ToappearinCommunicationsinAlgebra ADMISSIBILITYCONDITIONSFORDEGENERATECYCLOTOMICBMW ALGEBRAS 0 1 0 FREDERICKM.GOODMAN 2 n ABSTRACT. Westudyadmissibilityconditionsfortheparametersofdegeneratecyclo- a tomicBMWalgebras. Weshowthattheu–admissibilityconditionofAriki,Mathas J andRuiisequivalenttoasimplemoduletheoreticcondition. 2 2 ] A Q 1. INTRODUCTION . ThecyclotomicBirman–Wenzl–Murakami(BMW)algebrasareBMWanaloguesof h t cyclotomic Hecke algebras[2,1], while the degeneratecyclotomicBMWalgebrasare a m BMWanaloguesofdegeneratecyclotomicHeckealgebras[10]. ThecyclotomicBMWalgebrasweredefinedbyHäring–Oldenburgin[9]andhave [ recentlybeenstudiedbythreegroupsofmathematicians: GoodmanandHauschild– 2 Mosley[6,7,8,4],Rui,Xu,andSi[14,12],andWilcoxandYu[16,17,15,18]. v DegenerateaffineBMWalgebraswereintroducedbyNazarov[11]underthename 3 5 affine Wenzl algebras. The cyclotomic quotients of these algebras were introduced 2 by Ariki, Mathas, andRuiin [3] andstudied furtherbyRui andSi in [13], underthe 4 namecyclotomicNazarov–Wenzlalgebras. (Wepropose torefertothesealgebrasas . 5 degenerateaffine(resp. degeneratecyclotomic) BMWalgebrasinstead,tobringthe 0 terminologyinlinewiththatusedfordegenerateaffineandcyclotomic Hecke alge- 9 0 bras.) : Apeculiar feature of the cyclotomic anddegenerate cyclotomic BMW algebras is v i thatit isnecessarytoimpose “admissibility" conditions ontheparametersentering X into the definition of the algebras in order to obtain a satisfactory theory. For the r a cyclotomicBMWalgebras,twoapparentlydifferentconditionswereproposed,oneby WilcoxandYu[16]andanotherbyRuiandXu[14].Werecentlyshowed[5]thatthetwo conditionsareequivalent.Moreover,accordingto[16],admissibilityisequivalenttoa simplemoduletheoreticcondition: theleftidealW e generatedbythe“contraction" 2 e inthetwo–strandalgebraisfreeofthemaximalpossiblerank. Itisnaturaltoaskforsimilarresultsregardingtheparametersofdegeneratecyclo- tomicBMWalgebras.Ariki,MathasandRui[3]introducedanadmissibilitycondition (calledu–admissibility)forthesealgebras,basedonaheuristicinvolvingtherankof theleftidealW e inthetwo–strandalgebra,butupuntilnowithasnotbeenshown 2 that their condition is equivalent to W e being free of maximal rank. In this note, 2 2000MathematicsSubjectClassification. 20C08,16G99,81R50. 1 2 FREDERICKM.GOODMAN weintroduceananalogueforthedegeneratecyclotomicBMWalgebrasoftheadmis- sibility condition ofWilcox andYu [16],we showthatthiscondition isequivalent to u–admissibility,andthatbothconditionsareequivalenttoW e beingfreeofmaximal 2 rank. 2. DEFINITIONS Fixapositiveintegern andacommutativeringR withmultiplicativeidentity. Let Ω={ω : a ≥0}beasequenceofelementsofR. a Definition 2.1(Nazarov [11]). The degenerateaffine BMW algebraWaff = Waff(Ω) is n n theunitalassociativeR–algebrawithgenerators{s ,e ,x : 1≤i <n and1≤ j ≤n} i i j andrelations: (1) (Involutions) s2=1,for1≤i <n. i (2) (Affinebraidrelations) (a) s s =s s if|i−j|>1, i j j i (b) s s s =s s s ,for1≤i <n−1, i i+1 i i+1 i i+1 (c) s x =x s ifj 6=i,i+1. i j j i (3) (Idempotentrelations) e2=ω e , for1≤i <n. i 0 i (4) (Commutationrelations) (a) s e =e s ,if|i−j|>1, i j j i (b) e e =e e ,if|i−j|>1, i j j i (c) e x =x e ,ifj 6=i,i+1, i j j i (d) x x =x x ,for1≤i,j ≤n. i j j i (5) (Skeinrelations) s x −x s =e −1,andx s −s x =e −1,for1≤i <n. i i i+1 i i i i i i+1 i (6) (Unwrappingrelations) e xae =ω e ,fora >0. 1 1 1 a 1 (7) (Tanglerelations) (a) e s =e =s e ,for1≤i ≤n−1, i i i i i (b) s e e =s e ,ande e s =e s ,for1≤i ≤n−2, i i+1 i i+1 i i i+1 i i i+1 (c) e e s =e s ,ands e e =s e ,for1≤i ≤n−2. i+1 i i+1 i+1 i i+1 i i+1 i i+1 (8) (Untwistingrelations) e e e =e ,ande e e =e ,for1≤i ≤n−2. i+1 i i+1 i+1 i i+1 i i (9) (Anti–symmetryrelations) e (x +x )=0,and(x +x )e =0,for1≤i <n. i i i+1 i i+1 i Definition2.2(Ariki,Mathas,Rui[3]). Fixanintegerr ≥1andelementsu ,...,u in 1 r R,ThedegeneratecyclotomicBMWalgebraW =W (u ,...,u )istheR–algebra r,n r,n 1 r Waff(Ω)/〈(x −u )...(x −u )〉. n 1 1 1 r Notethat,duetothesymmetryoftherelations,WaffhasauniqueR–linearalgebra n involution∗(thatis,analgebraanti-automorphismoforder2)suchthate∗=e ,s∗= i i i s ,andx∗=x foralli.Theinvolutionpassestocyclotomicquotients. i i i aff Lemma2.3(see[3],Lemma2.3). InthedegenerateaffineBMWalgebraW ,for1≤ n i <n anda ≥1,onehas a (2.1) s xa =xa s + xb−1(e −1)xa−b. i i i+1 i i+1 i i b=1 X DEGENERATECYCLOTOMICBMWALGEBRAS 3 Takingi =1inLemma2.3,pre–andpost–multiplyingbye andsimplifyingusing 1 therelationsgives: a a (2.2) ω e =(−1)aω e + (−1)b−1ω ω e + (−1)bω e a 1 a 1 b−1 a−b 1 a−1 1 b=1 b=1 X X Fora odd,thisgives a (2.3) 2ω e = (−1)b−1ω ω e −ω e , a 1 b−1 a−b 1 a−1 1 b=1 X whichisCorollary2.4in[3].Asnotedin[3],theidentityderivedfrom(2.2)incasea is evenisatautology. ConsiderthecyclotomicalgebraW (u ,...,u ),andleta denotethesignedele- r,n 1 r j mentarysymmetricfunctioninu ,...,u ,namely,a =(−1)r−jǫ (u ,...,u ).Thus, 1 r j r−j 1 r inthecyclotomic algebra,wehavetherelation r a xj =0. Multiplyingbyxa for j=0 j 1 1 anarbitrarya ≥0andpre–andpost–multiplyingbye gives P 1 r (2.4) a ω e =0. j j+a 1 j=0 X Corollary2.4. Considerthe cyclotomicalgebraW (u ,...,u ). If e isnot a torsion r,n 1 r 1 elementoverR,thenwehave: (1) 2ω = a (−1)b−1ω ω −ω ,forallodda ≥1,and a b=1 b−1 a−b a−1 r (2) a ω =0,foralla ≥0. j=0 Pj j+a DefinitiPon2.5. Saythattheparametersω (a ≥0)andu ,...,u areweaklyadmis- a 1 r sible,orthatthe groundringR is weaklyadmissible,if the relationsof Corollary2.4 hold. Weakadmissibility isanon–trivialitycondition forthecyclotomic algebras; ifthe groundringisafield,andweakadmissibilityfails,thene =0,andthecyclotomical- 1 gebrareducestoaspecializationofthedegeneratecyclotomicHeckealgebra,see[3], pages60–61. Inthefollowing,weusethenotationδ =1ifP istrueandδ =0ifP isfalse. (P) (P) Lemma2.6. InthedegenerateaffineBMWalgebra,fora ≥1,wehave a (2.5) s xae =(−1)axae + (−1)b−1ω xb−1e −δ xa−1e . 1 1 1 1 1 a−b 1 1 (aisodd) 1 1 b=1 X Proof. Takei =1inequation(2.1). Post–multiplybye ,andsimplify,usingtherela- 1 (cid:3) tions. 3. u–ADMISSIBILITY Thedefinitionofu–admissibilityismotivatedbyTheorem3.2below,whichises- sentiallycontainedin[3],althoughnotexplicitlystatedthere. 4 FREDERICKM.GOODMAN Lemma3.1. LetR beanygroundringwithparametersω fora ≥0andu ,...,u .Let a 1 r W denotethetwostranddegeneratecyclotomicBMWalgebradefinedoverR.Then 2,R (1) TheleftidealW e equalstheR–spanof{e ,x e ,...,xr−1e }. 2,R 1 1 1 1 1 1 (2) W isspannedoverR by{xae xb, xaxbs , xaxb :0≤a,b ≤r−1} 2,R 1 1 1 1 2 1 1 2 Proof. UsingLemma2.3,andthedefiningrelationsofthealgebra,oneseesthatthe span of {e ,x e ,...,xr−1e } is invariant under left multiplication by the generators 1 1 1 1 1 x ,e ,s ,andthatx xae =−xa+1e . Thisprovespart(1). Part(2)issimilar,see[3], 1 1 1 2 1 1 1 1 (cid:3) Proposition2.15. Theorem3.2([3]). LetF beafieldofcharacteristic6=2,withparametersω fora ≥0 a andu ,...,u .Assumethattheu aredistinctandu +u 6=0foralli,j.LetW =W 1 r i i j 2,F bethedegeneratecyclotomicBMWalgebradefinedoverF withparametersω fora ≥0 a andu ,...,u .Thenthefollowingconditionsareequivalent: 1 r (1) {e ,x e ,...,xr−1e }⊆We islinearlyindependentoverF,ande We 6=0. 1 1 1 1 1 1 1 1 (2) Foralla ≥0,ω = r γ ua,where a i=1 i i P u +u (3.1) γ =(2u −(−1)r) i j, i i u −u i j j6=i Y andsomeω isnon–zero. a (3) W admitsamoduleM withbasis{v ,x v ,...,xr−1v }suchthate M =Fv . 0 1 0 1 0 1 0 Proof. Theimplication(1) =⇒ (3)isobvious. Assumecondition(3). Wehavev =e m forsomem ∈M,soe xjv =e xje m = 0 1 1 0 1 1 ω e m =ω v for1≤j ≤r −1. Moreover,someω 6=0sincee M 6=(0). Definep ∈ j 1 j 0 j 1 i x −u W byp = 1 j .Thenp2=p , p =1,andx p =u p .Definem ∈M by 1,F i u −u i i i i 1 i i i i i j j6=i m =p v .ThYenm 6=0bytheassumedPlinearindependenceof{v ,x v ,...,xr−1v }, i i 0 i 0 1 0 1 0 x m = u m , and m = ( p )v = v . It follows that {m ,...,m } is linearly 1 i i i i i i i 0 0 1 r independent, since the m areeigenvectors forx with distinct eigenvalues. Define i 1 P P κ andc inF bye m =κ v =κ m ,ands m = c m . (Itwillbeshown j i,j 1 j j 0 j i i 1 j i i,j i thatκ =γ ,whereγ isdefinedabove.) Notethate M 6=(0)impliesthatκ 6=0for j j j 1 j P P somej. TheargumentcontinuesasintheproofofTheorem3.2in[3],pp.65–67.Applythe identityx s −s x −e +1=0tom toderiveaformulaforc , 1 1 1 2 1 j i,j c =(κ −δ )/(u +u ). i,j j i,j i j Nextapplytheidentitye =s e tom toget 1 1 1 i d d κ −1 κ j k (3.2) κ m =e m =s e m =κ + m , i j 1 i 1 1 i i j 2u u +u j=1 j=1 j k6=j j k X Xn X o DEGENERATECYCLOTOMICBMWALGEBRAS 5 fori =1,...,r. Since atleastoneκ isnon–zero,matchingcoefficientsin(3.2)gives i theequations d κ 1 k (3.3) =1+ , u +u 2u j k j k=1 X forj =1,...,r. Nowitisshownin[3],page66,thattheuniquesolutiontothissystem ofequationsisκ =γ for1≤j ≤1.Finally,wehave j j j j j j (3.4) ω v =e x v =e x ( m )=e u m =( u γ )v . j 0 1 1 0 1 1 i 1 i i i i 0 i i i X X X Thisshows(3) =⇒ (2). Finally, (2) =⇒ (1) byTheoremAof [3],namelyassuming (2), W hasanR–basis thatincludes{e ,x e ,...,xr−1e },sothelattersetislinearlyindependent. (cid:3) 1 1 1 1 1 Theelementsγ appearinginTheorem3.2arerationalfunctionsinu ,...,u with j 1 r singularitiesatu =u ,butitisshownin[3]thattheelements γ ua arepolyno- i j i i i mialsinu ,...,u ,asfollows: 1 r P Let u ,...,u and t be algebraically independent indeterminants over Z. Define 1 r symmetricpolynomialsq (u)inu ,...,u by a 1 r r 1+u t i = q (u)ta. 1−u t a i=1 i a≥0 Y X Thepolynomialsq areknownasSchurq–functions.Letγ (u)bedefinedby(3.1)with a j u replacedbyu . Moreover,letη (u)= r γ (u)ua fora ≥0. Then([3],Lemma i i a i=1 j j 3.5) P 1 1 (3.5) η (u)=q (u)− (−1)rq (u)+ δ . a a+1 a a,0 2 2 Inparticulartheη arepolynomialsinu ,...,u . a 1 r Corollary3.3. LetF beafieldofcharacteristic6=2,withparametersω fora ≥0and a u ,...,u . Assume thattheu aredistinctandu +u 6=0 for all i,j. Let W = W 1 r i i j 2,F bethedegeneratecyclotomicBMWalgebradefinedoverF withparametersω fora ≥ a 0 and u ,...,u . If {e ,x e ,...,xr−1e } ⊆ We is linearly independent over F, and 1 r 1 1 1 1 1 1 e We 6=0,then 1 1 1 1 (3.6) ω =q (u ,...,u )− (−1)rq (u ,...,u )+ δ . a a+1 1 r a 1 r a,0 2 2 Thismotivatesthefollowingdefinition,whichmakessenseforarbitraryu ,...,u : 1 r Definition 3.4 ([3]). Let R be a commutative ring with parameters ω (a ≥ 0) and a u ,...,u .Supposethat2isinvertibleinR.Saythattheparametersareu–admissible 1 r if 1 1 (3.7) ω =q (u ,...,u )+ (−1)r−1q (u ,...,u )+ δ a a+1 1 r a 1 r a,0 2 2 foralla ≥0. 6 FREDERICKM.GOODMAN 4. ADMISSIBILITY WefixagroundringR withparametersω (a ≥0)andu ,...,u . Weconsiderthe a 1 r twostranddegeneratecyclotomic BMWalgebraoverR,W =W (u ,...,u )andwe 2,r 1 r writee =e ,s =s ,andx =x . 1 1 1 Lemma4.1. Supposethat{e,xe,...,xr−1e}islinearlyindependentover R. Thenthe parametersω (a ≥0)andu ,...,u areweaklyadmissibleandsatisfythefollowing a 1 r relations: r−j−1 (4.1) ω a =−2δ a +δ a , µ µ+j+1 (r−jisodd) j (jiseven) j+1 µ=0 X for0≤j ≤r−1. Proof. Since{e,xe,...,xr−1e}isassumedlinearlyindependentoverR,inparticulare isnotatorsionelementoverR,andhenceR isweaklyadmissiblebyCorollary2.4. Ifr =1,(4.1)reducestothesingle equationω +2a −1=0,whichfollows from 0 0 (sx+xs+1−e)e =0,togetherwithx =u =−a andse =e.Assumer ≥2.Wehave 1 0 0=(sx−x s+1−e)xr−1e =(sx+xs+1−e)xr−1e. 2 Applytheidentityx(xr−1e)=− r−1a xje aswellastheidentity(2.5)andsimplify. j=0 j Thisgives: P r−1 r−2 r−j−2 0=−a e− (−1)ja xje + a xje+ (−1)j ω a xje 0 j j+1 µ µ+j+1 j=1 0≤j≤r−2 j=0 µ=0 X X X X jeven r−1 r−1 +(−1)r a xje−δ xr−1e+ ω xje j (riseven) r−1−j j=0 j=1 X X +xr−1e−ω e, r−1 where the three lines of the display correspond to evaluation of sxxr−1e, xsxr−1e, and (1−e)xr−1e. Because {e,xe,...,xr−1e} is assumed to be linearly independent, thecoefficientofxje iszeroforeachj,0≤j ≤r−1.Extractingthecoefficientsyields (4.1).Hereonehastotreatthethreecasesj =0, 1≤j ≤r−2,andj =r−1separately, (cid:3) buttheresultinallthreecasesisthesame. Definition4.2. Saythattheparametersω (a ≥0)andu ,...,u areadmissible(or a 1 r thatthegroundringR isadmissible) ifthe relations(4.1)holdfor0≤ j ≤r −1and r a ω =0holdsforalla ≥0. µ=0 µ µ+a RPemark4.3. Admissibility isanalogoustotheadmissibility conditionofWilcoxand Yu for the cyclotomic BMW algebras [16]. Our terminology differs from that in [3], whereadmissibilitymeansessentiallywhatwehavecalledweakadmissibility. DEGENERATECYCLOTOMICBMWALGEBRAS 7 Lemma 4.4. There exist universal polynomials H (u ,...,u ) (a ≥ 0), symmetric in a 1 r u ,...u ,suchthatwheneverR isanadmissiblering,onehas 1 r (4.2) ω =H (u ,...,u ) a a 1 r fora ≥0. Proof. Thesystemofrelations(4.1)isaunitriangularlinearsystemofequationforthe variablesω ,...,ω .Infact,ifwelisttheequationsinreverseorderthenthematrix 0 r−1 ofcoefficientsis 1 a 1 r−1 a a 1 r−2 r−1 . ... ... ... a a ... a 1 1 2 r−1 Solvingthesystemforω0,...,ωr−1 givesthesequantitiesaspolynomialfunctionsof r a ,...,a ,thussymmetricpolynomialsinu ,...,u . Therelations a ω = 0 r−1 1 r j=0 j j+m 0,forallm ≥0yield(4.2)fora ≥r. (cid:3) P 5. EQUIVALENCE OFADMISSIBILITY CONDITIONS In thissection we will show thatadmissibility andu–admissibility areequivalent (assumingthat2isinvertibleinthegroundring,sothatu–admissibilitymakessense.) First, we will obtain the polynomials H of Lemma 4.4 explicitly in terms of the a Schurq–functions.ConsideringthegeneratingfunctionfortheSchurq–functions, r 1+u t (5.1) i = q (u)ta, 1−u t a i=1 i a≥0 Y X wehave r r (5.2) (1−u t) q (u)ta = (1+u t). i a i ! ! i=1 a≥0 i=1 Y X Y Takingintoaccountthatq (u)=1,wealsohave 0 r r r (5.3) (1−u t) q (u)ta = (1+u t)− (1−u t). i a i i ! ! i=1 a≥1 i=1 i=1 Y X Y Y Matchingcoefficientsin(5.2)andwritingintermsofthesignedelementarysymmet- ricfunctionsa (u)gives i r−j−1 (5.4) q (u)a (u)=(−1)r−j−1a (u), for0≤j ≤r−1, µ µ+j+1 j+1 µ=0 X and,moreover, r (−1)ra (u) ifa =0, (5.5) a (u)q (u)= 0 µ µ+a µ=0 (0 ifa ≥1. X 8 FREDERICKM.GOODMAN Doingthesamewith(5.3)yields r−j−1 (5.6) q (u)a (u)=−2δ a (u), for0≤j ≤r −1. µ+1 µ+j+1 (r−j isodd) j µ=0 X Ifweset 1 1 η (u)=q (u)+ (−1)r−1q (u)+ δ , a a+1 a a,0 2 2 then,using(5.4)and(5.6),weget r−j−1 (5.7) η (u)a (u)=−2δ a (u)+δ a (u), µ µ+j+1 (r−jisodd) j (jiseven) j+1 µ=0 X for0≤j ≤r−1.From(5.5),weobtain r (5.8) a (u)η (u)=0, foralla ≥0. µ µ+a µ=0 X Lemma5.1. Let R be commutativeringinwhich2isinvertible. Letω (a ≥0)and a u ,...,u beparametersinR. Thentheparametersareadmissible,if,andonlyif,they 1 r areu–admissible. Proof. By definition, the parameters are u–admissible if ω = η (u ,...,u ) for all a a 1 r a ≥0.Itfollowsfrom(5.7)and(5.8)thatu–admissibleparametersareadmissible. On the other hand, if the parameters are admissible, then the relations (4.1) for r 0≤j ≤r −1and a ω =0fora ≥0uniquelydeterminetheω foralla ≥0 µ=0 µ µ+a a assymmetricpolynomialfunctionsofu ,...,u . Butaccordingto(5.7)and(5.8),the 1 r P elementsη (u ,...,u )satisfythesamerelations.Henceω =η (u ,...,u )fora ≥0, a 1 r a a 1 r (cid:3) sotheparametersareu–admissible. Theorem5.2. LetR beacommutativeringwithparametersω (a ≥0)andu ,...,u . a 1 r Supposethat 2isinvertibleinR.ConsiderthetwostranddegeneratecyclotomicBMW algebraoverR,W =W (u ,...,u ).Thefollowingareequivalent: 2,r 1 r (1) {e ,x e ,...,xr−1e }islinearlyindependentoverR. 1 1 1 1 1 (2) {xae xb,xaxbs ,xaxb :0≤a,b ≤r−1}islinearlyindependentoverR. 1 1 1 1 2 1 1 2 (3) Theparametersareadmissible. (4) Theparametersareu–admissible. Proof. Lemma4.1gives(1) =⇒ (3). Lemma5.1gives(3)⇐⇒(4). Theimplication(4) =⇒ (2)ispartofthemainresult(TheoremA)of[3].Finally(2) =⇒ (1)istrivial. 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