Similarity versus Coincidence Rotations of Lattices

SIMILARITY VERSUS COINCIDENCE R OT A TIONS OF LA TTICES SVENJA GLIED AND MICHAEL BAAKE Abstra t. The groups of similarit y and oinidene rotations of an arbitrary lattie Γ in d - dimensional Eulidean spae are onsidered. It is sho wn that the group of similarit y rotations on tains the oinidene rotations as a normal subgroup. F urthermore, the struture of the orresp onding fator group is examined. If the dimension d is a prime n um b er, this fator group is an elemen tary Ab elian d -group. Moreo v er, if Γ is a rational lattie, the fator group is trivial ( d o dd) or an elemen tary Ab elian 2 -group ( d ev en). 1. Intr odution The lassiation of olour symmetries and that of grain b oundaries in rystals and qua- sirystals are in timately related to the existene of similar and oinidene sublatties of the underlying lattie of p erio ds or the orresp onding translation mo dule. It is th us of in terest to understand the orresp onding groups of isometries from a more mathematial p ersp etiv e. An example for the struture of the groups of oinidene rotations and similarit y rotations of planar latties is onsidered and the fator group of similarit y mo dulo oinidene rotations is alulated. More generally , for latties in d dimensions, w e sho w that the fator group is the diret sum of yli groups of prime p o w er orders that divide d . In the ase of rational latties, whi h inlude h yp erubi latties and all ro ot latties, this means that the fator group is either trivial or an elemen tary Ab elian 2 -group, dep ending on the parit y of d . 2. Coinidene R ot a tions A latti e in R d is a subgroup of the form Γ = Z b 1 ⊕ Z b 2 ⊕ . . . ⊕ Z b d , where { b 1 , . . . , b d } is a basis of R d . T w o latties Γ , Γ ′ in R d are alled  ommensur ate if their in tersetion Γ ∩ Γ ′ has nite index b oth in Γ and in Γ ′ . In this ase, w e write Γ ∼ Γ ′ . Commensurateness of latties is an equiv alene relation (f. [1℄). An elemen t R ∈ SO( d ) is alled a  oiniden e r otation of Γ , if Γ ∼ R Γ . W e th us dene SOC( Γ ) := { R ∈ SO( d ) | Γ ∼ R Γ } , whi h is a subgroup of SO( d ) . Example 2.1 (The square lattie Z 2 ) . As sho wn in Thm. 3.1 of [1℄, the oinidene rotations of Z 2 are preisely the sp eial orthogonal matries with rational en tries, SOC( Z 2 ) = SO(2 , Q ) . On the other hand, one an iden tify Z 2 with the Gaussian in tegers Z [ i ] , where i is the imaginary unit. Then, a rotation R ( ϕ ) with rotation angle ϕ orresp onds to a m ultipliation with the 1 2 SVENJA GLIED AND MICHAEL BAAKE omplex n um b er e iϕ ∈  Q ( i ) ∩ S 1  ≃ SO C( Z 2 ) ; see [5℄. Using the fat that Z [ i ] is a unique fatorisation domain, ea h oinidene rotation uniquely fatorises as (1) e iϕ = ε Y p ≡ 1(4)  ω p ω p  n p , where ε is a unit in Z [ i ] , n p ∈ Z with only nitely man y of them nonzero, p runs through the rational primes ongruen t to 1 (mo d 4 ), and p fatorises as p = ω p ω p in Z [ i ] with ω p /ω p not a unit. This sho ws that SOC( Z 2 ) is a oun tably generated Ab elian group. More preisely , SOC( Z 2 ) = C 4 × Z ( ℵ 0 ) , where C 4 denotes the yli group of order 4 (here generated b y i ) and Z ( ℵ 0 ) stands for the diret sum of oun tably man y innite yli groups, whi h are here generated b y the ω p /ω p with p ≡ 1 (mo d 4 ) (f. [5℄). 3. Similarity R ot a tions Let Γ ⊂ R d again b e a lattie. Dene SOS( Γ ) := { R ∈ SO( d ) | Γ ∼ αRΓ for some α > 0 } . The elemen ts of SOS( Γ ) are alled similarity r otations . SOS( Γ ) is a group (f. [4℄) and on tains SOC( Γ ) as a subgroup. Example 3.1 ( Z 2 ) . F or Z 2 , the group of similarit y rotations onsists preisely of the set of Z 2 -diretions, (2) SOS( Z 2 ) = n a | a |    0 6 = a ∈ Z [ i ] o . W e parametrise the Eulidean plane b y the omplex n um b ers C , and use SO(2) ≃ S 1 and Z 2 = Z [ i ] . T o sho w (2), let z ∈ Z [ i ] \ { 0 } . Sine Z [ i ] is a ring, one has | z | · z | z | Z [ i ] ⊂ Z [ i ] , so that z / | z | ∈ S OS( Z 2 ) . Con v ersely , let r ∈ SOS( Z 2 ) , meaning that r ∈ S 1 with λr Z [ i ] ∼ Z [ i ] for some λ > 0 . By Remark 4.2 b elo w, there exists a nonzero in teger t with tλr Z [ i ] ⊂ Z [ i ] . Sine 1 ∈ Z [ i ] , this yields tλr ∈ Z [ i ] , sa y tλr = v . Th us | tλ | = | v | , b eause r ∈ S 1 . This sho ws that r = v / | v | is a Z [ i ] -diretion. Ea h nonzero elemen t of SOS( Z 2 ) is th us of the form z / | z | with 0 6 = z ∈ Z [ i ] . Using unique fatorisation in Z [ i ] again, w e get z | z | =  1 + i √ 2  k Y p ≡ 1(4)  ω p √ p  ℓ p , where 0 6 k < 8 and ℓ p ∈ Z (other restritions as in (1)). One observ es that (1 + i ) / √ 2 is a primitiv e 8 th ro ot of unit y , hene it generates the yli group C 8 . F urthermore, one nds  ω p √ p  2 = ω 2 p ω p ω p = ω p ω p . SIMILARITY VERSUS COINCIDENCE R OT A TIONS OF LA TTICES 3 This sho ws that the generators of SOC( Z 2 ) = C 4 × Z ( ℵ 0 ) are the squares of the generators of SOS( Z 2 ) . Th us SOC( Z 2 ) =  x 2   x ∈ SOS( Z 2 )  =:  SOS( Z 2 )  2 . The follo wing more general result w as sho wn in [ 5℄: F or all ylotomi elds Q ( ξ n ) of lass n um b er one (exluding Q ), one has SOC( O n ) ≃ C N ( n ) × Z ( ℵ 0 ) , where O n = Z [ ξ n ] is the ring of in tegers in Q ( ξ n ) and N ( n ) = lcm( n, 2) . Returning to our example, w e nd the struture of the fator group to b e SOS( Z 2 ) / SOC( Z 2 ) ≃ ( C 8 /C 4 ) × C ( ℵ 0 ) 2 ≃ C 2 × C ( ℵ 0 ) 2 , where C ( ℵ 0 ) 2 stands for the diret sum of oun tably man y yli groups of order 2 . Hene, the fator group is the diret sum of yli groups of order 2 , whi h means that it is an elemen tary Ab elian 2 -group. More generally , for arbitrary latties in Eulidean d -spae, w e shall see b elo w that the group SOC is a normal subgroup of SOS , whene the fator group alw a ys exists. 4. F a tor Gr oup Throughout this setion, let Γ b e a lattie in R d , with d ≥ 2 . Denition 4.1. F or an arbitrary elemen t R ∈ SO( d ) , dene scal Γ ( R ) = { α ∈ R | Γ ∼ αR Γ } . Note that SOS( Γ ) = { R ∈ SO( d ) | scal Γ ( R ) 6 = ∅ } . Remark 4.2. If α ∈ scal Γ ( R ) , then there exists a nonzero in teger t su h that tαRΓ ⊂ Γ . Namely , if α ∈ scal Γ ( R ) , the group index [ αRΓ : ( Γ ∩ αRΓ )] = t is nite. Consequen tly , one has tαRΓ ⊂ ( Γ ∩ αRΓ ) ⊂ Γ . Lemma 4.3. F or R ∈ SO S( Γ ) , the fol lowing assertions hold. (1) b · scal Γ ( R ) ⊂ scal Γ ( R ) for al l b ∈ Q \ { 0 } (2) r Γ ∼ Γ with r ∈ R implies r ∈ Q (3) αβ − 1 ∈ Q for al l α, β ∈ scal Γ ( R ) Pr o of. Let α ∈ scal Γ ( R ) . F or b = b 1 /b 2 with b 1 , b 2 ∈ Z \ { 0 } , one nds b 1 b 2 αRΓ ∼ 1 b 2 αRΓ ∼ 1 b 2 Γ ∼ Γ . This pro v es (1). In order to sho w (2), let r ∈ R with r Γ ∼ Γ . By Remark 4.2 , there exists a nonzero in teger k with k r Γ ⊂ Γ . No w, let γ ∈ Γ b e represen ted in terms of a basis { γ 1 , . . . , γ d } of Γ as γ = P d i =1 c i γ i , with c i ∈ Z . On the other hand, k r γ an b e represen ted as k r γ = P d i =1 a i γ i , where a i ∈ Z . Th us d X i =1 k r c i γ i = d X i =1 a i γ i . 4 SVENJA GLIED AND MICHAEL BAAKE By assumption, Γ spans R d , so that { γ 1 , . . . , γ d } forms an R -basis of R d . Therefore, one has k r c i = a i , yielding r = a i c − 1 i k − 1 ∈ Q . Finally , (3) is obtained from (2) as follo ws. By assumption, one has β R Γ ∼ Γ ∼ αR Γ . Multiplying with 1 /β giv es RΓ ∼ α β RΓ , whi h ompletes the pro of.  Denote b y R • (b y Q • ) the m ultipliativ e groups formed b y the nonzero real (rational) n um b ers. Dene a map η : SOS( Γ ) − → R • / Q • b y R 7− → [ α ] , where [ · ] denotes the equiv alene lasses of R • / Q • and α is an arbitrary elemen t of scal Γ ( R ) . This map is w ell-dened due to the fat that scal Γ ( R ) is non-empt y for R ∈ SOS( Γ ) and b y Lemma 4.3(3). Lemma 4.4. The map η is a gr oup homomorphism with Ker( η ) = SO C( Γ ) . Pr o of. Let R, S ∈ SOS( Γ ) and  ho ose α ∈ scal Γ ( R ) and β ∈ scal Γ ( S ) . W e need to sho w that αβ ∈ scal Γ ( RS ) . By assumption, one has Γ ∼ αR Γ ∼ αR ( β S Γ ) = αβ R S Γ . Th us αβ ∈ scal Γ ( RS ) , hene η is a group homomorphism. It remains to sho w that Ker( η ) = SOC( Γ ) . F or R ∈ SOC( Γ ) , the set scal Γ ( R ) on tains 1 , whi h means R ∈ Ker( η ) . Con- v ersely , if S ∈ Ker( η ) , one has scal Γ ( S ) ⊂ Q . Let µ ∈ scal Γ ( S ) . Due to Lemma 4.3(1), w e ha v e 1 = µ − 1 µ ∈ scal Γ ( S ) , whi h pro v es S ∈ SO C( Γ ) .  Sine SOC( Γ ) is the k ernel of a group homomorphism, it is a normal subgroup of SOS( Γ ) , so that the fator group SOS( Γ ) / S OC( Γ ) an b e onsidered. It is isomorphi to the image of η , whi h is a subgroup of R • / Q • and th us Ab elian. T o examine the struture of the fator group SOS( Γ ) / S OC( Γ ) , w e need the follo wing result from the theory of Ab elian groups. Theorem 4.5. L et G b e a  ountable A b elian gr oup. (1) If a prime numb er p exists suh that x p = 1 for al l x ∈ G , then G is the dir e t sum of sub gr oups of or der p . (2) If a p ositive inte ger n exists suh that x n = 1 for al l x ∈ G , then G is the dir e t sum of yli gr oups of prime p ower or ders that divide n . Pr o of. See [6, Thms. 5.1.9 and 5.1.12℄.  Remark 4.6. Let R ∈ SOS( Γ ) . F or all elemen ts α ∈ R with αRΓ ⊂ Γ , one has | α d | = [ Γ : αR Γ ] ∈ N . This follo ws via the determinan ts of basis matries of the latties in v olv ed. Consequen tly , α is an algebrai n um b er. Theorem 4.7. The gr oup SOS( Γ ) / SOC( Γ ) is  ountable. F urthermor e, it is the dir e t sum of yli gr oups of prime p ower or ders that divide d . Pr o of. W e onsider again the group homomorphism η : SOS( Γ ) − → R • / Q • . Let R ∈ SO S( Γ ) . This implies η ( R ) = [ α ] for some elemen t α ∈ scal Γ ( R ) . Due to Remark 4.2, there exists a nonzero in teger t with tαRΓ ⊂ Γ . F urthermore, one has η ( R ) = [ tα ] . By Remark 4.6 , tα is an algebrai n um b er. This means that all elemen ts of η (SOS( Γ )) are represen ted b y SIMILARITY VERSUS COINCIDENCE R OT A TIONS OF LA TTICES 5 algebrai n um b ers. Th us, sine the set of algebrai n um b ers is oun table, also the group SOS( Γ ) / S OC( Γ ) is oun table. A ording to Remark 4.6 , one has ( tα ) d ∈ Q , whi h yields (3) η ( R ) d = [ tα ] d = [( tα ) d ] = [1] in R • / Q • . Using the group isomorphism η (SOS( Γ )) ≃ S OS( Γ ) / SOC( Γ ) , this sho ws that the order of ea h elemen t of SOS( Γ ) / S OC( Γ ) divides d . Theorem 4.5(2) then implies that the group SOS( Γ ) / S OC( Γ ) is the diret sum of yli groups of prime p o w er orders. Conse- quen tly , the prime p o w er order of ea h yli group divides d .  Corollary 4.8. If d = p is a prime numb er, the fator gr oup SOS( Γ ) / S OC( Γ ) is an elemen- tary A b elian p -gr oup, i.e., it is the dir e t sum of yli gr oups of or der p .  Corollary 4.9 (Rational Latties) . L et Γ b e a latti e in R d suh that h x, x i ∈ Q for al l x ∈ Γ , wher e h · , · i denotes the standar d s alar pr o dut in R d . L atti es satisfying the ab ove pr op erty ar e also  al le d rational (f. [2℄) . F or these latti es, the gr oup SOS( Γ ) / SOC( Γ ) is an elementary A b elian 2 -gr oup when d is even. If d is o dd, one has SOS( Γ ) = SO C( Γ ) . Either way, one has (SOS( Γ )) 2 ⊂ SOC( Γ ) . Pr o of. Let R ∈ SO S( Γ ) . By Remark 4.2, there exists a nonzero real n um b er α su h that αRΓ ⊂ Γ . By assumption, one has h αRγ , αRγ i ∈ Q for all γ ∈ Γ . Hene α 2 ∈ Q , sa y α 2 = r /s , where r , s ∈ Z \ { 0 } . Sine sα 2 = r ∈ Z and αRΓ ⊂ Γ , one gets Γ ⊃ sαR ( αR Γ ) = sα 2 R 2 Γ ⊂ R 2 Γ , whene r R 2 Γ ⊂  Γ ∩ R 2 Γ  . Th us b oth [ Γ : r R 2 Γ ] and [ R 2 Γ : r R 2 Γ ] are nite. This implies Γ ∼ R 2 Γ , so that R 2 is a oinidene rotation of Γ . Consequen tly , (SOS ( Γ )) 2 ⊂ SOC( Γ ) . This means that ev ery elemen t of the fator group SOS( Γ ) / S OC( Γ ) is of order 1 or 2 . Th us, the fator group is an elemen tary Ab elian 2 -group b y Theorem 4.5 (1). If d is o dd, set d = 2 m + 1 with m ∈ N . Then α ( α 2 ) m = α d ∈ Q yields α ∈ Q , b eause α 2 ∈ Q . Th us η ( R ) = [ α ] = [1] in R • / Q • for all R ∈ SOS( Γ ) , whene SOS( Γ ) / S OC( Γ ) is the trivial group.  5. Outlook In view of P enrose tilings and similar mo dels, where the translation mo dule is not a lattie, it is desirable to generalise the ab o v e notions of similarit y and oinidene rotations from latties to mo dules. Some progress has b een made in this diretion for ertain mo dules o v er subrings S of the rings of in tegers of real algebrai n um b er elds. More preisely , similar results [3 ℄ to those presen ted here hold for S -mo dules of rank d that span R d . A kno wledgements The authors are grateful to U. Grimm, C. Hu k, R.V. Mo o dy and P . Zeiner for v aluable disussions and ommen ts on the man usript. This w ork w as supp orted b y the German Resear h Counil (DF G), within the CR C 701 . S.G. w ould lik e to thank the IUCr for nanial supp ort to attend ICQ10. 6 SVENJA GLIED AND MICHAEL BAAKE Referenes [1℄ Baak e, M.: Solution of the oinidene problem in dimensions d ≤ 4 . In: The Mathematis of L ong-R ange Ap erio di Or der (Ed. R.V. Mo o dy), p. 944. NA TO-ASI Series C 489 , Klu w er, Dordre h t 1997; revised v ersion: arXiv:math/0605222 [math.MG℄ . [2℄ Con w a y , J.H.; Rains, E.M.; Sloane, N.J.A.: On the existene of similar sublatties, Canad. J. Math. 51 (1999) 13001306. [3℄ Glied, S.: Similarit y and oinidene isometries for mo dules, in preparation. [4℄ Baak e, M.; Grimm, U.; Heuer, M.; Zeiner, P .: Coinidene rotations of the ro ot lattie A 4 , Europ. J. Com binatoris, in press. [math.MG℄ . [5℄ Pleasan ts, P .A.B.; Baak e, M.; Roth, J.: Planar oinidenes for N -fold symmetry , J. Math. Ph ys. 37 (1996) 10291058, revised v ersion: arXiv:math/0511147 [math.MG℄ . [6℄ Sott, W.R.: Group Theory , Pren tie-Hall, Englew o o d Clis 1964. F akul t ä t für Ma thema tik, Universit ä t Bielefeld, Postf a h 100131, 33501 Bielfeld, Germany E-mail addr ess : {sglied,mbaake}math.uni-biele feld. de URL : http://www.math.uni-bielefeld.de /baak e/