THE DIMENSION FORMULA FOR THE RING OF CODE POLYNOMIALS IN GENUS 4

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1 Oura, M. Osaka J. Mah. (), - THE DIMENSION FORMULA FOR THE RING OF CODE POLYNOMIALS IN GENUS MANABU OURA (Receved March, ). Inroducon The purpose of hs paper s o sudy he dmenson formula for he nvaran rng C[/ α for aef^] **, whch may be consdered as he rng of code polynomals n genus. We also gve all characersc polynomals of elemens n H. The man ngreden s he deermnaon of he conjugacy classes of he symplecc group /(,). Our resul wll be useful for he nvesgaon of he Segel modular forms n genus four. We recall from [, ] ha he fne group H g s (up o ±) jus he mage of he modular group Γ g = Sp(g,Z) under he hea represenaon (of ndex ) and ha he rng of modular forms of even wegh s gven by Λ(Γ g ) () = Θ [Γ r *] = (qyj*- / {relaon})", fc where N denoes he normalzaon n s feld of fracons and "relaon" are he hea relaons. However, he generaors and he dmenson formulas for A(Γ g ) are known only for genus g<. On he oher hand, he nvaran rng C[/ α ] Hg may be consdered as he rng of code polynomals n genus g. In [], here s he defnon of he g-h wegh polynomal for codes (codes mean he bnary lnear codes) and he connecons among codes, laces, he nvaran rngs of he fne groups, and he heory of modular forms were suded (cf.[], [], [], [], []). In parcular, was shown ha he nvaran rng of he group <// g, >, whch s he subrng of C[/ α ] H «, s generaed by he g-h wegh polynomals for self-dual doubly-even codes, where s he prmve -h roo of uny. Ths nvaran rng corresponds o he rng of he modular forms of weghs dvsble by. The auhor would lke o hank Prof. Banna for suggesng hs work and for hs encouragemen. He owes hanks o Prof. Runge for hs crcal commens on he orgnal manuscrp. Furhermore, he auhor would lke o hank he referee for provdng hm wh many mprovemens houghou he whole paper.

2 M. OURA. On he group H g In hs secon, we sudy he group H g. In addon o Runge, he group H g has been suded by several auhors, for example, see [], []. Le V be he g-dmensonal vecor space over he feld of wo elemens,.e., K=Ff. For x,ye V, le x-y denoe he usual do produc. Se g '~ v and for a symmerc g x g marx S /) s :=dag(/ S[Λ] wh aev), where S[ά]:=aS*a. Le /^:=<^,D s S runs over all symmerc marces n Ma gxg (Z)> be he subgroup of Gl( C) generaed by he elemens T and he D. g s We ge for he nvarn rng C[/ a ] Hg he dmenson formula Φn.(= Σ (dιmc[/j d^ =_^, where C[/ fl ]? g s he rf-h homogeneous par of C[/ α ] Hg, Φ Hg ( s called he Molen seres s of H g. The followng lemma gves he smplfcaon we use. Lemma. (Lemma. []). One has an exac sequence where N g :=(,Dl,T~ l DlT g y and he homomorphsm φ s gven by he conjugaon of H g on he F -vecor space Λ^/</>. Π Therefore we have he obvous formula where {Q} are he conjugacy classes of Sp(g,) and z s some elemen of H g wh φ(z )ec. Our compuaon s done usng (.). Fnally we ls he szes of groups.

3 CODE POLYNOMIALS IN GENUS In our case fe = ), \N \ =, =, \Sp(,)\ =,,, =, \H \=,,,, =. REMARK. H <//? C >, // are he reflecon groups No., No., No. n [], respecvely (cf. Proposon. []).. On /(,) and s conjugacy classes In hs secon, we sudy he symplecc group Sp(,). Ths s denfed wh he Chevalley group of ype (C ) over he feld of wo elemens. We gve all he characersc polynomals of elemens n //. We remark ha we canno read off represenaves of he conjugacy classes of S/?(,) from Alas [] alhough s he good reference for he fne smple groups. As s sad, X,) s one of he Chevalley groups. The properes of such groups are known and we descrbe wha we need. Le Δ = {±f, ±ξ±ξj (/</)! </',./< } be he roo sysem of ype (C ), and choose a = ξ ί ξ, b = ξ ξ^ c = ξ ξ, d=ξ for a fundamenal sysem Π of roos. We denoe by Δ" " he se of posve roos wh respec o Π. We wre an elemen αα -f βb + yc + δd of Δ + as aβyδ. For example, we wre for a + c + d. Ej s he elemenary marx of sze x wh n he (/J)-enry. For < j <, x ξ -</=!+ E^ + E +ja + ( <y ), Then X,) s generaed by x r (reδ) and s known o be one of he fne smple groups (cf.[]). Le X r be he group generaed by x r and pu ^ = A Γ Ar Ar Ar Ύ ^o ^ ^o ^^ ooo^ oo^ o^ ^ ^ ^ Then B s a Sylow -subgroup of S/?(,) and s normal n /?(,). Pu n r = x r x_ r x r and V=<«r reδ>. Λf s somorphc o he Weyl group of ype (C ). We fx he followng correspondence: ~ :=[,,,,,,,],

4 M. OURA <E ~ := [,,,,,,,], ~ := [,,,,,,,], -^ <->-_!:= [,,,,,,,], -ί ~:z:= [,,,,,,,], - ^^-:= [,,,,,,,], -ξ ^-:= [,,,,,,,]. An elemen of N s unquely deermned by s naural acon on,,,. If nεn sasfes In = a?«= /?, «= y, n = δ, hen we denoe n by w(α,/?,y,(). For example, we have In he followng, we gve he conjugacy classes of S/?(,) and all characersc polynomals of elemens n /. To deermne he conjugacy classes of S/?(,), we have o show () () No wo elemens n he ls are conjugae n S/?(,), where C Sp(>) (c,) denoes he cenralzer group of c { n /?(,). These saemens are proved usng GAP []. Then we compue x deermnans of sze x. Snce H s a subgroup of SU(\Z\^^ (Proposon. []), he polynomals have he ype Σ :δl^ έ/ wh α = α = l, ά =α, ά = α, ά = α, ά =, ^ = ^!!, ά =α, ά Ί a g, and ά s = a s (A bar denoes complex conjugaon). There are boxes below and he ί'-h box (<z<) gves he characersc polynomals of elemens n z ( V. In each box, he frs column gves he mulplcy of each polynomal. The nex columns gve he values of α,α,,α, respecvely. For example, n he frs box, he n he frs column means ha here are occurences of he polynomal n he form + ( /)ί /f + (

5 CODE POLYNOMIALS IN GENUS Table. -: + :' - - : - + : -: : - -* order =, co =, \C Sp (Λ,)M\ = -- : : : - -+: : : -: :' : + : - - : :' - - : - : + : - : -: +:' - - : - + -: : :' -:' : -: order =, GI = rooo, \C Sp (e..)(cι)\ = -- : : : : :' : -: : - : - + :' : + : - - :' :' + : - :' + :' - : - - S order =, = #» \Csp fι) (c ) = ι order =, c = α?oιoo*o, Cs P (,)( c )l = ι order =, c = :* xo, \Cs P (,)(c*)\ = ι ί «z

6 M. OURA order =, c = xooo^mo, \C Sp (B.)M\ = - - -ι - " - - ι' order =, cβ = r ιoo*oooι, \C Sp( ι )(c ) = ' - + ' ι -I I S -S - I Sι - - ' ι order =, c = n(,,l,), \C Sp(>} (c )\ = - - S - -S ι ι order =, c β = (xooιm(-, -,, -)), C p( ) (c ) = - ι lo - - order =, c = xoooιn(l,,, -), C ί p ( ( )(c ) = -ί z S - - -ι order =, CJQ = xoooιn(,, -, -), C' Spί ) (cιo) = S - -S

7 CODE POLYNOMIALS IN GENUS - + ' order =, c n = :*, l^ P (.)(cιι) = z ι' « » order =, = # * ^, C'sp(,)( c ) = : ί order =, = #^^^» I^Sjpίβ,)(c) = ' - -' order =, cι = ^oooaoxoo, C's P (,)( c ) = z order =, = #^*, C'sp(,)( c ) = ι order =, GIG = xoιooa:oooι^ιιιo^ιιιι, C's p (,)( c ) = - -- lo lo - -ι - - lo lo

8 M. OURA order =, c = *^, Csp(,)(c) = ί - z - - S order =, cβ = #oιoo^oooι#π, \C p(b,)( c ι*)\ = ί ' order =, cg = :rooo#ooo^oo^» I^p(,)( c ) = f + ~ " ' - -I- + - order =, c o = a ooo^ooo^ooo, I^Spίβ, )(cθ) = order =, = aoιoo*ooιo*oon*ιιιo> C*p(,)( c ) = - *

9 CODE POLYNOMIALS IN GENUS OΓdeΓ =, C = ^^^^^,! C 'pί,)( c ) = lo -lo - -I I - order =, c = *ooo^oonn(l,, -,), C f Sp(>)(c) = order =, C = #ooo'on(-,-, -, ), C l p S (()(c ) = f - order =, c = n(-,, -,), C p () (c ) = - + ι' S - - S-S order =, c β = n(-, -, -, ), \C Sp( - ί ) (c ) =» z + order =, c = n(-,,l,), C r )(c ) = Sp() ι + - -ι Ϊ I - + ; I -f lo -- lo --- W - lo

10 M. OURA order =, c = zooιι^oθn(, -,, -), C p( ) (c ) = ι - - ' order =, c = ^ooo # # n(l,, -,), C r p ( > )(c ) = S order =, c = a:oon(l, l -,), C Sp(ι) (c ) = - z' -ι ϊ order =, c ι = :r ooffooιι*oθrι(-, -,, ), C*s p(ι )(c ι) = order =, c = :rooιo#oooι*on(l,,, ), C ί ( )(c ) = Sp ) » order =, c = xon(-l, -,, -), C p ( )(c ) = f

11 CODE POLYNOMIALS IN GENUS order =, = xoooι^oon^oιιι^oo^o^(l»,, -), I -f Iβ - ι - -f <?Sp(β,)(c) = order =, c = :^^^^(, -,,), Cp(,)( c ) = order =, = ^oooι^oon^oιιι^oo^o^(,,, -), \C Sp l f' I )(cβ) = order =, c? = (acoooιn(- f, -, -)), C r Sp( )(c) = S S S + order =, c = (a"oooaoo^o^on(l, -,,)), C' p (( )(c ) = I - - order =, c g = (^ooon(,,,)), C p ( )(c) = - - order =, = (aooιo^oo^on^ooa:on(, -,,)), C f p ( f )(cθ) = I υ - -

12 M. OURA order =, c J = α: on(-,,, -), C Sp(ι )(c ι) = Ί order =, c = ^ooo^ooo^ooo, C'sp(,)( c ) = + ι f Ί «- - z - + order =, = orooo^ooo^ooo^j ^p(,)( c ) z - - order =, c * = xooo^ooo^ooo^ooo, C'sp(,)( c ) = - ί f -H - + f - -! + «' -I f - -f f - - f -+ -» + ' - -f order =, c = ^ooo^ooo^ooo^oo, I^Spίβ^ί^s)! = -ί f W ' _ lo - - -» - -f W -I - - -' - ί -f < - --f ι' -- order _, c β = #oιoo^ooιo^oooι#ooιι#ιooo> C^p( f )(c ) = f ί - - ί ί -ί ί ί f - - ί

13 CODE POLYNOMIALS IN GENUS order =, = ίcθl^^ίpθ^^^» I^Sρ(,)( c ) = - order =, c = :rooora(-, -l,,), \C S Pί,)(c) = o order =, = #ιι^π(-, -, -, ), \C Sp ( S )( C ) = I - order =, c = xooιm(-,,-l f -), C'sp(,)(cθ) = ι - ι - O order =, c ι = xooιn(-, -, -, ), C Sp() (c ι) = I order =, c = x on(-,, -, -), \C Sp(Bι ) ι ί (c ) =

14 M. OURA order =, c s = ^oooλ oonn(,,, -), C Sp(( )(c) = ' -lo «- - lo' -I order =, c = x oon(-, -,, ), Cs P (,)(c) = ί order =, = xoooιn(,, -, -), C f p( )(c) = ί - - order =, c = #^(,,,), \C Sp (,)(C) = order =, c = :r ol^oon(-, -,, ), Cs P (,)^) = ί ι ' - - ί ί - order =, c = xo#oθra(,,, -), \C Sp(> )(c s)\ = order =, = xoooι^oozόιon(l,,,), Cp( f )(c) =

15 CODE POLYNOMIALS IN GENUS order =, c o = zoooι*ooιxo (-l,,, -), Cs^β^ί^Go)! = I order =, c ι = r ol^ol^o^on(l,,, -), C p()) (c ι ) = + ' ' ' ' ' ' ' order =, cβ = ^^ n(l,,, ), \C Sp f,)(c) = * + ' - ' ϊ + ' order =, CGS = foooι#ooιια;oιιson(, -,, ), \C Sp(,)(C) = order =, c = (#ooo^ooιι^oθn(-, -,, -)), C ί p( ι.)(c) = - - ' order =, c = (xoιιι*oθn(- f, -, -I)), C ί (c p() ) = ' -

16 M. OURA order =, c e = ^ooιo^ooιιxoιιι^ooaon(~, -,,), \C Sp (& )(cββ)\ = - - order =, c = xooιιn(-,, l,-), C Sp(βι) (cβ) = f order =, c β = ^ooι^(-,, -, ), \C Sj X.)(C) = order =, c = (*oonzoon(, -, -, -)), C p( - - -I )(C) = order =, c = aooι^(-, -,, -), C' Sp(f) (co) = I order =, c J = xoooι^ooιιn(-, -,,), C f ί, ( )(c ι) = order =, c = (*oooι*ooιιπ(-, -,,)), C p( )(C) = *

17 CODE POLYNOMIALS IN GENUS order =, c = x ooιn(-,,-, -), C Sp( ) (c ) = order =, en - *ooo*oon(,, -,), Csp(,)( c ) = order =, c = a?ιma?ιn(-, -,, ), C pfβ.)(c) = order =, c?e = #oooι#ooιπ(-, -, -, ), \C [B SP,)(C *)\ = - I ' order =, = ίόoo^oon^oθn(-, -,, -), C f p( )(c) = f order =, c = a?om*oon(-,, -, -), C Sp( ) (c) = ί order =, c = *oo*oon(,-, -, -), C p(β) (c) =

18 M. OURA order =, c = zoon(-,, -, ), Cs P (,)(cθ) = ϊ ' x ι REMARK. The deermnaon of he conjugacy classes of Chevalley groups were suded by several auhors. For example, see [], [], [].. Man resul We have obaned he surjecve homomorphsm φ:h -+Sp(* ) wh Kerφ = ΛΓ and a se {c} <<o of represenaves of conjugacy classes of >$p(,) n he precedng secons. Our man resul can be saed as follows: Theorem.. The Molen seres of H s gven by + / + f + ί + / + ί + f where D = ( - - ί ) (l - ί χ - - r )(l - r χ - ί χ - βs ) x - ί - f + ί + f + ί -h U + H + Γ - ί + f + / + ί + f + ί / -f^ -/ +ί

19 CODE POLYNOMIALS IN GENUS + / +f + + * + f + * + f + ί +? + < + f + ί + / + * + ί +? + / + ί + / + ί + ί + ί + ί + / + ί + / + / +? + ί + ί + / + ί + / + ί + ί + / + ί + ί + ί + ί +ί + / + / + / + ; +? + ί + / + ί + ί + ί + ί + ί + / + ί + ί + ί + < + / +ί +ί + / +ί + * + / +ί + ί + ί +ί + ί + < + ί +? + / + / + ί + / + ί + ί + ί +ί + ί + / + ί + ί + *

20 M. OURA f + f + ί + ί - < + ί + ί. Π References [] M. Broue and M. Enguehard: Polynόmes des pods de cerans codes e funcons hea de cerans reseaux, Ann. Scen. EC. Norm. Sup. (), -. [] B. Chang: The conjugae classes ofchevalley groups of ype (G \ J. Algebra (),. [] Conway, Curs, Noron, Parker and Wlson: Alas of fne groups, Oxford Unversy Press,. [] W. Duke: On codes and Segel modular forms, In. Mah. Research Noces (), -. [] W. Ebelng: Laces and Codes, A course parally based on lecures by F. Hrzeburch, Veweg,. [] H. Enomoo: The conjugacy classes of Chevalley groups of ype (G ) over fne felds of characersc or, J. Fac. Sc. Unv. Tokyo Sec. IA (), -. [] S.P. Glasby: On he fahful represenaons, of degree ", of ceran exensons of -groups by orhogonal and symplecc groups, J. Ausral. Mah. Soc. (Seres A) (),. [] W.C. Huffman: The bwegh enumeraor of selforhogonal bnary codes, Dsc. Mah. (), -. [] B. Runge: Codes and Segel modular forms, Dsc. Mah. (),. [] B. Runge: On Segel modular forms, par I, J. Rene angew. Mah. (), -. [] B. Runge: On Segel modular forms, par II, Nagoyya Mah. J. (), -. [] B. Runge: The Schoky deal, n "Abelan Varees" (Proceedngs of he Inernaonal Conference held n Eggloffsen, Germany, Ocober, ), Waler de Gruyer, Berln-New York,, pp.. [] M. Schόner, e. al: GAP: Groups, Algorhms and Programng, Lehrsuhl D fur Mahemk, RWTH Aachen,. [] G.C. Shephard and J.A. Todd: Fne unary reflecon groups, Canad. J. Mah. (), -. [] K. Shnoda: The conjugacy classes of Chevalley groups of ype (F ) over fne felds of characersc, J. Fac. Sc. Unv. Tokyo (), -. [] N.J.A. Sloane: Error-correcng codes and nvaran heory:new applcaons of a nneeenhcenury echnque, Amer. Mah. Mahly (), -. Graduae School of Mahemacs Kyushu Unversy Hakozak --, Hgash-ku, Fukuoka, -, Japan e-mal address: ohura@mah.kyushu-u.ac.jp