ I ° I £:r> if ^ ifr i; [(nice —'2 Ẹ jev„{, n) V'„(/n) 'i - E „ { tỉí) — (- E /=0 = A=0 ‘h e o r e m Ful ni \ n let (ĩup‘(m) = fu(rn, E /=0 = f„{tn)B{Z/p)'\_ Then B(Z/pyi '2';,,íỉ,('0 ~ \/ — \ / z,,.,„(//,), »e/íífí) ^'n(0 lemw.irn) = J ,(ĩ ì)B(Z /] )Yl rind Y],(j) \J ~ ie\\,{ni) the Criniphell-Sclick Ị lir ( ’ai iiphrll-S(‘lic'k sninniaii(ls are (les cri br d in d e t a i l s in [2] a n d [7 hvof Fro m L n i i i u a 2.3 VV(' havp Mm) = ieinv„{ni) ('K' the / „ ( / » ) s (í/ G H^o) ) io n n an orl-hogonal spt of idempotPiits with x ^ / u ( w ) ' j , \ C L „ { Z /}))] H p i k' o , = in the proof of the Theorem is conipletf'd Ộ In 8], Harris and K uhn follow the ronstiiK'tions of tlie inedviciblp representations S \ (resp 5^) ior A € A (resp À G A') of Fi,[M„{Z/p)] (resp Fi,{GLri{Z/p)]) as given by Jam es and Kerber t 9] chapter - in paitic'ulai, exercise 8.4 of their book, whpiv A = { (A i , A„) I < A, < /> - 1, < Ẳ- < 7/}, A' = {(À 1, ,A„) I < A*, < - 1,1 < A- < 7) - 1,0 < A„ < p - } N g ĩ i y e n G ia Di% ỊO th(' stal)l(' sumiiiaìKỈ o ĩ l ì { Z / p Ỵ Ị coiK'spoiulin^ tí) 5(A) (l■í^sp 'r iu ' A'a s HK' tlu' iu(l('coiuỊ)osahle s u iu in a iu ls oí B { Z / p ) ' ị aiul th(' (2.5) V ( \, A'(a .A,.)A,, s HK' s p l i t t e d ov('i tli(' X \ < A„ < / ; - aiul 0) — ' ^ ( > ' \ „ - 1, 0) ^ ( A i A „ _ , ; ; - ! ) • Foi each A e A', i € I let a'y^ 1)(' the nuuilx'i of times tli(' n'picsciitatioii Fj,\F*y,]Ị,{iì) occ 14 s \ lien ftii' J G ill a couipositioii sf'i ics for y „ (j) (2.6) Harris has proved that ([7], G): V ^ ■ Af=A' One of the fundam ental piohlpiiiK tliat has been Ix'ing iuteist(*d in at picsoiit in the piohlciu fiiidiag a stable splitting of B { Z / p ) ' ị is tơ ciptennịiií' L o m i n a ([5j) The eigenvalĩìcs fur the fiction of f ) oil ^iic Wcyl inodilic v r “ riie T is a seniistandỉìid n-tỉihỉeaii u f content , íị,) »nd li(T) = ) win ^0k) ( iijod p - H(ne the notions ơt' Wevl mo(hil(‘ and sPuiisraiKlaicl a - t a h le a u ot content [ị3\ A , ) (' b(' found in [8 L e m m a ( [5 ]) T h e eigenvalnes fur th e ac t io n ui f) uii Sịx) where m.x = Ai + A2 + ■■■+ r>Ki ^vitìì A = ( À i A„ ) From Theorom 2.4 (2 5) (2.6) Leiiinm 2.7 and Lem m a 2.8, T h e o r e m If ni\ ^ wit h J = m x { m u d Ị) — have tlie following th fo if II (mud p — Ì) then X \ is nut a suuiiLiíiììd uf Zii ni{ii), su It IS Itut siiuniinnd u f Y „ { j ) fur Hiich J € V'„(///)■ For A = (Ai .A„) where A, = or /) - nix = (mod /) - 1) and Wf> have rlic tolowi corollary: C o r o l l a r y A'(Ai A„) foi X, = p - 1, fuv example 01 1) 1, m e the sminnHitds / v , (,„od r - ) ^'n(j)- REFERENCES G Carlsson Equivaiiarit stable hom otopy and Segal’s Burnside ring conjecttue, Ann Mat 120(1984), 189-224 H.E.A.Cam pbell and P.S.Soliok Polynom ial algebras over the Steeruod algebra, Comme Math Helv , 65 (1990), 171-180 H.Davenport Bases for finite fields, J.L ond on Math.Soc , 65(1968), 21-39 Nguven Gia Diiih M odular represeiitation of some linear groups over a finite field, Viet Math., Vol 24, N (1996) 143-154 Nguyen Gia Dinh On certain stable v/edge sum m ands of B { Z / p ) ’J^, A cta m atheriiatica Vi namica, Vol.21, N o.l (1996), 1-13 J c Harris, Stable spUttings of classifying spaces, Ph.D thesis, University of Chicago, 1985 f l i t t i n g B ( Z / p ) " f r o m the m ultip lica iiv e group ►J c Harris, On certain s t a b l e wedge suniiimnds o f B ( Z / p ) ’l , Can ’ 104-118 13 J M ath., 44(1) (1992) 'ị I.e Harris and N.J Kuhn, Stalììe decoiiipositiuns o f clitssifyiiig spaces o f finite abelian p-grotips M ath Proc; Camb Phil Soc., 103 (1988), 427-449 , G Jam es and A Kerber, The Representation Theory of Sym r netnc Group, Encyclopedia of Mat[i and its A pplications,16,Atidisoii-Wesley, 1981 ^p CHI KHOA HOC DHQGHN, KHTN, t x v , n^s - 1999 PH Ả N RÃ B { Z / P ) ^ T Ừ NHĨM NHẢN TÍN H CỦA CÁ C T R Ư Ờ N G CON TR O N G CÁC T R Ư Ờ N G HỬƯ HẠN Nguyễn Gia Định Khon Toán - Đại học K H - Đni học Htiế Giả sử p luọt số a gu vên tố, f), triròrng p ” phần tử, K trường pỊ,n i K ' = A'-\ {()} Iihỏiu Iihảu tính cùa K Bầiig cách sừ dụng lý tliuyết biểu dipii lũy đ ẩn g «■11 troiift vành nhóm ('liúiig ta đ a phản hoạch B { Z / p Y ị thành hạug tủ ỊỌUÍ^ số ôn địiih đipii kiện đe rnỗi liạiif> từ Iiáy khóiig chửa hạng từ khơng khai ạru ilưưc  ... k' o , = in the proof of the Theorem is conipletf'd Ộ In 8], Harris and K uhn follow the ronstiiK'tions of tlie inedviciblp representations S (resp 5^) ior A € A (resp À G A') of Fi,[M„{Z/p)]... Soc., 103 (1988), 427-449 , G Jam es and A Kerber, The Representation Theory of Sym r netnc Group, Encyclopedia of Mat[i and its A pplications,16,Atidisoii-Wesley, 1981 ^p CHI KHOA HOC DHQGHN, KHTN,... piohlpiiiK tliat has been Ix'ing iuteist(*d in at picsoiit in the piohlciu fiiidiag a stable splitting of B { Z / p ) ' ị is tơ ciptennịiií' L o m i n a ([5j) The eigenvalĩìcs fur the fiction of f )