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 286|ALGEBRAIC

Symmetric Functions of Several Systems of Quantities.—It will suffice to consider two systems of quantities as the corresponding theory for three or more systems is obtainable by an obvious enlargement of the nomenclature and notation.

Taking the systems of quantities to be

we start with the fundamental relation

As shown by Schläfli this equation may be directly formed and exhibited as the resultant of two given equations, and an arbitrary linear non-homogeneous equation in two variables. The right-hand side may be also written

$$ 1 + \Sigma \alpha_1x + \Sigma \beta_1y + \Sigma \alpha_1 \alpha_2 x^2 + \Sigma \beta_1y \beta_2 x y + \Sigma \beta_1y \beta_2 y^2 + \cdots $$

The most general symmetric function to be considered is

$$ \Sigma \alpha_1^{p_1} \beta_1^{q_1} \alpha_2^{p_2} \beta_2^{q_2} \alpha_3^{p_3} \beta_3^{q_3} \cdots $$

conveniently written in the symbolic form $$ \left ( \overline{p_1 q_1} \overline{p_2 q_2} \overline{p_3 q_3} \cdots \right ). $$

Observe that the summation is in regard to the expressions obtained by permuting the $$n$$ suffixes $$1, 2, 3,\cdots n$$. The weight of the function is bipartite and consists of the two numbers $$\Sigma p$$ and $$\Sigma q$$; the symbolic expression of the symmetric function is a partition into biparts (multiparts) of the bipartite (multipartite) number $$\overline{\Sigma p, \Sigma q}$$. Each part of the partition is a bipartite number, and in representing the partition it is convenient to indicate repetitions of parts by power symbols. In this notation the fundamental relation is written $$ \begin{aligned} & && (1+ \alpha_1 x + \beta_1 y) (1 + \alpha_2 x + \beta_2 y ) (1 + \alpha_3 x + \beta_3 y ) \cdots \\ &=&& 1 + ( \overline{10}) x + ( \overline{01} ) y + ( \overline{10}^2) x^2 + + ( \overline{10} \overline{01} )xy ( \overline{01}^2) y^ 2 \cdots \\ & && + (\overline{10}^2 ) x^3 + (\overline{10}^2 \overline{01} ) x^2 y + (\overline{10} \overline{01}^2 ) xy^2 + (\overline{01}^3 ) y^3 + \cdots \\ \end{aligned} $$

where in general cq,?=:(lOp 01«). All symmetric functions are expressible in terms of the quantities aPj in a rational integral form ; from this property they are termed elementary functions ; further they are said to be singleunitary since each part of the partition denoting apq involves but a single unit. The number of partitions of a biweight pq into exactly y biparts is given (after Euler) by the coefficient of a^x^if in the expansion of the generating function l -ax.-ay. l-ax*.-axy .l-ay*. 1 - ax* -ax^y. -axy*. 1-ayZ... The partitions with one bipart correspond to the sums of powers in the single system or unipartite theory ; they are readily expressed in terms of the elementary functions. For write {pq)=spq and take logarithms of both sides of the fundamental relation ; we obtain s x io + Soiy='2{a.iX + ply) s20a:2 + 2snxy + s02y2 = 2(aja: + ^y)2, &c., and s io* + soi2/ + 2snxy + s02y2) + ... ~log (l + a1oX + a01y + ...+apqxPyv + ...) From this formula we obtain by elementary algebra (_ _ )2-1 <2* ~ .n ... % -’'Pi'lFPi'hp!q! corresponding to Waring’s formula for the single system. The analogous formula appertaining to n systems of quantities v'hieh expresses in terms of elementary functions can be at once written down. Fx. gr. We can verify the relations

FORMS

and thence derive the formula— (-)iHv-ia^ pp.qf. J l pf.qf. J 7r1!7r2!...VriV/2'-- ’ which expresses the elementary functions in terms of the single bipart functions. The similar theorem for n systems of quantities can be at once written down. It will be. shown later that every rational integral symmetric function is similarly expressible. The Function hm.—As the definition of hpq we take l + h10x + hQ1y+ ... +hpjxPyQ +... 1 (1 — ape — Piy){l — ape - P2y)... ’ and now expanding the right-hand side

the summation being for all partitions of the biweight. Further writing 1 + hl(pc + h01y+... + hpqXPyv +... _ 1 1 --«oi2/+••• + (-)p+qaPgXpyi +... ’ we find that the effect of changing the signs of both x and y is merely to interchange the symbols a and h ; hence in any relation connecting the quantities hm with the quantities apq we are at liberty to interchange the symbols a and h. By the exponential and multinomial theorems we obtain the results— jp-fa—1 1 JStt)! a 71'1 a7To (-) K ttAtiv,!.. l>i'h p2>h and in this a and h are interchangeable. (ff + g-PL ,S?r-l(S^--l)!, 2 7r1!7r2!... Mi /Y/a' 7T > 1 ,ri 7. - S /(yi + gl-l)! /(g>2 + g2-l)P,r2 1 ^Zl pf.qf. J l p2q2... J 7r1! 7t 2 'Vi Uoi, ...)=/is converted into exp{yd10 + vdm)f where dw =

, <*01 =

’

The rule over exp will serve to denote that yd^ + vd01 is to be raised to the various powers symbolically as in Taylor’s theorem. q Writing Dpg= pq ——d^d u o’, exp(ydw + j'(f01) = (1 + /xDjo + J'Doi + ... + yWDpq +...)/; now, since the introduction of the new quantities y, v results in the addition to the function {Px^p^p^...) of the new terms hPlvqi(P-2<h Psrh ■ • •) + yP-vqPi(li P-M‘s ■ • •) + y^VMlFMh-- •)+•••, we find, P^h ih% ■■■)=(ih<h Wh ■■■) 1 and thence • -(Pi'll P'/D'P'/ls- ■ ■)= 1 J S a 3 + 3a while Dri/=0 unless the part rs is involved in /. We may then 8o- io VTo 3o’ state that is an operation which obliterates one part pq when S = a 21 “L oi" a20*0i_ an®io+ a~ such part is present, but in the contrary case causes the function The formula actually gives the expression of {pq) by means of to vanish. From the above Dpq is an operator of order pq, but it S6p3»r3( Lions 01 is convenient for some purposes to obtain its expression in the form of a number of terms, each of which denotes pq successive (To^oT?), which is one of the partitions of {Jq). This is the true stand- linear operations ; to accomplish this write d point from which the theorem should be regarded. It is but a ud'p<i s particular case of a general theory of expressibility. U'P+r.q+s To invert the formula we may write and note the general result.t 1 +a10a; + a01y + ... + appefyi +... exp (m10(fi0 + m01cf0 + •■•+'inpgdpq +...) ~ exp (MjQtfjg + M01<f01 + ... + ^ft-pqdpq + ...) ; — exP + soi2/)— + ^snxy + s02F2) + •■•}) where the multiplications on the left- and right-hand sides of the Vienna Transactions, t. iv. 1852. t Phil. Trans. R. S. London, 1890, p. 490.