ALGEBRAIC
FORMS
285
Symbolic Represeoitation of Symmetric Functions.—Denote the is transformed into the operator by the substitution aLS aS a.9 a a a (a0, a2, ...as, ...) —( o> ^o i> ^o 2> •••)> elementary symmetric function ky ^ 5 ’ • * ’ at pleasure ; so that the theory of the general operator is coincident with that then, taking oi equal to co , we may write of the particular operator dv For example, the theory of inl + a1x + apc‘i+...=(l+pix)(l+p3x)... = eaix = ea.2x=ea&:=... variants may be regarded as depending upon the consideration of the symmetric functions of the differences of the roots of the where equation a —A= s— / flPZ'-Ps — a0xn - (” )a1xn~1 + (” )aixn~2 - ... = 0 ; s! s! Further, let and such functions satisfy the differential equation m 1 + 1^ + bpt? +... + bmx ={ + <t1x)(1 + a pc)... (1 + <rmx) ; a0Ca-f- '2.02 -t -yOiO-^•> -f-... -t* TICtn—p ^n ~ 0. so that For such functions remain unaltered when each root receives the 1+ + atfr + ... = { + /,2<r1)(l1 + P2<ri)-• • = e<TW > e<r2a same infinitesimal increment h ; but writing x-h for x causes 1 + ai<r ‘> 2 -b a2<r 1 + ... = (1 + ^2^2) ( + P&i)- ■ ■ ~ a a to ®o> 2> 3> ••• become respectively a , a^ha^, a^ + ^h^, a3 + Sha2, ... and/(a0, a1} a2, a3,...) becomes 0 l+a1<xm + ana;n + ... = {l + p1<rm){f+ p2am)...=e<Tma'm ) y"t ^^ (V-tf.a 1 "b "b 3(X20a3 “b . • • )/*, and, by multiplication,i l(l + ax<T + a2<r + ...) = H(l + b1p->rb3pi+ ... + bmpn and hence the functions satisfy the differential equation. The <T P important result is that the theory of invariants is from a certain _ „<rial + a-2“2+ • ■ + a'ma.m point of view coincident with the theory of non-unitary symmetric functions. On the one hand we mayn staten that non-unitary by brackets ( ) and [ ] symmetric functions of the quansymmetric functions of the roots of a0x - axx ~^ + a^n~‘2‘ - ... = 0, Denote tities p and <r respectively. Then are symmetric functions of differences of the roots ot 1 + aqtl] + altl2] + «2[2] -b a?[l3] + aia2[21] -b a3[3] + ... a0aj”-l!(”)a1a3”-1 + 2!(”)a2a3n-2- ... = 0 ; + apfyfyz ■ ■ aprlP1P‘2Ps-' -Pm] + ’ • • and on the other hand that symmetric functions of the differ= 1 + &2(1) + 6?(12) + 62(2) + &?(13) + &A(21) + HZ) + ences of the roots of + bqibfbf. .Lq™(Hlmm~qm-1.-Hnqi) + ... - ("Vpe”-1 -b (” )a#Sn-‘2‘ - ... = 0, + cr2a2. • + <rmam are non-unitary symmetric functions of the roots of Expanding the right-hand side by the exponential theorem, and L +l_ lArv> ... =0. then expressing the symmetricwefunctions of oq, cr2,_...<rm, which 2! arise, in terms of &2> H obtain by comparison with the An important notion in the theory of linear operators in middle series the symbolical representation of all symmetric functions in brackets ( ) appertaining to the quantities p^ p2, P3, — general is that of MaciVIalion s Tfiwl/t'il'LTicciT O'pcTCitoT ( . Theory of a Multilinear partial Differential Operator with Applications to To obtain particular theorems the quantities oq, oq, o-3,...crm are the Theories of Invariants and Reciprocants, ” Proc. Land. Math. auxiliaries which are at our entire disposal. Thus to obtain Soc., t. xviii. (1886), pp. 61-88). It is defined as having four Stroll’s theory of seoninvariants put elements, and is written bi = <rx + (t2+ ... +<rm=:[l] = 0 ; n) we then obtain the expression of non-unitary symmetric functions of the quantities p as functions of differences of the symbols am ^’2) * * • = 1 [^a0dan + {IJ- + V) (rn-lim 0 Ex. gr. bl^) with m-2 must be a term in m ! am-2Wi2l u J a 4 + Si'H(m-l)! 1 !, 0 ®2 + fZ. o 1! oi*0 (m-2) 2 ! ° ®1J a»+2 e<Tiai+<roa2 _ e<ri(ai ~ <*2) — ... -f icj-f (oj - Ct2) + ..., ,f m! .... m -1 m! m-2 + (/t+3y) and since bl=cr$ we must have (m-l)~!T!ao a3 + (m_2)!i!i!«0 ^2 m! m-3a s_9 + (22) = Kr(ai ~ “2)4 = " ^(“^2 + “i«D + ^“1 (m-3)!3!“° °/ an-b3 = 2a4 - 2aLa3 + al "]• as is well known. l- k fa . The operators Again, if oq, aq, (r3...<rm be the m, mth roots of —1, bx = b2—... the coefficient of a^a^a^... being = bm—x — 0 and bm=l, leading to «o3aI + «l3a2+— a0dai + 2alda2+- are seen to be (1, 0 ; 1,1) 1 + (m) -b (mr) + (m3) +... = ^iai+^+.. -b<~ and (1, 1 ; 1, 1) respectively. Also the operator of the Theory and a a m of Pure Reciprocants (see Sylvester Lectures on the New Theory .’. (m‘)=: of Reciprocants, Oxford, 1888) is ' ' ms!;(<r2 i-b<r2o2-b...-bam, m)* , (4, 1 ;2, l) = h4ao0C(1 + lO«oal3a2+6(2aOa2 + «l)3a3+-”} It will be noticed that (yu, v ; m, ri) = p{, 0 ; m, n) + i'(0, 1 ; m, n). The importance of the operator consists in the fact that taking any two operators of the system (yu, v ; m, n) ; (yu1, v1 ; m1, n1). the operator equivalent to (/4, v ; m, ri)(y.1, vl; on1, n1) - (/a1, id ; m1, n1)^, v ; m, n), known as the “alternant” of the two operators, is also an operator of the same system. We have the theorem {y.,v,m,ri)(y}, jdjm1, ti1) — (/d, v1; m1, nx){y., v ;m, «) = (^1, J'i! mi, nfj ; where 1 1 1 1 Ml = (m + m-l)|^-1(ya + «. i')-£(/x + ?i»' )j , 1 . .1 ,,1!' H m-1. m1——M*' -1 2 > 1 p )' v,1 = (n ' 1 - ■n.)i' ' mjon m1=on1 + m-l, n1 = n +n, and we conclude that qud “alternation” the operators of the system form a “group.” It is thus possible to study simultaneously all the theories which depend upon operations of the group.
and we see further that (oq^i + <72a2 + • ■ • +vanishes identically unless &=0 mod m. If m be infinite and 1 -b Zqa; + Zq*2 + ... = (1 + <ri*)(l +°'2a::)”*=:e^ia:;=e^2X= ••• > we have the symbolic identity eo-2a2+0-2012+V3«3+- • • — eP101+P202+P303+- • •, and (oqcti + <r 2a2 + cr 303 + .. .)p - (piPi + p.fl2 + pA + . ■ .f . Instead of the above symbols we may use equivalent differential operators. Thus let ba = ufa0 + 2a + Safo^ + ... and let a, b, c, ... be equivalent quantities. Any function of differences of Sa, d^, dc, ... being formed the expansion being carried out, an operand a0 or b0 or c0... being taken and b, c,... being subsequently put equal to a, a non-unitary symmetric function will be produced. Ex. gr. (Sa - 5J)2(5(i - Sc) = (S2 - 28aSb + 52)(5„ - Sc) = 5* - 2d2a8b + 5Jl - 5% + 28a8b8c - 5% = 6a, - iaJ)x + 2a,bo - 2a,,c. + 2ai&2Ci_ 2b2cx = 2(al-3a1a2 + 3a?) = 2(3). The whole theory of these forms is consequently contained imI plicitly in the operation 5.
