Page:EB1911 - Volume 01.djvu/667

 we may write in general

$$ \text{D}_sf = \Sigma \text{D} (p_1 p_2 p_3 \ldots)f, $$

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

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

and hence

$$ \begin{align} \text{D}_2f = (21^4) (1^5) (1^3) + (21^3) (1^5) (1^4) + (&21^3) (21^2) (1^5) + (21^3)^2 (1^4) \\ &+ (21^4) (21^2) (1^4), \end{align} $$

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

$$ \text{D}_5 \text{D}^4_2 \text{D}^3_1 (21^3) (21^4) (1^5) = \text{A}; $$

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:—

$$ \begin{align} &\text{D}^4_2 \text{D}^3_1 (1^4) (1^4) (1^3) = \text{A} \\ &\text{D}^3_2 \text{D}^3_1 \{ (1^3) (1^3) (1^3) + 2(1^4) (1^3) (1^2) \} = \text{A} \\ &\text{D}^2_2 \text{D}^3_1 \{ 5(1^3) (1^2) (1^2) + 2(1^4) (1^2) (1) + 2(1^3) (1^3) (1) \} = \text{A} \\ &\text{D}_2 \text{D}^3_1 \{ 12(1^2) (1^2) (1) + 7(1^3) (1) (1) + 2(1^4) (1) + 6(1^3)(1^2) \} = \text{A} \\ &\phantom{\text{D} \,} \text{D}^3_1 12(1)^3 = \text{A}, \end{align} $$

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  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

$$ a_0x^n - 1!\tbinom n1 a_1x^{n-1} + \tbinom n2 a_2x^{n-2} - \ldots = 0; $$

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 symmetric functions of the roots of $$a_0x^n - a_1x^{n-1} + a_2x^{n-2} - \ldots = 0,$$ are symmetric functions of differences of the roots of

$$ a_0x^n - 1! \tbinom n1 a_1x^{n-1} + 2! \tbinom n2 a_2x^{n-2} - \ldots = 0; $$

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

$$ a_0x^n - \tbinom n1 a_1x^{n-1} + \tbinom n2 a_2x^{n-2} - \ldots = 0, $$

are symmetric functions of the roots of

$$ a_0x^n - \frac{a_1}{1!} x^{n-1} + \frac{a_2}{2!} x^{n-2} - \ldots = 0. $$

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

$$ (\mu, \nu; m, n) $$ $$ \begin{align} = \frac{1}{m} \bigg[ &\mu a^m_0 \delta_{an} + (\mu + \nu) \frac{m!}{(m-1)!1!} a^{m-1}_0 a_1 \delta_{an+1} \\ &+ (\mu + 2\nu) \bigg\{ \frac{m!}{(m-1)!1!} a^{m-1}_0 a_2 + \frac{m!}{(m-2)!2!} a^{m-2}_0 a^2_1 \bigg\} \delta_{an+2} \\ &+ (\mu + 3\nu) \bigg\{ \frac{m!}{(m-1)!1!} a^{m-1}_0 a_3 + \frac{m!}{(m-2)!1!1!} a^{m-2}_0 a_1 a_2 \\ &\qquad \qquad \qquad \qquad \qquad \qquad \quad \;\; + \frac{m!}{(m-3)!3!} a^{m-3}_0 a^3_0 \bigg\} \delta_{an+3} \\ &+ \ldots \bigg], \end{align} $$

the coefficient of $$a^{k_0}_0 a^{k_1}_1 a^{k_1}_1 \ldots$$ 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

$$ (4, 1; 2, 1) = \frac{1}{2} \bigg\{ 4a^2_0 \delta_{a1} + 10a_0 a_1 \delta_{a2} + 6(2a_0a_2 + a^2_1) \delta_{a3} + \ldots \bigg\}. $$

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

$$ \begin{align} \mu_1 &= (m^1+ m - 1) \bigg\{ \frac{\mu^1}{m^1} (\mu + n^1 \nu) - \frac{\mu}{m} (\mu^1 + n \nu^1) \bigg\}, \\ \nu_1 &= (n^1 - n) \nu^1 \nu + \frac{m - 1}{m^1} \mu^1 \nu - \frac{m^1 - 1}{m} \mu \nu^1, \\ m_1 &= m^1 + m - 1, \\ n_1 &= n^1 + n, \end{align} $$

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 $$\tfrac{a^s_1}{s!}, \tfrac{a^s_2}{s!}, \tfrac{a^s_3}{s!}, \ldots$$ at pleasure; then, taking $$n$$ equal to $$\infty,$$ we may write

$$ 1 + a_1x + a_2x^2 + \ldots = (1 + \rho_1x) (1+\rho_2x) \ldots = e^{a_{1^x}} = e^{a_{2^x}} = e^{a_{3^x}} = \ldots $$

where

$$ a_s = \sum \rho_1 \rho_2 \ldots \rho_3 = \tfrac{a^s_1}{s!}, \tfrac{a^s_2}{s!}, \tfrac{a^s_3}{s!}, \ldots. $$

Further, let

$$ 1 + b_1x + b_2x^2 + \ldots + b_mx^m = (1 + \sigma_1x) (1 + \sigma_2x) \ldots (1 + \sigma_mx); $$

so that

$$ \begin{align} 1 + a_1 \sigma_1 + a_2 \sigma^2_1 + \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^2_m + \ldots = (1 + \rho_1 \sigma_m) (1 + \rho_2 \sigma_m) \ldots &= e^{\sigma_m a_m}; \end{align} $$

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

$$ 1 + a_1[1] + a^2_1[1^2] + a_2[2] + a^3_1[1^3] + a_1a_2[21] + a_3[3] + \ldots $$ $$ + a_{p_1} a_{p_2} a_{p_3} \ldots a_{p_m} \big[ p_1 p_2 p_3 \ldots p_m \big] + \ldots $$ $$ \begin{align} &= 1 + b_1(1) + b^2_1(1^2) + b_2(2) + b^3_1(1^3) + b_1b_2(21) + b_3(3) + \ldots \\ &\qquad \quad \; + b^{q_1}_1 b^{q_2}_2 b^{q_3}_3 \ldots b^{q_m}_m (m^{q_m} m-1^{q_{m-1}} \ldots 2^{q_2}1^{q_1}) + \ldots \\ &= e^{\sigma_1 a_1 + \sigma_2 a_2 .. + \sigma_m a_m}. \end{align} $$

Expanding the 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 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

$$ e^{\sigma_1 a_1 + \sigma_2 a_2} = e^{\sigma_1 (a_1 - a_2)} = \ldots + \frac{1}{4!} \sigma^4_1 (a_1 - a_2)^4 + \ldots, $$

and since $$b^2_2 = \sigma^4_1$$ we must have

$$ \begin{align} (2^2) &= \frac{1}{24} (a_1 - a_2)^4 = \frac{1}{24} (a^4_1 + a^4_2) - \frac{1}{6} (a^3_1 a_2 + a_1 a^3_2) + \frac{1}{4} a^2_1 a^2_2 \\ &= 2a_4 - 2a_1a_3 + a^2_2 \end{align} $$

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

$$ \therefore (m^s) = \frac{1}{ms!} (\sigma_1 a_1 + \sigma_2 a_2 + \ldots + \sigma_m a_m)^{sm}, $$