Page:EB1911 - Volume 01.djvu/678

form of infinite order. In this case the ground forms, called also perpetuants, have been enumerated and actual representative seminvariant forms established. Putting $n$ equal to ∞, in a generating function obtained above, we find that the function, which enumerates the asyzygetic seminvariants of degree $\theta$, is

${\frac {1}{1-z^{2}.~1-z^{3}.~1-z^{4}.\ldots 1-z^{\theta }}}$

that is to say, of the weight $w$, we have one form corresponding to each non-unitary partition of $w$ into the parts 2, 3, 4,...$\theta$. The extraordinary advantage of the transformation of $\Omega$ to association with non-unitary symmetric functions is now apparent; for we may take, as representative forms, the symmetric functions which are symbolically denoted by the partitions referred to. Ex. gr., of degree 3 weight 8, we have the two forms $(3^{2}2)$, $a(2^{4})$. If we wish merely to enumerate those whose partitions contain the figure $\theta$, and do not therefore contain any power of $a$ as a factor, we have the generator

${\frac {z^{\theta }}{1-z^{2}.~1-z^{3}.~1-z^{4}.~\ldots 1-z^{\theta }}}$.

If $\theta =2$, every form is obviously a ground form or perpetuant, and the series of forms is denoted by $(2),~(2^{2}),~(2^{3}),\dots (2^{\kappa +1})\dots$. Similarly, if $\theta =3$, every form $(3^{\kappa +1}2^{\lambda })$ is a perpetuant. For these two cases the perpetuants are enumerated by

${\frac {z^{2}}{1-z^{2}}}$, and ${\frac {z^{3}}{1-z^{2}.~1-z^{3}}}$

respectively.

When $\theta =4$ it is clear that no form, whose partition contains a part 3, can be reduced; but every form, whose partition is composed of the parts 4 and 2, is by elementary algebra reducible by means of perpetuants of degree 2. These latter forms are enumerated by ${\frac {z^{4}}{1-z^{2}.~1-z^{4}}}$; hence the generator of quartic perpetuants must be

${\frac {z^{4}}{1-z^{2}.~1-z^{3}.~1-z^{4}}}-{\frac {z^{4}}{1-z^{2}.~1-z^{4}}}={\frac {z^{7}}{1-z^{2}.~1-z^{3}.~1-z^{4}}}$;

and the general form of perpetuants is $(4^{\kappa +1}~3^{\lambda +1}~2^{\mu })$.

When $\theta \geq 5$, the reducible forms are connected by syzygies which there is some difficulty in enumerating. Sylvester, Cayley and MacMahon succeeded, by a laborious process, in establishing the generators for $\theta =5$, and $\theta =6$, viz.:

${\frac {z^{15}}{1-z^{2}.~1-z^{3}.~1-z^{4}.~1-z^{5}}}$, ${\frac {z^{31}}{1-z^{2}.~1-z^{3}.~1-z^{4}.~1-z^{5}.~1-z^{6}}}$;

but the true method of procedure is that of Stroh which we are about to explain.

Method of Stroh.—In the section on “Algebraic Forms” , it was noted that Stroh considers

$(\sigma _{1}\alpha _{1}+\sigma _{2}\alpha _{2}+\ldots +\sigma _{\theta }\alpha _{\theta })^{w}$,

where $\sigma _{1}+\sigma _{2}+\ldots +\sigma _{\theta }=0$ and ${\frac {\alpha _{1}^{s}}{s{\text{!}}}}={\frac {\alpha _{2}^{s}}{s{\text{!}}}}=\ldots ={\frac {\alpha _{\theta }^{s}}{s{\text{!}}}}=a_{s}$ symbolically, to be the fundamental form of seminvariant of degree $\theta$ and weight $w$; he observes that every form of this degree and weight is a linear function of such symbolic expressions. We may write

$(1+\sigma _{1}\xi )(1+\sigma _{2}\xi )\ldots (1+\sigma _{\theta }\xi )=1+{\text{A}}_{2}\xi ^{2}+{\text{A}}_{3}\xi ^{3}+\ldots +{\text{A}}_{\theta }\xi ^{\theta }$.

If we expand the symbolic expression by the multinomial theorem, and remember that any symbolic product $\alpha _{1}^{\pi _{1}}\alpha _{2}^{\pi _{2}}\alpha _{3}^{\pi _{3}}\dots$ retains the same value, however the suffixes be permuted, we shall obtain a sum of terms, such as $w{\text{!}}~{\frac {\alpha _{1}^{\pi _{1}}}{\pi _{1}{\text{!}}}}{\frac {\alpha _{2}^{\pi _{2}}}{\pi _{2}{\text{!}}}}{\frac {\alpha _{3}^{\pi _{3}}}{\pi _{3}{\text{!}}}}\ldots \Sigma \sigma _{1}^{\pi _{1}}\sigma _{2}^{\pi _{2}}\sigma _{3}^{\pi _{3}}\ldots$, which in real form is $w{\text{!}}~a_{\pi _{1}}a_{\pi _{2}}a_{\pi _{3}}\ldots \Sigma \sigma _{1}^{\pi _{1}}\sigma _{2}^{\pi _{2}}\sigma _{3}^{\pi _{3}}\ldots$; and, if we express $\Sigma \sigma _{1}^{\pi _{1}}\sigma _{2}^{\pi _{2}}\sigma _{3}^{\pi _{3}}\ldots$ in terms of ${\text{A}}_{2},~{\text{A}}_{3},\dots$, and arrange the whole as a linear function of products of ${\text{A}}_{2},~{\text{A}}_{3},\dots$, each coefficient will be a seminvariant, and the aggregate of the coefficients will give us the complete asyzygetic system of the given degree and weight.

When the proper degree $\theta$ is < $w$ a factor $a_{0}^{w-\theta }$ must be of course understood.

Ex. gr.

${\frac {1}{2{\text{!}}}}(\sigma _{1}\alpha _{1}+\sigma _{2}\alpha _{2}+\sigma _{3}\alpha _{3}+\sigma _{4}\alpha _{4})^{2}={\frac {\alpha _{1}^{2}}{2{\text{!}}}}\Sigma \sigma _{1}^{2}+\alpha _{1}\alpha _{2}\Sigma \sigma _{1}\sigma _{2}$
$=a_{2}(-2{\text{A}}_{2})+a_{1}^{2}{\text{A}}_{2}=(a_{1}^{2}-2a_{2}){\text{A}}_{2}=(2){\text{A}}_{2}\equiv a_{0}^{2}(2){\text{A}}_{2}$.

In general the coefficient, of any product ${\text{A}}_{\pi _{1}}{\text{A}}_{\pi _{2}}{\text{A}}_{\pi _{3}}\ldots$, will have, as coefficient, a seminvariant which, when expressed by partitions, will have as leading partition (preceding in dictionary order all others) the partition $(\pi _{1}\pi _{2}\pi _{3}\dots )$. Now the symbolic expression of the seminvariant can be expanded by the binomial theorem so as to be exhibited as a sum of products of seminvariants, of lower degrees if $\sigma _{1}\alpha _{1}+\sigma _{2}\alpha _{2}+\ldots +\sigma _{\theta }\alpha _{\theta }$ can be broken up into any two portions

$(\sigma _{1}\alpha _{1}+\sigma _{2}\alpha _{2}+\ldots +\sigma _{3}\alpha _{3})+(\sigma _{s+1}\alpha _{s+1}+\sigma _{s+2}\alpha _{s+2}+\ldots +\sigma _{\theta }\alpha _{\theta })$

such that $\sigma _{1}+\sigma _{2}+\ldots +\sigma _{s}=0$, for then

$\sigma _{s+1}+\sigma _{s+2}+\ldots +\sigma _{\theta }=0$

and each portion raised to any power denotes a seminvariant. Stroh assumes that every reducible seminvariant can in this way be reduced. The existence of such a relation, as $\sigma _{1}+\sigma _{2}+\ldots +\sigma _{\theta }=0$, necessitates the vanishing of a certain function of the coefficients ${\text{A}}_{2},~{\text{A}}_{3},\ldots {\text{A}}_{\theta }$, and as a consequence one product of these coefficients can be eliminated from the expanding form and no seminvariant, which appears as a coefficient to such a product (which may be the whole or only a part of the complete product, with which the seminvariant is associated), will be capable of reduction.

Ex. gr. for $\theta =2$, $(\sigma _{1}\alpha _{1}+\sigma _{2}\alpha _{2})^{w}$; either $\sigma _{1}$ or $\sigma _{2}$ will vanish if $\sigma _{1}\sigma _{2}={\text{A}}_{2}=0$; but every term, in the development, is of the form $(222\ldots ){\text{A}}_{2}^{{\frac {1}{2}}w}$ and therefore vanishes; so that none are left to undergo reduction. Therefore every form of degree 2, except of course that one whose weight is zero, is a perpetuant. The generating function is ${\frac {z^{2}}{1-z^{2}}}$.

For $\theta =3$, $(\sigma _{1}\alpha _{1}+\sigma _{2}\alpha _{2}+\sigma _{3}\alpha _{3})^{w}$; the condition is clearly $\sigma _{1}\sigma _{2}\sigma _{3}={\text{A}}_{3}=0$, and since every seminvariant, of proper degree 3, is associated, as coefficient, with a product containing ${\text{A}}_{3}$, all such are perpetuants. The general form is $(3^{\kappa }2^{\lambda }$ and the generating function ${\frac {z^{3}}{1-z^{2}.~1-z^{3}}}$.

For $\theta =4$, $(\sigma _{1}\alpha _{1}+\sigma _{2}\alpha _{2}+\sigma _{3}\alpha _{3}+\sigma _{4}\alpha _{4})^{w}$; the condition is

$\sigma _{1}\sigma _{2}\sigma _{3}\sigma _{4}(\sigma _{1}+\sigma _{2})(\sigma _{1}+\sigma _{3})(\sigma _{1}+\sigma _{4})={\text{A}}_{4}{\text{A}}_{3}=0$.

Hence every product of ${\text{A}}_{1}$, ${\text{A}}_{2}$, ${\text{A}}_{3}$, ${\text{A}}_{4}$, which contains the product ${\text{A}}_{4}{\text{A}}_{3}$ disappears before reduction; this means that every seminvariant, whose partition contains the parts 4, 3, is a perpetuant. The general form of perpetuant is $(4^{\kappa }3^{\lambda }2^{\mu })$ and the generating function

${\frac {z^{7}}{1-z^{2}.~1-z^{3}.~1-z^{4}}}$.

In general when $\theta$ is even and $=2\phi$, the condition is

$\sigma _{1}\sigma _{2}\ldots \sigma _{2\phi }{\text{II}}(\sigma _{1}+\sigma _{2}){\text{II}}(\sigma _{1}+\sigma _{2}+\sigma _{3})\ldots {\text{II}}(\sigma _{1}+\sigma _{2}+\ldots +\sigma _{\phi })=0$;

and we can determine the lowest weight of a perpetuant; the degree in the quantities $\sigma$ is

$2\phi +{\tbinom {2\phi }{2}}+{\tbinom {2\phi }{3}}+\ldots +{\frac {1}{2}}{\tbinom {2\phi }{\phi }}=2^{2\phi -1}-1=2^{\theta -1}-1$.

Again, if $\theta$ is uneven $=2\phi +1$, the condition is

$\sigma _{1}\sigma _{2}\ldots \sigma _{2\phi +1}{\text{II}}(\sigma _{1}+\sigma _{2}){\text{II}}(\sigma _{1}+\sigma _{2}+\sigma _{3})\ldots {\text{II}}(\sigma _{1}+\sigma _{2}+\ldots +\sigma _{\phi })=0$;

and the degree, in the quantities $\sigma$, is

$2\phi +1+{\tbinom {2\phi +1}{2}}+{\tbinom {2\phi +1}{3}}+\ldots +{\tbinom {2\phi +1}{\phi }}$
$=2^{2\phi }-1=2^{\theta -1}-1$.

Hence the lowest weight of a perpetuant is $2^{\theta -1}-1$, when $\theta$ is >2. The generating function is thus

${\frac {z^{2^{\theta -1}}-1}{(1-z^{2})(1-z^{3})(1-z^{4})\ldots (1-z^{\theta })}}$.

The actual form of a perpetuant of degree $\theta$ has been shown by MacMahon to be

$(\theta ^{\kappa _{\theta }+1}{\overline {,\theta -1}}^{\kappa _{\theta -1}+1}{\overline {,\theta -2}}^{\kappa _{\theta -2}+2}{\overline {,\theta -3}}^{\kappa _{\theta -3}+4},\ldots 3^{\kappa _{3}+2^{\theta -4}},2^{\kappa _{2}})$,

$\kappa _{\theta },\kappa _{\theta -1},\ldots \kappa _{2}$ being given any zero or positive integer values.

Simultaneous Seminvariants of two Binary Forms.—Taking the two forms to be

$a_{0}x_{1}^{p}+pa_{1}x_{1}^{p-1}x_{2}+p(p-1)a_{2}x_{1}^{p-2}x_{2}^{2}+\ldots +a_{p}x_{2}^{p}$,
$b_{0}x_{1}^{q}+qb_{1}x_{1}^{q-1}x_{2}+q(q-1)b_{2}x_{1}^{q-2}x_{2}^{2}+\ldots +b_{q}x_{2}^{q}$,

every leading coefficient of a simultaneous covariant vanishes by the operation of

$\Omega _{a}+\Omega _{b}=a_{0}{\frac {d}{da_{1}}}+a_{1}{\frac {d}{da_{2}}}+\ldots +a_{p-1}{\frac {d}{da_{p}}}+b_{0}{\frac {d}{db_{1}}}+b_{1}{\frac {d}{db_{2}}}+\ldots +b_{q-1}{\frac {d}{db_{q}}}$.

Observe that we may employ the principle of suffix diminution to obtain from any seminvariant one appertaining to a $(p-1)^{ic}$ and a $q-1^{ic}$, and that suffix augmentation produces a portion of a higher seminvariant, the degree in each case remaining unaltered. Remark, too, that we are in association with non-unitary symmetric functions of two systems of quantities which will be denoted by partitions in brackets $(~~)_{a}$, $(~~)_{b}$ respectively. Solving the equation

$(\Omega _{a}+\Omega _{b})u=0$,

by the ordinary theory of linear partial differential equations, we obtain $p+q+1$ independent solutions, of which $p$ appertain to $\Omega _{a}u=0$, $\Omega _{b}u=0$; the remaining one is ${\text{J}}_{ab}=a_{0}b_{1}-a_{1}b_{0}$, the leading coefficient of the Jacobian of the two forms. This constitutes an algebraically complete system, and, in terms of its members, all seminvariants can be rationally expressed. A similar theorem holds in the case of any number of binary forms, the mixed seminvariants being derived from the Jacobians of the several pairs of forms. If the seminvariant be of degree $\theta ,~\theta '$ in the coefficients, the forms of orders $p,~q$ respectively, and the weight $w$, the degree of the covariant in the variables will be $p\theta +q\theta '-2w=\epsilon$, an easy generalization of the theorem connected with a single form.