Page:EB1911 - Volume 01.djvu/665

II. The Theory Of Symmetric Functions

Consider $n$ quantities $a_{1},a_{2},a_{3},\dots a_{n}$.

Every rational integral function of these quantities, which does not alter its value however the $n$ suffixes $1,2,3,\dots n$ be permuted, is a rational integral symmetric function of the quantities. If we write $(1+a_{1}x)(1+a_{2}x)\dots (1+a_{n}x)=1+a_{1}x+a_{2}x^{2}+\dots +a_{n}x^{n}$, $a_{1},a_{2},\dots a_{n}$ are called the elementary symmetric functions.

$a_{1}=a_{1}+a_{2}+\dots +a_{n}=\Sigma a_{1}$
$a_{2}=a_{1}a_{2}+a_{2}a_{3}+a_{2}a_{3}+\dots =\Sigma a_{1}a_{2}$
⋅⋅⋅⋅⋅
$a_{n}=a_{1}a_{2}a_{3}\dots a_{n}$

The general monomial symmetric function is

$\Sigma a_{1}^{p_{1}}a_{2}^{p_{2}}a_{3}^{p_{3}}\dots a_{n}^{p_{n}}$,

the summation being for all permutations of the indices which result in different terms. The function is written

$(p_{1}p_{2}p_{3}\dots p_{n})$

for brevity, and repetitions of numbers in the bracket are indicated by exponents, so that $(p_{1}p_{1}p_{2})$ is written $(p_{1}^{2}p_{2}).$ The weight of the function is the sum of the numbers in the bracket, and the degree the highest of those numbers.

Ex. gr. The elementary functions are denoted by

$(1),(1^{2}),(1^{3}),\dots (1^{n})$,

are all of the first degree, and are of weights $1,2,3,\dots n$ respectively.

Remark.—In this notation $(0)=\Sigma a_{1}^{0}={\tbinom {n}{1}}$; $(0^{2})=\Sigma a_{1}^{0}a_{2}^{0}={\tbinom {n}{2}}$; ... $(0^{s})={\tbinom {n}{s}}$, &c. The binomial coefficients appear, in fact, as symmetric functions, and this is frequently of importance.

The order of the numbers in the bracket $(p_{1}p_{2}\dots p_{n})$ is immaterial; we may therefore always place them, as is most convenient, in descending order of magnitude; the numbers then constitute an ordered partition of the weight $w$, and the leading number denotes the degree.

The sum of the monomial functions of a given weight is called the homogeneous-product-sum or complete symmetric function of that weight; it is denoted by $h_{w}$; it is connected with the elementary functions by the formula

${\frac {1}{1-a_{1}x+a_{2}x^{2}-a_{3}x^{3}+\dots }}=1+h_{1}x+h_{2}x^{2}+h_{3}x^{3}+\dots$,

which remains true when the symbols $a$ and $h$ are interchanged, as is at once evident by writing $-x$ for $x$. This proves, also, that in any formula connecting $a_{1},a_{2},a_{3},\dots$ with $h_{1},h_{2},h_{3},\dots$ the symbols $a$ and $h$ may be interchanged.

Ex. gr, from $h_{2}=a_{1}^{2}-a_{2}$ we derive $a_{2}=h_{1}^{2}-h_{2}$.

The function $\Sigma a_{1}^{p_{1}}a_{2}^{p_{2}}\dots a_{n}^{p_{n}}$ being as above denoted by a partition of the weight, viz. $(p_{1}p_{2}\dots p_{n})$, it is necessary to bring under view other functions associated with the same series of numbers: such, for example, as

$\Sigma a_{1}^{p_{1}}a_{2}^{p_{3}}\Sigma a_{1}^{p_{2}}a_{2}^{p_{4}}\dots a_{n-2}^{p_{n-2}}=(p_{1}p_{3})(p_{2}p_{4}\dots p_{n-2})$.

The expression just written is in fact a partition of a partition, and to avoid confusion of language will be termed a separation of a partition. A partition is separated into separates so as to produce a separation of the partition by writing down a set of partitions, each separate partition in its own brackets, so that when all the parts of these partitions are reassembled in a single bracket the partition which is separated is reproduced. It is convenient to write the distinct partitions or separates in descending order as regards weight. If the successive weights of the separates $w_{1},w_{2},w_{3},\dots$ be enclosed in a bracket we obtain a partition of the weight $w$ which appertains to the separated partition. This partition is termed the specification of the separation. The degree of the separation is the sum of the degrees of the component separates. A separation is the symbolic representation of a product of monomial symmetric functions. A partition, $(p_{1}p_{1}p_{1}p_{2}p_{2}p_{3})=(p_{1}^{3}p_{2}^{2}p_{3})$, can be separated in the manner $(p_{1}p_{2})(p_{1}p_{2})(p_{1}p_{3})=(p_{1}p_{2})^{2}(p_{1}p_{3})$, and we may take the general form of a partition to be $(p_{1}^{\pi _{1}}p_{2}^{\pi _{2}}p_{3}^{\pi _{3}}\dots )$ and that of a separation $({\text{J}}_{1})^{j_{1}}({\text{J}}_{2})^{j_{2}}({\text{J}}_{3})^{j_{3}}\dots$ when ${\text{J}}_{1},{\text{J}}_{2},{\text{J}}_{3}\dots$ denote the distinct separates involved.

Theorem.— The function symbolized by $(n)$, viz. the sum of the n^th powers of the quantities, is expressible in terms of functions which are symbolized by separations of any partition $(n_{1}^{\nu _{1}}n_{2}^{\nu _{2}}n_{3}^{\nu _{3}}\ldots )$ of the number $n$. The expression is—

$(-)^{\nu _{1}+\nu _{2}+\nu _{3}+\ldots }{\frac {(\nu _{1}+\nu _{2}+\nu _{3}+\dots -1){\text{!}}}{\nu _{1}{\text{!}}\nu _{2}{\text{!}}\nu _{3}{\text{!}}\ldots }}(n)$
$=\sum (-)^{j_{1}+j_{2}+j_{3}+\ldots }{\frac {(j_{1}+j_{2}+j_{3}+\dots -1){\text{!}}}{j_{1}{\text{!}}j_{2}{\text{!}}j_{3}{\text{!}}\ldots }}({\text{J}}_{1})^{j_{1}}({\text{J}}_{2})^{j_{2}}({\text{J}}_{3})^{j_{3}}\dots$,

$({\text{J}}_{1})^{j_{1}}({\text{J}}_{2})^{j_{2}}({\text{J}}_{3})^{j_{3}}\dots$ being a separation of $(n_{1}^{\nu _{1}}n_{2}^{\nu _{2}}n_{3}^{\nu _{3}}\ldots )$ and the summation being in regard to all such separations. For the particular case $(n_{1}^{\nu _{1}}n_{2}^{\nu _{2}}n_{3}^{\nu _{3}}\ldots )=(1^{n})$

$(-)^{n}{\frac {1}{n}}(n)=\sum (-)^{j_{1}+j_{2}+j_{3}+\ldots }{\frac {(j_{1}+j_{2}+j_{3}+\dots -1){\text{!}}}{j_{1}{\text{!}}j_{2}{\text{!}}j_{3}{\text{!}}\ldots }}(1_{1})^{j_{1}}(1_{2})^{j_{2}}(1_{3})^{j_{3}}\dots$

To establish this write—

$1+\mu {\text{X}}_{1}+\mu ^{2}{\text{X}}_{2}+\mu ^{3}{\text{X}}_{3}+\dots ={\underset {a}{\text{II}}}(1+\mu a_{1}x_{1}+\mu ^{2}a_{1}^{2}x_{2}+\mu ^{3}a_{1}^{3}x_{3}+\dots )$,

the product on the right involving a factor for each of the quantities $a_{1},a_{2},a_{3}\dots$, and $\mu$ being arbitrary.

Multiplying out the right-hand side and comparing coefficients

${\text{X}}_{1}=(1)x_{1}$,
${\text{X}}_{2}=(2)x_{2}+(1^{2})x_{1}^{2}$,
${\text{X}}_{3}=(3)x_{3}+(21)x_{2}x_{1}+(1^{3})x_{1}^{3}$,
${\text{X}}_{4}=(4)x_{4}+(31)x_{3}x_{1}+(2^{2})x_{2}^{2}+(21^{2})x_{2}x_{1}^{2}+(1^{4})x_{1}^{4}$,
⋅⋅⋅⋅⋅⋅⋅
${\text{X}}_{m}=\Sigma (m_{1}^{\mu _{1}}m_{2}^{\mu _{2}}m_{3}^{\mu _{3}}\dots )x_{m_{1}}^{\mu _{1}}x_{m_{2}}^{\mu _{2}}x_{m_{3}}^{\mu _{3}}\dots$,

the summation being for all partitions of $m$.

Auxiliary Theorem.—The coefficient of $x_{l_{1}}^{\lambda _{1}}x_{l_{2}}^{\lambda _{2}}x_{l_{3}}^{\lambda _{3}}\dots$ in the product ${\frac {{\text{X}}_{m_{1}}^{\mu _{1}}{\text{X}}_{m_{2}}^{\mu _{2}}{\text{X}}_{m_{3}}^{\mu _{3}}\dots }{\mu _{1}{\text{!}}\mu _{2}{\text{!}}\mu _{3}{\text{!}}\dots }}$ is $\sum {\frac {({\text{J}}_{1})^{j_{1}}({\text{J}}_{2})^{j_{2}}({\text{J}}_{3})^{j_{3}}\dots }{j_{1}{\text{!}}j_{2}{\text{!}}j_{3}{\text{!}}\dots }}$ where $({\text{J}}_{1})^{j_{1}}({\text{J}}_{2})^{j_{2}}({\text{J}}_{3})^{j_{3}}\dots$ is a separation of $(l_{1}^{\lambda _{1}}l_{2}^{\lambda _{2}}l_{3}^{\lambda _{3}}\dots )$ of specification $(m_{1}^{\mu _{1}}m_{2}^{\mu _{2}}m_{3}^{\mu _{3}}\dots )$, and the sum is for all such separations.

To establish this observe the result.

${\frac {1}{p{\text{!}}}}{\text{X}}_{3}^{p}=\sum {\frac {(3)^{\pi _{1}}(21)^{\pi _{2}}(1^{3})^{\pi _{3}}}{\pi _{1}{\text{!}}\pi _{2}{\text{!}}\pi _{3}{\text{!}}}}x_{3}^{\pi _{1}}x_{2}^{\pi _{2}}x_{1}^{\pi _{2}+3\pi _{3}}$

and remark that $(3)^{\pi _{1}}(21)^{\pi _{2}}(1^{3})^{\pi _{3}}$ is a separation of $(3^{\pi _{1}}2^{\pi _{2}}1^{\pi _{2}+3\pi _{3}})$ of specification $(3^{p})$. A similar remark may be made in respect of

${\frac {1}{\mu _{1}{\text{!}}}}{\text{X}}_{m_{1}}^{\mu _{1}},{\frac {2}{\mu _{2}{\text{!}}}}{\text{X}}_{m_{2}}^{\mu _{2}},{\frac {3}{\mu _{3}{\text{!}}}}{\text{X}}_{m_{3}}^{\mu _{3}},\dots$,

and therefore of the product of those expressions. Hence the theorem.

Now

$\log(1+\mu {\text{X}}_{1}+\mu ^{2}{\text{X}}_{2}+\mu ^{3}{\text{X}}_{3}+\dots )$
$={\underset {a}{\Sigma }}\log(1+\mu a_{1}x_{1}+\mu ^{2}a_{1}^{2}x_{2}+\mu ^{3}a_{1}^{3}x_{3}+\dots )$

whence, expanding by the exponential and multinomial theorems, a comparison of the coefficients of $\mu ^{n}$ gives

$(n)\sum (-)^{\nu _{1}+\nu _{2}+\nu _{3}+\ldots -1}{\frac {(\nu _{1}+\nu _{2}+\nu _{3}+\ldots -1){\text{!}}}{\nu _{1}{\text{!}}\nu _{2}{\text{!}}\nu _{3}{\text{!}}\ldots }}x_{n_{1}}^{\nu _{1}}x_{n_{2}}^{\nu _{2}}x_{n_{3}}^{\nu _{3}}\dots$
$=\sum (-)^{\nu _{1}+\nu _{2}+\nu _{3}+\ldots -1}{\frac {(\nu _{1}+\nu _{2}+\nu _{3}+\ldots -1){\text{!}}}{\nu _{1}{\text{!}}\nu _{2}{\text{!}}\nu _{3}{\text{!}}\ldots }}{\text{X}}_{n_{1}}^{\nu _{1}}{\text{X}}_{n_{2}}^{\nu _{2}}{\text{X}}_{n_{3}}^{\nu _{3}}\dots$

and, by the auxiliary theorem, any term ${\text{X}}_{m_{1}}^{\mu _{1}}{\text{X}}_{m_{2}}^{\mu _{2}}{\text{X}}_{m_{3}}^{\mu _{3}}\dots$ on the right-hand side is such that the coefficient of $x_{n_{1}}^{\nu _{1}}x_{n_{2}}^{\nu _{2}}x_{n_{3}}^{\nu _{3}}\dots$ in ${\frac {1}{\mu _{1}{\text{!}}\mu _{2}{\text{!}}\mu _{3}{\text{!}}\ldots }}{\text{X}}_{m_{1}}^{\mu _{1}}{\text{X}}_{m_{2}}^{\mu _{2}}{\text{X}}_{m_{3}}^{\mu _{3}}\dots$ is

$\sum {\frac {({\text{J}}_{1})^{j_{1}}({\text{J}}_{2})^{j_{2}}({\text{J}}_{3})^{j_{3}}\dots }{j_{1}{\text{!}}j_{2}{\text{!}}j_{3}{\text{!}}\ldots }}$,

where since $(m_{1}^{\mu _{1}}m_{2}^{\mu _{2}}m_{3}^{\mu _{3}}\dots )$ is the specification of $({\text{J}}_{1})^{j_{1}}({\text{J}}_{2})^{j_{2}}({\text{J}}_{3})^{j_{3}}\dots$, $\mu _{1}+\mu _{2}+\mu _{3}+\dots =j_{1}+j_{2}+j_{3}+\dots$. Comparison of the coefficients of $x_{n_{1}}^{\nu _{1}}x_{n_{2}}^{\nu _{2}}x_{n_{3}}^{\nu _{3}}\dots$ therefore yields the result

$(-)^{\nu _{1}+\nu _{2}+\nu _{3}+\ldots }{\frac {(\nu _{1}+\nu _{2}+\nu _{3}+\ldots -1){\text{!}}}{\nu _{1}{\text{!}}\nu _{2}{\text{!}}\nu _{3}{\text{!}}\ldots }}(n)$
$=\sum (-)^{j_{1}+j_{2}+j_{3}+\ldots }{\frac {(j_{1}+j_{2}+j_{3}+\ldots -1){\text{!}}}{j_{1}{\text{!}}j_{2}{\text{!}}j_{3}{\text{!}}\ldots }}({\text{J}}_{1})^{j_{1}}({\text{J}}_{2})^{j_{2}}({\text{J}}_{3})^{j_{3}}\dots$,

for the expression of $\Sigma a^{n}$ in terms of products of symmetric functions symbolized by separations of $(n_{1}^{\nu _{1}}n_{2}^{\nu _{2}}n_{3}^{\nu _{3}}\dots )$.

Let $(n)_{a},(n)_{x},(n)_{\text{x}}$ denote the sums of the n^th powers of quantities whose elementary symmetric functions are $a_{1},a_{2},a_{3},\dots$; $x_{1},x_{2},x_{3},\dots$; ${\text{X}}_{1},{\text{X}}_{2},{\text{X}}_{3},\dots$ respectively: then the result arrived at above from the logarithmic expansion may be written

$(n)_{a}(n)_{x}=(n){\text{x}}$,

exhibiting $(n)_{\text{x}}$ as an invariant of the transformation given by the expressions of ${\text{X}}_{1},{\text{X}}_{2},{\text{X}}_{3},\dots$ in terms of $x_{1},x_{2},x_{3},\dots$.

The inverse question is the expression of any monomial symmetric function by means of the power functions $(r)=s_{r}$.

Theorem of Reciprocity.—If

${\text{X}}_{m_{1}}^{\mu _{1}}{\text{X}}_{m_{2}}^{\mu _{2}}{\text{X}}_{m_{3}}^{\mu _{3}}\dots =\dots +\theta (s_{1}^{\sigma _{1}}s_{2}^{\sigma _{2}}s_{3}^{\sigma _{3}}\dots )x_{l_{1}}^{\lambda _{1}}x_{l_{2}}^{\lambda _{2}}x_{l_{3}}^{\lambda _{3}}\dots +\dots$,

where $\theta$ is a numerical coefficient, then also

${\text{X}}_{s_{1}}^{\sigma _{1}}{\text{X}}_{s_{2}}^{\sigma _{2}}{\text{X}}_{s_{3}}^{\sigma _{3}}\dots =\dots +\theta (m_{1}^{\mu _{1}}m_{2}^{\mu _{2}}m_{3}^{\mu _{3}}\dots )x_{l_{1}}^{\lambda _{1}}x_{l_{2}}^{\lambda _{2}}x_{l_{3}}^{\lambda _{3}}\dots +\dots$.

We have found above that the coefficient of $(x_{l_{1}}^{\lambda _{1}}x_{l_{2}}^{\lambda _{2}}x_{l_{3}}^{\lambda _{3}}\dots )$ in the product ${\text{X}}_{m_{1}}^{\mu _{1}}{\text{X}}_{m_{2}}^{\mu _{2}}{\text{X}}_{m_{3}}^{\mu _{3}}\dots$ is

${\mu _{1}{\text{!}}\mu _{2}{\text{!}}\mu _{3}{\text{!}}\ldots }\sum {\frac {({\text{J}}_{1})^{j_{1}}({\text{J}}_{2})^{j_{2}}({\text{J}}_{3})^{j_{3}}\dots }{j_{1}{\text{!}}j_{2}{\text{!}}j_{3}{\text{!}}\ldots }}$,

the sum being for all separations of $l_{1}^{\lambda _{1}}l_{2}^{\lambda _{2}}l^{\lambda _{3}}\dots )$ which have the specification $(m_{1}^{\mu _{1}}m_{2}^{\mu _{2}}m^{\mu _{3}}\dots )$. We can multiply out this expression so as to obtain a series of monomials of the form $\theta (s_{1}^{\sigma _{1}}s_{2}^{\sigma _{2}}s_{3}^{\sigma _{3}}\dots )$. It can be shown that the number $\theta$ enumerates distributions of a certain nature defined by the partitions $(m_{1}^{\mu _{1}}m_{2}^{\mu _{2}}\dots )$, $(s_{1}^{\sigma _{1}}s_{2}^{\sigma _{2}}\dots )$,