Page:EB1911 - Volume 01.djvu/667

we may write in general

$${\text{D}}_{s}f=\Sigma {\text{D}}(p_{1}p_{2}p_{3}\ldots )f,$$

the summation being for every partition $(p_{1}p_{2}p_{3}\ldots )$ of $s,$ and ${\text{D}}(p_{1}p_{2}p_{3}\ldots )f$ being $=\Sigma ({\text{D}}p_{1}f_{1})({\text{D}}p_{2}f_{2})({\text{D}}p_{3}f_{3})f_{4}\ldots f_{m}.$

Ex. gr. To operate with ${\text{D}}_{2}$ upon $(21^{3})(21^{4})(1^{5}),$ we have

$${\begin{aligned}{\text{D}}_{(2)}f&=(1^{3})(21^{4})(1^{5})+(21^{3})(1^{4})(1^{5})\\{\text{D}}_{(1^{2})}f&=(12^{2})(21^{3})(1^{5})+(21^{3})(21^{3})(1^{4})+(21^{2})(21^{4})(1^{4}),\end{aligned}}$$

and hence

$${\begin{aligned}{\text{D}}_{2}f=(21^{4})(1^{5})(1^{3})+(21^{3})(1^{5})(1^{4})+(&21^{3})(21^{2})(1^{5})+(21^{3})^{2}(1^{4})\\&+(21^{4})(21^{2})(1^{4}),\end{aligned}}$$

Application to Symmetric Function Multiplication.—An example will explain this. Suppose we wish to find the coefficient of $(52^{4}1^{3})$ in the product $(21^{3})(21^{4})(1^{5})$.

Write

$$(21^{3})(21^{4})(1^{5})=\ldots +{\text{A}}(52^{4})(1^{3})+\ldots ;$$

then

$${\text{D}}_{5}{\text{D}}_{2}^{4}{\text{D}}_{1}^{3}(21^{3})(21^{4})(1^{5})={\text{A}};$$

every other term disappearing by the fundamental property of ${\text{D}}_{s}.$ Since

$${\text{D}}_{5}(21^{3})(21^{4})(1^{5})=(1^{3})(1^{4})(1^{4}),$$

we have:—

$${\begin{aligned}&{\text{D}}_{2}^{4}{\text{D}}_{1}^{3}(1^{4})(1^{4})(1^{3})={\text{A}}\\&{\text{D}}_{2}^{3}{\text{D}}_{1}^{3}\{(1^{3})(1^{3})(1^{3})+2(1^{4})(1^{3})(1^{2})\}={\text{A}}\\&{\text{D}}_{2}^{2}{\text{D}}_{1}^{3}\{5(1^{3})(1^{2})(1^{2})+2(1^{4})(1^{2})(1)+2(1^{3})(1^{3})(1)\}={\text{A}}\\&{\text{D}}_{2}{\text{D}}_{1}^{3}\{12(1^{2})(1^{2})(1)+7(1^{3})(1)(1)+2(1^{4})(1)+6(1^{3})(1^{2})\}={\text{A}}\\&{\phantom {{\text{D}}\,}}{\text{D}}_{1}^{3}12(1)^{3}={\text{A}},\end{aligned}}$$

where ultimately disappearing terms have been struck out. Finally ${\text{A}}=6\cdot 12=72.$

The operator $d_{1}=a_{0}\delta a_{1}+a_{1}\delta a_{2}+a_{2}\delta a_{3}+\ldots$ which is satisfied by every symmetric fraction whose partition contains no unit (called by Cayley non-unitary symmetric functions), is of particular importance in algebraic theories. This arises from the circumstance that the general operator

$$\lambda _{0\iota }a_{0}\delta a_{1}+\lambda _{1}a_{1}\delta a_{2}+\lambda _{2}a_{2}\delta a_{3}+\ldots$$

is transformed into the operator $d_{1}$ by the substitution

$$(a_{0},a_{1},a_{2},\ldots a_{s},\ldots )=(a_{0},\lambda _{0}a_{1},\lambda _{0}\lambda _{1}a_{2},\ldots ,\lambda _{0}\lambda _{1}\ldots \lambda _{s-1}a_{s},\ldots ),$$

so that the theory of the general operator is coincident with that of the particular operator $d_{1}.$ For example, the theory of invariants may be regarded as depending upon the consideration of the symmetric functions of the differences of the roots of the equation

$$a_{0}x^{n}-1!{\tbinom {n}{1}}a_{1}x^{n-1}+{\tbinom {n}{2}}a_{2}x^{n-2}-\ldots =0;$$

and such functions satisfy the differential equation

$$a_{0}\delta a_{1}+2a_{1}\delta a_{2}+3a_{2}\delta a_{3}+\ldots +na_{n-1}\delta a_{n}=0.$$

For such functions remain unaltered when each root receives the same infinitesimal increment $h;$ but writing $x-h$ for $x$ causes $a_{0},a_{1},a_{2},a_{3},\ldots$ to become respectively $a_{0},a_{1}+ha_{0},a_{2}+2ha_{1},a_{3}+3ha_{2},\ldots$ and $f(a_{0},a_{1},a_{2},a_{3},\ldots )$ becomes

$$f+h(a_{0}\delta _{a1}+2a_{1}\delta _{a2}+3a_{2}\delta _{a3}+\ldots )f,$$

and hence the functions satisfy the differential equation. The important result is that the theory of invariants is from a certain point of view coincident with the theory of non-unitary symmetric functions of the roots of $a_{0}x^{n}-a_{1}x^{n-1}+a_{2}x^{n-2}-\ldots =0,$ are symmetric functions of differences of the roots of

$$a_{0}x^{n}-1!{\tbinom {n}{1}}a_{1}x^{n-1}+2!{\tbinom {n}{2}}a_{2}x^{n-2}-\ldots =0;$$

and on the other hand that symmetric functions of the differences of the roots of

$$a_{0}x^{n}-{\tbinom {n}{1}}a_{1}x^{n-1}+{\tbinom {n}{2}}a_{2}x^{n-2}-\ldots =0,$$

are non-unitary symmetric functions of the roots of

$$a_{0}x^{n}-{\frac {a_{1}}{1!}}x^{n-1}+{\frac {a_{2}}{2!}}x^{n-2}-\ldots =0.$$

An important notion in the theory of linear operators in general is that of MacMahon’s multilinear operator (“Theory of a Multilinear partial Differential Operator with Applications to the Theories of Invariants and Reciprocants,” Proc. Lond. Math. Soc. t. xviii. (1886), pp. 61-88). It is defined as having four elements, and is written

$$(\mu ,\nu ;m,n)$$ $${\begin{aligned}={\frac {1}{m}}{\bigg [}&\mu a_{0}^{m}\delta _{an}+(\mu +\nu ){\frac {m!}{(m-1)!1!}}a_{0}^{m-1}a_{1}\delta _{an+1}\\&+(\mu +2\nu ){\bigg \{}{\frac {m!}{(m-1)!1!}}a_{0}^{m-1}a_{2}+{\frac {m!}{(m-2)!2!}}a_{0}^{m-2}a_{1}^{2}{\bigg \}}\delta _{an+2}\\&+(\mu +3\nu ){\bigg \{}{\frac {m!}{(m-1)!1!}}a_{0}^{m-1}a_{3}+{\frac {m!}{(m-2)!1!1!}}a_{0}^{m-2}a_{1}a_{2}\\&\qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;+{\frac {m!}{(m-3)!3!}}a_{0}^{m-3}a_{0}^{3}{\bigg \}}\delta _{an+3}\\&+\ldots {\bigg ]},\end{aligned}}$$

the coefficient of $a_{0}^{k_{0}}a_{1}^{k_{1}}a_{1}^{k_{1}}\ldots$ being ${\frac {m!}{k_{0}!k_{1}!k_{2}!\ldots }}.$ The operators $a_{0}\delta _{a1}+a_{1}\delta _{a2}+\ldots ,a_{0}\delta _{a1}+2a_{1}\delta _{a2}+\ldots$ are seen to be $(1,0;1,1)$ and $(1,1;1,1)$ respectively. Also the operator of the Theory of Pure Reciprocents (see Sylvester Lectures of the New Theory of Reciprocants, Oxford, 1888) is

$$(4,1;2,1)={\frac {1}{2}}{\bigg \{}4a_{0}^{2}\delta _{a1}+10a_{0}a_{1}\delta _{a2}+6(2a_{0}a_{2}+a_{1}^{2})\delta _{a3}+\ldots {\bigg \}}.$$

It will be noticed that

$$(\mu ,\nu ;m,n)=\mu (1,0;m,n)+\nu (0,1;m,n).$$

The importance of the operator consists in the fact that taking any two operators of the system

$$(\mu ,\nu ;m,n);(\mu ^{1},\nu ^{1};m^{1},n^{1}),$$

the operator equivalent to

$$(\mu ,\nu ;m,n)(\mu ^{1},\nu ^{1};m^{1},n^{1})-(\mu ^{1},\nu ^{1};m^{1},n^{1})(\mu ,\nu ;m,n),$$

where

$${\begin{aligned}\mu _{1}&=(m^{1}+m-1){\bigg \{}{\frac {\mu ^{1}}{m^{1}}}(\mu +n^{1}\nu )-{\frac {\mu }{m}}(\mu ^{1}+n\nu ^{1}){\bigg \}},\\\nu _{1}&=(n^{1}-n)\nu ^{1}\nu +{\frac {m-1}{m^{1}}}\mu ^{1}\nu -{\frac {m^{1}-1}{m}}\mu \nu ^{1},\\m_{1}&=m^{1}+m-1,\\n_{1}&=n^{1}+n,\end{aligned}}$$

and we conclude that quâ “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.

Symbolic Representation of Symmetric Functions.—Denote the elementary symmetric function $a_{s}$ by ${\tfrac {a_{1}^{s}}{s!}},{\tfrac {a_{2}^{s}}{s!}},{\tfrac {a_{3}^{s}}{s!}},\ldots$ at pleasure; then, taking $n$ equal to $\infty ,$ we may write

$$1+a_{1}x+a_{2}x^{2}+\ldots =(1+\rho _{1}x)(1+\rho _{2}x)\ldots =e^{a_{1^{x}}}=e^{a_{2^{x}}}=e^{a_{3^{x}}}=\ldots$$

where

$$a_{s}=\sum \rho _{1}\rho _{2}\ldots \rho _{3}={\tfrac {a_{1}^{s}}{s!}},{\tfrac {a_{2}^{s}}{s!}},{\tfrac {a_{3}^{s}}{s!}},\ldots .$$

Further, let

$$1+b_{1}x+b_{2}x^{2}+\ldots +b_{m}x^{m}=(1+\sigma _{1}x)(1+\sigma _{2}x)\ldots (1+\sigma _{m}x);$$

so that

$${\begin{aligned}1+a_{1}\sigma _{1}+a_{2}\sigma _{1}^{2}+\ldots =(1+\rho _{1}\sigma _{1})(1+\rho _{2}\sigma _{1})\ldots &=e^{\sigma _{1}a_{1}},\\1+a_{1}\sigma _{2}+a_{2}\sigma _{2}^{2}+\ldots =(1+\rho _{1}\sigma _{2})(1+\rho _{2}\sigma _{2})\ldots &=e^{\sigma _{2}a_{2}},\\\cdot \qquad \quad \;\cdot \qquad \quad \;\cdot \qquad \quad \;\cdot \qquad \quad \;\cdot \qquad \quad \;\cdot \quad &\qquad \;\cdot \\1+a_{1}\sigma _{m}+a_{2}\sigma _{m}^{2}+\ldots =(1+\rho _{1}\sigma _{m})(1+\rho _{2}\sigma _{m})\ldots &=e^{\sigma _{m}a_{m}};\end{aligned}}$$

and, by multiplication,

$$\mathop {\Pi } _{\sigma }(1+a_{1}\sigma +a_{2}\sigma ^{2}+\ldots )=\mathop {\Pi } _{\rho }(1+b_{1}\rho +b_{2}\rho ^{2}+\ldots +b_{m}\rho ^{m}),$$ $$=e^{\sigma _{1}a_{1}+\sigma _{2}a_{2}+..+\sigma _{m}a_{m}}.$$

Denote by brackets $(\;)$ and $[\;]$ symmetric functions of the quantities $\rho$ and $\sigma$ respectively. Then

$$1+a_{1}[1]+a_{1}^{2}[1^{2}]+a_{2}[2]+a_{1}^{3}[1^{3}]+a_{1}a_{2}[21]+a_{3}[3]+\ldots$$ $$+a_{p_{1}}a_{p_{2}}a_{p_{3}}\ldots a_{p_{m}}{\big [}p_{1}p_{2}p_{3}\ldots p_{m}{\big ]}+\ldots$$ $${\begin{aligned}&=1+b_{1}(1)+b_{1}^{2}(1^{2})+b_{2}(2)+b_{1}^{3}(1^{3})+b_{1}b_{2}(21)+b_{3}(3)+\ldots \\&\qquad \quad \;+b_{1}^{q_{1}}b_{2}^{q_{2}}b_{3}^{q_{3}}\ldots b_{m}^{q_{m}}(m^{q_{m}}m-1^{q_{m-1}}\ldots 2^{q_{2}}1^{q_{1}})+\ldots \\&=e^{\sigma _{1}a_{1}+\sigma _{2}a_{2}..+\sigma _{m}a_{m}}.\end{aligned}}$$

Expanding the right-hand side by the exponential theorem, and then expressing the symmetric functions of $\sigma _{1},\sigma _{2},\sigma _{3},\ldots \sigma _{m},$ which arise, in terms of $b_{1},b_{2},\ldots b_{m},$ we obtain by comparison with the middle series the symbolical representation of all symmetric functions in brackets $(\;)$ appertaining to the quantities $\rho _{1},\rho _{2},\rho _{3},\ldots$ To obtain particular theorems the quantities $\sigma _{1},\sigma _{2},\sigma _{3},\ldots \sigma _{m}$ are auxiliaries which are at our entire disposal. Thus to obtain Stroh’s theory of seminvariants put

$$b_{1}=\sigma _{1}+\sigma _{2}+\ldots +\sigma _{m}=[1]=0;$$

we then obtain the expression of non-unitary symmetric functions of the quantities $\rho$ as functions of differences of the symbols $a_{1},a_{2},a_{3},\ldots$

Ex. gr. $b_{2}^{2}(2^{2})$ with $m=2$ must be a term in

$$e^{\sigma _{1}a_{1}+\sigma _{2}a_{2}}=e^{\sigma _{1}(a_{1}-a_{2})}=\ldots +{\frac {1}{4!}}\sigma _{1}^{4}(a_{1}-a_{2})^{4}+\ldots ,$$

and since $b_{2}^{2}=\sigma _{1}^{4}$ we must have

$${\begin{aligned}(2^{2})&={\frac {1}{24}}(a_{1}-a_{2})^{4}={\frac {1}{24}}(a_{1}^{4}+a_{2}^{4})-{\frac {1}{6}}(a_{1}^{3}a_{2}+a_{1}a_{2}^{3})+{\frac {1}{4}}a_{1}^{2}a_{2}^{2}\\&=2a_{4}-2a_{1}a_{3}+a_{2}^{2}\end{aligned}}$$

as is well known.

Again, if $\sigma _{1},\sigma _{2},\sigma _{3}\ldots \sigma _{m}$ be the $m,m^{th}$ roots of $-1,b_{1}=b_{2}=\ldots =b_{m-1}=0$ and $b_{m}=1,$ leading to

$$1+(m)+(m^{2})+(m^{3})+\ldots =e^{\sigma _{1}a_{1}+\sigma _{2}a_{2}+..+\sigma _{m}a_{m}}$$

and

$$\therefore (m^{s})={\frac {1}{ms!}}(\sigma _{1}a_{1}+\sigma _{2}a_{2}+\ldots +\sigma _{m}a_{m})^{sm},$$