APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 4, 351 35 (1997) ARTICLE NO. HA9701 Platonic Orthonormal Wavlts 1 Murad Özaydin and Tomasz Przbinda Dpartmnt of Mathmatics, Univrsity of Oklahoma, Norman, Oklahoma 73019 Communicatd by Ph. Tchamitchian Rcivd Jun 0, 1995; rvisd Fbruary 10, 1997 W classify all orthonormal wavlts which occur in th L spac of th facs of a platonic solid. 1997 Acadmic Prss 0. INTRODUCTION Lt G b th ax / b group. Thus G Å x 1 with th group structur givn by (a, b)(a, b ) Å (aa, ab / b), (a, b) 01 Å (a 01, 0a 01 b) ((a, b), (a, b ) G). This group acts on th ral lin by th formula (a, b)x Å ax / b ((a, b) G, x ) and on th spac L ( ) of squar intgrabl functions on : s(a, b) (x) Å ÉaÉ 01/ ((a, b) 01 x) Å ÉaÉ 01/ (a 01 (x 0 b)), L ( ). (0.1) It is asy to s that th formula (0.1) dfins a unitary rprsntation (s, L ( )) of G. It is wll known, and asy to chck, that this rprsntation is irrducibl. Indd, suppos T is a boundd oprator on L ( ) which commuts with s(g). Thn it commuts with all translations. Thrfor F T F 01, th conjugat of T by a Fourir transform F, coincids with multiplication by a function. Sinc this multiplication commuts with all dilatations, th function is constant. Thus T is a constant multipl of th idntity. Lt H b th dyadic ax / b subgroup of G, i.., H Å {( j, k l ); j, k, l }. This subgroup is nithr discrt nor dns in G. Rcall [3] that an orthonormal wavlt is a unit vctor L ( ) such that th orbit s(h) contains an orthonormal basis of L ( ). In fact, this orthonormal basis can b writtn as s(d)rs(t), whr 1 Rsarch partially supportd by NSF Grant DMS 904488. 351 103-503/97 $5.00 Copyright 1997 by Acadmic Prss All rights of rproduction in any form rsrvd.
35 ÖZAYDIN AND PRZEBINDA D Å {( j, 0); j } H is th subgroup of dilations, and T Å {(1, k); k } H is th subgroup of translations. Not that th subst DrT H is not a subgroup. In this articl w tak a closr look at this aspct of th thory of wavlts whn G is a finit group, whr thr sms to b no good analog of D, and all analytic problms ar absnt. Nvrthlss som applications to coding thory sm to b in sight. Spcifically w ar intrstd in th following situation. Lt G b a finit group and lt (s, V) b an irrducibl unitary (or orthogonal, if th ground fild is ) rprsntation of G. Lt H b a subgroup of G. DEFINITION 0.. An H-wavlt in V is a unit vctor V whos stabilizr in H is trivial, and such that s(h) is an orthonormal basis of V. Lt (l, L (H)) dnot th lft rgular rprsntation of H: l(g)f(h) Å f(g 01 h), h, g H, f L (H). Lt V b an H-wavlt. Thn th map L (H) r L (V) f r f(h)s(h) hh is a bijctiv isomtry, which intrtwins th actions of H on both spacs. Thus th rstriction of s to H is quivalnt to l: sé H à l. Suppos u V is anothr H-wavlt. Thn th map V r V c h s(h)u r c h s(h) hh hh is a unitary bijction which commuts with th action of H. Lt U(V) dnot th group of unitary oprators on th Hilbrt spac V. Th abov argumnt shows that U(V) H, th cntralizr of H in U(V), acts simply transitivly on th st of orthonormal wavlts. Lt x s Å tr s(g), g G b th charactr of s. This is a conjugation invariant function on G, which dtrmins th rprsntation s. Th following lmma ssntially rducs our problm to charactr thory and is fundamntal for th rst of th papr. LEMMA 0.3. Th following conditions ar quivalnt: (a) th rprsntation (s, V)has an H-wavlt; (b) (sé H,V)is quivalnt to th lft rgular rprsntation (l, L (H)); (c) ÉHÉ, if h Å 1 x s (h) Å 0, if h H"{1}. Morovr, th group U(V) H acts simply transitivly on th st of all H-wavlts in V. In particular this st is homomorphic to th group U(V) H. Proof. All th assrtions of th lmma with th xcption of (c) implis (a) follow immdiatly from th prcding discussion.
PLATONIC ORTHONORMAL WAVELETS 353 Assuming (c) s rstrictd to H is th lft rgular rprsntation, bcaus it is dtrmind by its charactr. A tchnical problm is that th givn innr product on V may not agr with th standard on on L (H). Lt V 1 V rrr V k b th dcomposition of V into isotypic componnts. Thn th V j s ar orthogonal with rspct to both innr products, by Schur s lmma; s [4]. Each V j can b writtn as U j W j, whr H acts irrducibly on U j, W j is an auxiliary spac with trivial H- action. On U j thr is, up to a positiv scalar multipl, a uniqu H-invariant innr product (, ) j. Th rstriction of th innr product from V to V j is of th form (, ) j (,) j, whr (, ) j is an innr product on W j. Th group GL(W j ) acts transitivly on innr products on W j. Thrfor w can choos g j GL(W j ) so that j I Uj g j is an isomtry btwn th two innr product spacs L (H) and V. Clarly th lft rgular rprsntation L (H) has an H-wavlt, say. Thn ( j I j g j ) is an H-wavlt for V. COROLLARY 0.4. If th rprsntation (sé H,V)has H-wavlts thn dim V H Å 1. (Hr V H stands for th spac of H-invariants in V). TRIVIAL EXAMPLES. (a) If dim V Å 1, thn any unit vctor in V is an H-wavlt for H Å {1}, and this is th only subgroup of G admitting wavlts. (b) Lt G b th cyclic group of ordr 3. This group acts on via rotations by p/3 and th rsulting rprsntation is irrducibl (ovr ). Sinc G has no subgroups of ordr, thr ar no wavlts in this rprsntation. Lt Symm(n) dnot th symmtric group of all prmutations on n objcts and lt Alt(n) dnot th altrnating group of vn prmutations. Ths groups act on n (or n ) by prmuting th coordinats. If n 4, thy act irrducibly on th subspac V consisting of vctors whos coordinats add up to zro (i.., th orthogonal complmnt of th diagonal). This is th (so-calld) standard rprsntation of Symm(n) and Alt(n), which shall b dnotd, from now on, by (r, V). Th standard innr product inducd from n is invariant, and, up to a scalar multipl, this is th only invariant innr product, bcaus th rprsntation is irrducibl ovr. THEOREM 0.5. Th standard rprsntation of Alt(n) has an H-wavlt if and only if n is not congrunt to 3 modulo 4. Proof. If n is congrunt to 3 modulo 4, thn th ordr of H Å dim V Å n 0 1is of th form (k / 1) for som natural numbr k. So thr must b an lmnt of ordr in H. As a prmutation this lmnt has to b a product of k / 1 disjoint transpositions, by Lmma 0.3(b). But this lmnt is not an vn prmutation, contradicting that H was a subgroup of Alt(n). Convrsly, if n is not congrunt to 3 modulo 4, thn n 0 1 is not congrunt to modulo 4. Lt n 0 1 Å k m with m odd, so k Å 0ork. Considr th abstract group H Å / 1 / 1 rrr 1 / 1 /m, whr thr ar k two-cyclic factors /, so th ordr of H is n 0 1. Th mbdding of H into Symm(n 0 1) via th lft action of H on itslf f: H r Symm(n 0 1)
354 ÖZAYDIN AND PRZEBINDA has its imag containd in Alt(n 0 1). Indd, for any nonidntity lmnt h H, f(h) has a disjoint cycl dcomposition consisting of cycls with lngth qual to th ordr of h, which is odd whn n 0 1 is odd (k Å 0), or thr is an vn numbr of such cycls (k ). Now f(h) can b rgardd as a subgroup of Alt(n), stabilizing th nth objct. Th charactr of th standard rprsntation of Alt(n) rstrictd to f(h) is as in Lmma 0.3(c), thrfor thr is a f(h)-wavlt. Rmark. Th proof abov shows that thr xist H-wavlts for th standard rprsntation of Symm(n) for vry n. In th rst of this articl w considr th group G of rotational symmtris of a platonic solid S, a polyhdron in 3 whos facs ar rgular polygons. An lmntary rfrnc for ths objcts is [1]. Complt information on th rprsntations of ths groups is in []. Th bst rfrnc for our purposs is [4, 1.], which unfortunatly is not yt availabl. W shall show that vry irrducibl unitary rprsntation (s, V) of Ghas an H-wavlt and w shall dscrib ach on. W may assum that dim V. Thn ach (s, V) is a complxification of an irrducibl orthogonal rprsntation. Thus w may work ovr th rals,. Lt F S dnot th st of facs of S and lt L (F S ) b th ral L spac of F S quippd with th counting masur. Lt p dnot th prmutation rprsntation of G on L (F S ): p(g)f(f) Å f(g 01 f) (g G, f L (F S ), f F S ). W shall dscrib th dcomposition of p into irrducibl componnts and xhibit th wavlts as lmnts of L (F S ). In othr words, w figur out all possibl ways of assigning ral numbrs to th facs of a platonic solid so that for som subgroup H of rotations, moving th solid by lmnts of H givs an orthonormal basis of an irrducibl subrprsntation (whos dimnsion coincids with th ordr of H) of th spac of functions on th facs. W bgin with th following. THEOREM 0.. Lt G b th group of rotational symmtris of a platonic solid and lt (s, V) b an irrducibl unitary (or orthogonal) rprsntation of G. Thn, up to conjugation, thr is only on subgroup H G for which th H-wavlts xist. Proof. Th group G Å Alt(4) or Symm(4) or Alt(5). (i) G Å Alt(4). Th only rprsntation of dimnsion gratr than 1 is th standard rprsntation (r, V), of dimnsion 3. Th charactr is givn by th following tabl: Conjugacy classs 1 (13) (13) (1)(34) x r 3 0 0 01. All subgroups of ordr 3 ar conjugat in G to H Å»(13), th subgroup gnratd by th cycl (13). Sinc th charactr x r vanishs on H"{1} th statmnt follows from Lmma 0.3. (ii) G Å Symm(4). This group has thr irrducibl rprsntations of dimnsion gratr than on: r-th standard on, r Å dt r both of dimnsion 3 and d of dimnsion. Hr is th charactr tabl:
PLATONIC ORTHONORMAL WAVELETS 355 Conjugacy classs 1 (1) (13) (1)(34) (134) x r 3 1 0 01 01 x r= 3 01 0 01 1 x d 0 01 0. Sinc all subgroups of G of ordr 3 ar conjugat to»(13), prvious argumnt suffics for th rprsntations r and r. Thr ar two conjugacy classs of lmnts of ordr in G: (1) and (1)(34). But (1)(34) is in th krnl of d, so its orbits ar trivial. On th othr hand th charactr x d vanishs on H Å»(1), so th statmnt follows from Lmma 0.3. (iii) G Å Alt(5). This group has 4 irrducibl rprsntations of dimnsion gratr than 1. Lt m Å (1 / 5)/ b th goldn man, and lt m V Å (1 0 5)/. (This is th conjugat of m in ( 5) ovr ). Hr is th charactr tabl: Conjugacy classs 1 (1)(34) (13) (1345) (1354) x a 3 01 0 m mv x b 3 01 0 mv m x r 4 0 1 01 01 x g 5 1 01 0 0. As bfor H Å»(13) for th rprsntations a and b, H Å»(1345) for g and H Å {1, (1)(34), (13)(4), (14)(3)} for r, up to conjugacy. (In fact th last group is th only subgroup of ordr 4, bcaus it is a Sylow subgroup). 1. THE TETRAHEDRON Lt G Å Alt(4) and lt (r, V) b th standard rprsntation of G, ovr. Lt 1 Å (01, 1, 1, 01), Å (1, 01, 1, 01), 3 Å (1, 1, 01, 01). Thn V Å 1 3. Dfin a scalar product on V by dclaring 1,, 3 to b an orthonormal basis. LEMMA 1.1. Lt H Å»(13). Th H-wavlts for th rprsntation (r, V)ar givn by th formula 3 ((1 / cos(x)) 1 / (1 0 cos(x) / 3 sin(x)) / (1 0 cos(x) 0 3 sin(x))3 ), whr Å{1and x. Th group O(V) H is isomorphic to SO() 1 O(1), a disjoint union of two circls. Proof. Clarly { 1,, 3 } is a singl H-orbit. Thus 1 is an H-wavlt. Th group O(V) H prsrvs th dirct sum dcomposition V Å ( 1 / / 3 ) ( 1 / / 3 ) and, hnc, is isomorphic to SO() 1 O(1). In particular on can calculat that th O(V) H -orbit of 1 consists of th vctors listd in th lmma.
35 ÖZAYDIN AND PRZEBINDA Lt T b th ttrahdron whos dgs hav midpoints { i, for i Å 1,, 3. Th group G prsrvs T and this way may b idntifid with th group of rotational symmtris of T. Thr is a G-intrtwining isomtry V r L (F T ) (1.) which assigns to V a function f L (F T ) whos valu on a fac f is qual to th scalar product of with th vctor of lngth 3/ prpndicular to f and pointing outward. Hr w list ths vctors: 1 (0 1 0 0 3 ), 1 (0 1 / / 3 ), 1 ( 1 0 / 3 ), 1 ( 1 / 0 3 ). (1.3) Sinc th sum of th vctors (1.3) is zro, th imag L (F T ) r of th map (1.) consists of functions whos valus sum up to zro. Th orthogonal complmnt L (F T ) 1 consists of constant functions. Thus L (F T ) Å L (F T ) 1 L (F T ) r. (1.4) PROPOSITION 1.5. Th wavlts which occur in L (F T ) ar: (a) th constant functions Å{1/ in L (F T ) 1, for H Å {1}; (b) th function f L (F T ) r which taks th valus 0, (1 0 4 cos(x)), (1 / cos(x) 0 3 sin(x)), (1 / cos(x) / 3 sin(x)) on th facs prpndicular to th vctors (1.3), rspctivly, for H Å»(13), Å{1, x. Proof. Part (a) is obvious. For part (b) w apply th map (1.) to th vctors of Lmma 1.1.. THE CUBE Hr G Å Symm(4) and (r, V) is th obvious xtnsion of th standard rprsntation of Alt(4) constructd in th prvious sction. Lt sgn dnot th sign rprsntation of G (composition of r with th dtrminant on End(V)). Lt r Å sgn r. This notation is consistnt with th on usd in th proof of Thorm 0.. Lt C b th cub in V whos facs hav cntrs at { i, for i Å 1,, 3, so th vrtics ar { 1 { { 3. Undr th action r th group G prsrvs C and may b idntifid with th group of rotational symmtris of C.
PLATONIC ORTHONORMAL WAVELETS 357 Thr is a G-intrtwining isomtry V r L (F C ) (.1) which assigns to V a function f L (F C ) whos valu on a fac f is qual to th scalar product of with th vctor of lngth / prpndicular to f and pointing outward. Hr w list ths vctors: 1,, 3, 0 1, 0, 0 3. (.) It is clar that th imag of th map (.1), L (F C ) r=, consists of all functions which tak opposit valus on antipodal facs. Th trivial rprsntation occurs as th subspac L (F C ) 1 of constant functions. Th orthogonal complmnt, L (F C ) d,ofl (F C ) r= L (F C ) 1 consists of functions which tak th sam valus on antipodal facs and all of whos valus sum to zro. It is asy to s that dim L (F C ) d Å and that G acts on it irrducibly. Thus, L (F C ) Å L (F C ) 1 L (F C ) d L (F C ) r= (.3) is th dcomposition into irrducibl componnts. PROPOSITION.4. Th wavlts which occur in L (F C ) ar: (a) th constant functions Å{ / in L (F C ) 1, for H Å {1}; (b) th function f L (F C ) r= which taks th valus (1 / cos(x)), (1 0 cos(x) / 3 sin(x)), 0 (1 0 cos(x) 0 3 sin(x)), (1 / cos(x)), 0 (1 0 cos(x) / 3 sin(x)), 0 (1 0 cos(x) 0 3 sin(x)) on th facs corrsponding to th vctors (.), rspctivly, for H Å»(13), Å {1, x ; (c) th function f L (F C ) d which taks th valus 31 ( 3 0 1), 31 ( 3 0 1), 31 (0 3 0 1), 31 (0 3 0 1), on th facs corrsponding to th vctors (.) rspctivly, for H Å»(1), 1 Å{1, Å{1. 31 3, 31 3
358 ÖZAYDIN AND PRZEBINDA Proof. Part (a) is obvious. Part (b) follows by applying th map (.1) to vctors of Lmma 1.1. For part (c) w procd as follows. Lt f L (F C ) d b an H-wavlt (H Å»(1) ). Lt x, y, z b th thr possibl valus of th function f. Thn x / y / z Å 0, x / y / z Å 1, y / xz Å 0. (.5) Th solutions of th systm (.5) ar 3 3 3 (x, y, z) Å{ ( 301), 0 ( 3 / 1),. 3 Sinc r Å sgn r, th wavlts for r coincid with th wavlts for r. 3. THE OCTAHEDRON Hr G Å Symm(4) acts on V via r as in Sction. Lt O b th rgular octahdron whos facs ar cntrd on th vrtics of th cub C. Thr is a G-intrtwining isomtry L (F C ) r L (F O ) (3.1) which assigns to a function f C L (F C ) a function f O L (F O ) whos valu on a fac f F O is qual to 3/ tims th sum of th valus of f C on th thr facs of C adjacnt to th vrtx of C corrsponding to th fac f. Th imags of L (F C ) 1 and L (F C ) r= ar nonzro and, hnc, irrducibl. W dnot thm by L (F O ) 1 and L (F O ) r=, rspctivly. Th imag of L (F C ) d is zro. Th functions in L (F O ) r= hav opposit valus on antipodal facs. Th dimnsion of th spac of all such functions is 4. Hnc th orthogonal complmnt of L (F O ) r= in this spac has dimnsion 1. Dnot it by L (F O ) sgn. Multiplication by a nonzro function from L (F O ) sgn maps L (F O ) r= to L (F O ) r, th r-isotypic componnt of L (F O ). Th lmnts of L (F O ) r hav th sam valus on antipodal facs. Altogthr w hav th dcomposition into irrducibl componnts L (F O ) Å L (F O ) 1 L (F O ) sgn L (F O ) r L (F O ) r=. (3.) By composing th map (3.1) with th map (.1) w gt V r L (F O ) (3.) which assigns to V a function f L (F O ) whos valu on a fac f is qual to th scalar product of with th vctor of lngth /4 prpndicular to f and pointing outward. Hr w list ths vctors:
PLATONIC ORTHONORMAL WAVELETS 359 4 ( 1 / / 3 ), 4 (0 1 / / 3 ), 4 ( 1 0 / 3 ), 4 (0 1 0 / 3 ), 0 4 ( 1 / / 3 ), 0 4 (0 1 / / 3 ), 0 4 ( 1 0 / 3 ), 0 4 (0 1 0 / 3 ). (3.4) PROPOSITION 3.5. Th wavlts which occur in L (F O ) ar: (a) th constant functions Å{ /4 in L (F O ) 1, for H Å {1}; (b) th function f L (F O ) sgn which taks th valus 4, 0 4, 4, 0 4, 0 4, 4, 0 4, 4 on th facs corrsponding to th vctors (3.4), rspctivly, for H Å {1}, Å{1; (c) th function f L (F O ) r= which taks th valus 4, (1 0 4 cos(x)), 1 1 (1 / cos(x) 0 3 sin(x)), 1 (01 0 cos(x) 0 3 sin(x)), 0 4, 0 (1 0 4 cos(x)), 1 0 1 (1 / cos(x) 0 3 sin(x)), 1 (1 / cos(x) / 3 sin(x)) on th facs corrsponding to th vctors (3.4), rspctivly, for H Å»(13), Å {1, x. (d) th function f L (F O ) r which taks th valus 4, (1 0 4 cos(x)), 1 1 (1 / cos(x) 0 3 sin(x)), 1 (01 0 cos(x) 0 3 sin(x)), 4, 0 (1 0 4 cos(x)), 1 1 (1 / cos(x) 0 3 sin(x)), 1 (1 / cos(x) / 3 sin(x)) on th facs corrsponding to th vctors (3.4), rspctivly, for H Å»(13), Å {1, x. Proof. Parts (a) and (b) ar obvious. Part (c) is obtaind by applying map (3.3) to th vctors of Lmma 1.1. Part (d) follows from (c) via multiplication by an lmnt of L (F O ) sgn.
30 ÖZAYDIN AND PRZEBINDA 4. THE DODECAHEDRON Hr G Å Alt(5). Rcall th goldn man m Å (1 / 5)/ and th conjugat m V Å (1 0 5)/. In th thr-dimnsional spac V dfind in Sction, lt D b th dodcahdron whos facs ar cntrd on th vctors 0m / 3, 1 / m 3, 0 1 / m 3, 0m 1 0, 0m 0 3, m 1 0 (4.1) and on th antipodal vctors. Th group Alt(4) is containd in G as th stabilizr of th numbr 5. Lt s Å (1345). Thn Alt(4) and s gnrat G. Rcall th standard rprsntation (r, V) of Alt(4). On can xtnd this rprsntation to a rprsntation a of G on V by ltting s act via rotations by p/5 about th axis passing through midpoints of som two opposit facs. Spcifically, a(s): 0m / 3 r 0m / 3 a(s): 1 / m 3 r 0 1 / m 3 r 0m 1 0 r 0m 0 3 r m 1 0 r 1 / m 3. (4.) Thus th matrix of a(s) with rspct to th basis { 1,, 3 }is 1 0mV 1 01 0m 1 m mv. (4.3) m mv W us th rprsntation a to idntify G with th group of rotational symmtris of D and to gt th rprsntation p on L (F D ). Thr is a G-intrtwining isomtry V r L (F D ) (4.4) which assigns to V a function f L (F D ) whos valu on a fac f is qual to th scalar product of with th vctor of lngth 1 prpndicular to f and pointing outward. W obtain ths vctors by multiplying th vctors (4.1) by 1/ 1 / m. If w rplac m by mv in th formulas (4.) (or (4.3)) w gt anothr xtnsion of r to a rprsntation b of G on V. Sinc tr b(s) Å mv x m Å tr a(s), ths two rprsntations ar not quivalnt. Undr th action b, G dos not prmut th vctors (4.1). In othr words it dos not prsrv th dodcahdron D. Howvr it dos prsrv a diffrnt dodcahdron DV (of th sam siz) whos facs ar cntrd on th conjugats (m rplacd by mv ) of th vctors (4.1): 0mV / 3, 1 / mv 3, 0 1 / mv 3, 0mV 1 0, 0mV 0 3, mv 1 0 (4.5) and th antipodal vctors. This dfins a bijction of facs: F D f } fu F DV.
PLATONIC ORTHONORMAL WAVELETS 31 Lt G act on V by b. Lt V r L (F D ) (4.) b th map which assigns to V a function f L (F D ) whos valu on a fac f F D is qual to th scalar product of with th vctor of lngth 1 prpndicular to th corrsponding fac fu F DV and pointing outward of DV. Th imags of (4.4) and (4.) ar both nonzro and, hnc, irrducibl. Morovr, thy fill up th subspac of L (F D ) consisting of functions having opposit valus on antipodal facs. Th orthogonal complmnt consists of functions having th sam valus on antipodal facs. This last subspac splits into an orthogonal dirct sum of th trivial rprsntation (constant functions) and th fiv-dimnsional vrtx rprsntation g. Thus, altogthr, PROPOSITION 4.8. L (F D ) Å L (F D ) a L (F D ) b L (F D ) 1 L (F D ) g. (4.7) Th wavlts which occur in L (F D ) ar: (a) th constant functions Å{ 3/ in L (F D ) 1, for H Å {1}; (b) th function f L (F D ) g which taks th valus 3 0, 10 0 cos(x) 0 cos(y), 10 / mv cos(x) / m cos(y) / 51/4 m 1/ sin(x) / 5 1/4 (0mV ) sin(y) 1/, 10 / m cos(x) / mv cos(y) / 51/4 m 01/ sin(x) 0 5 1/4 (0mV ) sin(y) 01/, 10 / m cos(x) / m V cos(y) 0 5 1/4 m 01/ sin(x) / 5 1/4 (0mV ) sin(y) 01/, 10 / m V cos(x) / m cos(y) 0 5 1/4 m 1/ sin(x) 0 5 1/4 (0mV ) sin(y) 1/ on th facs corrsponding to th vctors (4.1), rspctivly and th sam valus on antipodal facs, for H Å»(1345), Å{1,x,y. Th group O(L (F D ) g ) H, and hnc th varity of all ths wavlts, is isomorphic to SO() 1 SO() 1 O(1), i.., th surfac of a disjoint union of two donuts. (c) th function f L (F D ) a which taks th valus / 1 / (m 0 1)cos(x) 0 3(m / 1)sin(x), m / 0m / 1 / ( 0 m)cos(x) 0 3m sin(x), m / m
3 ÖZAYDIN AND PRZEBINDA 0 1 0 (m / )cos(x) 0 3m sin(x), m / m 0 1 / (1 0 m)cos(x) 0 3 sin(x), m / 0m m / 0m 0 1 / (m / 1)cos(x) / 3(1 0 m)sin(x), m / m 0 1 / (m / 1)cos(x) 0 3 sin(x) on th facs corrsponding to th vctors (4.1), rspctivly and th opposit valus on antipodal facs, for H Å»(13), Å{1,x. (d) Th function f L (F D ) b obtaind by rplacing m by mv in (c). Proof. Part (a) is trivial. Part (c) is obtaind by applying map (4.4) to th vctors of Lmma 1.1. Similarly part (d) follows by applying map (4.). For (b) w procd as follows. Dnot by x 0 /, x 1 /, x /, x 3 /, x 4 /, x 5 / th valus of a function f L (F D ) g on facs corrsponding to th vctors (4.1). Thn x 0 Å0x 1 0x 0rrr 0 x 5. Morovr, f is an H-wavlt if and only if x 0 / x 1 / rrr / x 5 Å 1, and x 0 / x 1 x s(1) / x x s() / rrr / x 5 x s(5) Å 0 for any cyclic rotation s of th indics 1,, 3, 4, 5. Ths quations ar quivalnt to x 1 / x / rrr / x 5 Å 5/ (x 1 / x / rrr / x 5 ) Å 1/ x 1 (x 3 0 x ) / x (x 4 0 x 3 ) / x 3 (x 5 0 x 4 ) / x 4 (x 1 0 x 5 ) / x 5 (x 0 x 1 ) Å 0. (4.9) On solution of (4.9) is x 1 Å 1 0 4 5, x Å x 3 Å x 4 Å x 5 Å 1 / 5. This givs on wavlt f 1 L (F D ) g. Dfin a rprsntation of th group H Å»s on 5 following matrix by idntifying s with th 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0.
PLATONIC ORTHONORMAL WAVELETS 33 Thn th column vctor 1 Å column(1, 0, 0, 0, 0) 5 is an H-wavlt and w hav an isomtry 5 r L (F D ) g (4.10) snding 1 to f 1 (s th Introduction). Th matrix s has fiv distinct ignvalus. Hnc th group O(5) H is isomorphic to SO() 1 SO() 1 O(1). This group can b calculatd xplicitly. This givs th varity of H-wavlts in 5 : O(5) H 1. Th imag of this orbit undr th map (4.10) is dscribd in (b). 5. THE ICOSAHEDRON Lt G Å Alt(5) and lt I b th icosahdron whos facs ar cntrd on th vrtics of th dodcahdron D. Th only rprsntation of G which dos not occur in L (F D ) is th standard rprsntation (r, V). Lt 5 b quippd with th usual scalar product. Thn V is th subspac consisting of vctors whos coordinats add up to zro. Lt 1 Å 0 1, 1, 1, 0 1,0, Å 3 Å 1,01,1,01,0, 1,1,01,01 5 5 5 5,0, 4 Å 10, 10, 10, 10, 0 4 10 5. (5.1) Th abov vctors form an orthonormal basis of V. Lt h 1 Å (1)(34), h Å (13)(4), h 3 Å (14)(3) and lt H Å {1, h 1, h, h 3 }. W idntify V with 4 using th basis (5.1). Thn th lmnts h 1, h, h 3 ar idntifid with th matrics: h 1 Å 01 0 0 0 0 01 0 0 0 0 1 0 0 0 0 1, h Å 01 0 0 0 0 1 0 0 0 0 01 0 0 0 0 1, h 3 Å 1 0 0 0 0 01 0 0 0 0 01 0 0 0 0 1. (5.) Hnc th group O(4) H consists of all matrics with {1 on th diagonal. This is a maximal commutativ -group in O(4). A vctor Å (x, y, z, w) 4 is an H-wavlt if and only if x / y / z / w Å 1, 0x 0 y / z / w Å 0, 0x / y 0 z / w Å 0, Th solutions of (5.3) ar x Å{ 1, yå{ 1, zå{ 1, wå{ 1. x 0 y 0 z / w Å 0. (5.3)
34 ÖZAYDIN AND PRZEBINDA Hnc th H-wavlts in V ar x 1 / y / z 3 / w 4 : { 1 1 (m, m, m, 3mV 01, 0), { 5 (m, m, 3mV 01, m, 0), 5 { 1 5 (m, 3m V 01, m, m, 0), { 1 5 (3m V 0 1, m, m, m, 0), { 1 5 (m V, mv, mv,3m01, 0), { 1 5 (m V, mv,3m01, mv, 0), { 1 5 (m V,3m01, mv, mv, 0), { 1 5 (3m 0 1, m V, mv, mv, 0), (5.4) whr m Å (1 / 5)/, th goldn man, and mv Å (1 0 5)/ ar as in th prvious sction. Obsrv that all ths wavlts (5.4) ar obtaind as th H-orbits of th vctors { and { V, whr is th first vctor in th list (5.4). Now w com back to th icosahdron I. Rcall that th group Alt(4) is containd in Alt(5) Å G as th stabilizr of th numbr 5. It has fiv orbits in F I. Gomtrically this mans that on may color th facs of I using fiv diffrnt colors in such a way that th fiv facs around ach vrtx ar all of diffrnt color. Thn th midpoints of th four facs of a givn color ar th vrtics of a ttrahdron. Lt us fix on vrtx in I. Givn a vctor (a, b, c, d, ) V w put th numbrs a, b, c, d, on th facs around this vrtx in a cyclic ordr. W put ths numbrs also on th othr facs rspcting th coloring. Thus on numbr corrsponds to ach ttrahdron. Aftr normalizing w gt a G-intrtwining isomtry: V r L (F I ). (5.5) Th imag of (5.5) is irrducibl but it dos not fill up th whol isotypic componnt L (F I ) r. Lt A { : L (F I ) r L (F I ) (5.) b th orthogonal projction on th {1-ignspac of th antipodal map A. Thn L (F I ) r is a dirct sum of th projctions of th imag of (5.5) undr A / and A 0. Thr is anothr natural G-intrtwining map L (F D ) r L (F I ) (5.7) which assigns to a function f D L (F D ) a function f I L (F I ) whos valu on a fac f I F I is qual to th sum of th valus of f D on th facs of D adjacnt to th vrtx dfining f I. Th adjoint to (5.7) is th map L (F I ) r L (F D ) (5.8) which assigns to a function f I L (F I ) a function f D L (F D ) whos valu on a fac
PLATONIC ORTHONORMAL WAVELETS 35 f D F D is qual to th sum of th valus of f I on th facs of I adjacnt to th vrtx of I corrsponding to f D in th D } I duality. It is clar that th krnl of (5.8) coincids with L (F I ) r. Thus L (F I ) Å L (F I ) a L (F I ) b L (F I ) 1 L (F I ) g L (F I ) r, (5.9) whr th first four summands ar th imag of (5.7). Th last on is th krnl of (5.8). It consists of two copis of r. Th discussion abov vrifis th following. PROPOSITION 5.10. (a) Th wavlts which occur in L (F I ) a,l (F I ) b,l (F I ) 1,L (F I ) g ar obtaind by applying th map (5.7) to th corrsponding wavlts on th dodcahdron D, and normalizing. (b) Th wavlts in L (F I ) r ar obtaind by applying th map (5.5) to th vctors (5.4), thn applying A / or A 0 and normalizing. REFERENCES 1. C. T. Bnson and L. C. Grov, Finit Rflction Groups, Springr-Vrlag, Nw York, 1985.. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parkr, and R. A. Wilson, Atlas of Finit Groups, Clarndon Prss, Oxford, 1985. 3. I. Daubchis, Tn Lcturs on Wavlts, SIAM, Philadlphia, 199. 4. R. Goodman, R. How, and N. Wallach, Rprsntations and invariants for th classical groups. [prliminary vrsion] 5. A. A. Kirillov, Elmnts of th Thory of Rprsntations, Springr-Vrlag, Nw York, 197.