JOURNAL OF RESEARCH of the National Bu rea u of S ta nda rd s- B. Mathe m atica l Scie nces Vol. 79B, Nos . I a nd 2, J a nuary·June 19 75 Linearly Independent Sets of Isotropic Cartesian Tensors of Ranks up to Eight Elliot A. Kearsley Institute fOl' Mate l'ials R eseal'c h, National Bureau of Standards Wa s hin~ton, D.C. 20234 and J effre y T. Fong Institute for Basic Standal'd s , National 8l11'eau of Standa rds Wa s hin~ton , D.C. 20234 T hi s pa per contains a com plete li s ting of isotropic Cart esia n tc nsors of ranks up to e ig ht with the ir associa ted red uc tio n e q uations fo r o bt aining linearl y ind epe nde nt sets whe ne ve r the red uc ti on is call ed for. In pa rticula r. the li sting is co mpil ed onl y for isot ropic ten sors assoc ia ted with t he rota tion group 0 +(3) of the th ree·dime n s iona l under lyin g vector Sl)ace. Based on a n id e ntit y origi nally due to Ca pelli ( 1887), red uc tion equ atio ns fo r te nsors of odd ra nks beginning a t rank fiv e a nd e ve n ra nks beginning a t ra nk e ig ht a re s how n to b e nont rivial. S ignif ic ance of the computa ti ona l re s ult in both pure and a pplie d mathe matics is discu ssed . Key word s: Alge bra : a lt e rna ting te nsor; Cart esia n te nsor : C a pe lli 's ide ntity; gro up re present a ti on: in varia nt: isutropi c te n sor: Kronec kt-'r de lt a: or! hogoll a l ~ r ou p: te nso r. 1. Introduction With respect to some of the most importa nt groups of line ar tra nsforma ti ons, We yl [1]1 has written a definitive trea ti se on the alge braic proble m of decomposing a space of tensors of a given rank into its irreducible invariant subspaces. In the course of proving his main results , -1.\1 .\ ,\II i1, .·. I ( //I ,\ ,I I /i, 1111 <1 11 1:) ", 72 I Fi~ lJ rt ·" in "r;lC'k t ' l ~ judi"al l"' Ih ,. lil eralllrt' r t" ft,rt-' ti l' (-' s a t ttw (·nd "f thi .. p a ~)f>r. 49 571 -4 53 0 - 75 - 4 Weyl employed a powerful formal instrume nt known as Capelli's identity [1 , p. 39]. Based on a theorem proved by Weyl [1 , pp. 53-56], and by applying a special form of Capelli's ide ntity, we first demonstrate a nontrivial problem. The problem is that of cons tructing explicit linearly independent sets of isotropic Cartesian tensors of odd ranks beginning at five and e ve n ranks beginning at eight. Our algebraic results including all reduc tion equations are then given in full. An application of some of these results in mechanics of deformable media is described in a companion paper [2]. 2. Statement of the Problem To facilitate our application in mechanics, we adopt the three-dime n sional Euclidean space as the underlying vector space in Cartesian coordinates. With respect to a fix ed set of base vectors, each Cartesian te nsor of rank n has 3/1 real components denoted by Tij ... (1/ times), each s ubscript varying from 1 to 3. Let A = [a ij ] be a rotation matrix satisfying the conditions ATA= 1 and det A=+ 1. A Cartesian tensor of rank n, with respect to the three-dimensional prope r orthogonal group 0 + (3), satisfies by definition the following transformation law: (new) (old) T jmq .. ( n times) T;kp .. ( 0 l;me8) a;j a km a pq" .(n times), (2.1) where repeated indices de note summation according to the usual conve ntion with each index ranging from 1 to 3, and the a's are based on the rotation matrix with which new base vectors are expressed in te rms of the old ones. Within the context of this paper, we define a Cartesian tensor of rank n as isotropic if it satisfies the additional requirement that all components of the te nsor re main constant for arbitrary rotations of the coordinate axes_ 2 It is not difficult to show that there is no isotropic Cartesian tensor of rank 1; all isotropic Cartesian tensors of rank 2 are in the form of 'A 8 ;j, 'A being an arbitrary scalar; and all isotropic Cartesian te nsors of rank 3 are scalar multiples of the alternating tensor E ;jk which equals + 1, 0, - 1 for cyclic, acyclic, anti-cyclic pe rmutations of the three indices i, j , k. 3 As proved in Weyl [1 , pp. 53-56], every isotropic Cartesian tensor of even rank is expressible as a linear combination of products of the Kronecke r deltas, 8 ;j' 8 km' etc., 4 and every isotropic Cartesian tensor of odd rank is given by a linear combination of terms formed of prod ucts of an appropriate number of Kronec ke r deltas with an alternating te nsor. Such products of Kronecker deltas with or without the alternating tensor E;jk will be refe rred to as fundamental isotropic Cartesian tensors (abbrev. FICT). For every rank n > 1, the total number N(n) of FICT's can be calculated from the following formulas: n For n eve n, N(n) (2.2) G) 2 0 / 2 n! For n odd, N(n) (2.3) 3 (n;3) ! 2( n- 3)/2 2.\1an y wril~r s prf" £f"T to U !'i f" Iht' It-'rm " Iwmitrl)pi(''' or "gYTotropiC''' for ! " n SOTS in variant undt· r rol a tions of thl' prupf'T ()rlliogol1ai group. 3St'f". t-'.g . • Jt·f(rf'Ys l :~, "~p o &6 .6Hj. 4Thf' sa mt-' Tt-' S U It f or ,'yen rank (t-'nSOTS was givf'n appare ntly first hy Fra nklin [4] in 192:\. 50 EXf't'pt for /I = 2,.),4, and 6, Racah [5] showed that N(n) in general exceed s M (n), the total numbe r of linearly inde pe nde nt FICT's. Us ing the theor y of group representation of the three­ dime nsional rotation group, 0+(3), he de rive d an expli c it formula for the number M(n) as follow s: [,,; 11 (2.4) M (n) = 1,~ JG~)-~(2k~ 1)](~} The following table shows tha t for odd ra nks beginning at 5 and e ve n ranks beginning at 8, the re exist linear combina tions among FICT's which are ide ntically ze ro: T .\ I!I .f: I . Rank II 2 3 4 5 6 7 8 9 10 II N umber of Di s tinct F ICT's N(II) I I 3 10 15 105 lOS 1.260 945 17.325 N umlwr of l.inearl y Independ e nt Flc r"s M(II) I I 3 6 15 36 91 232 60:3 1.511S The problem now is to find these linear combina tio ns, know n as reduc ti on e quations, fo r ranks 5, 7, and 8, for use in expli c it applicati ons suc h as those described in [2]. 3. Capelli's Identity To obt ain the reductio n equati ons for isotropic tensors of odd ranks beginning a t fi ve a nd even ra nk s beginning a t e ight, we rely on special cases of a powe rful forma l in s trume nt know n as Capelli 's id e ntity. s In partic ular, we need only to apply the following two id entities: I) ip I) iq I)i' ojp oj q OJ, EjjkEpqro (3.1) otp I) tq 0.,. I) ip I) iq 0 ;,- ° i, I) jp ojq OJ, OJ, = 0 . (3.2) I) tp ot q I) k, 0", I) mp I) mq Om, Om, 5 "t>yl [1. pp. :W·4 71 intrmIU('t"d C:qwlli'!' idf'ntit)' 10 prov(" a fundallH'lItal t lwort'm un orthogonal LIl"ariant ;;. II. p. S:q. For our pllrp,, ;;.t"~ hf'Tf' , Wf'~ r ... f'q u ali"l1 (C). I ) in ll. p . S.11 l'ouHil"f'!'\ as thl' ha si~ for (, tHllpu t a tion . 51 Ide ntity (3 .1) can be ve rified directly by expa nding the 3 X 3 d e te rminant. Ide ntity (3.2) follows from the fact that the re are only three possible values of the indices in a three·dime nsional space, and since the re are four columns on the left hand side, the indices of at least two columns must be equal. 6 He nce the dete rminant on the left of (3.2) must vanish. In light of (3. 1), consider the d e t e rmin a ~t of (3.2) e xpande d in the minors of the first column: E qro (8 ip E jkm - 8 jp E ikm + 8 kp E ijm - 8 mp E ijk ) = O. (3.3) Since the first factor of this equation does not vanish ide ntically, the linear combination of the following four isotropic tensors of rank fiv e must: (3.4) Equation (3.4) can b e c harac terized by the inde x p, whic h is distinguished by occurring in each of the four Kronecke r deltas.