COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.
COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.
COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.
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94<br />
hence<br />
C<br />
*<br />
a@ = C atma,<br />
t<br />
and so x = a,m,.<br />
John McKay Construction of characters of a Jinite group 95<br />
We have the following situation.<br />
The entries in the ith column of M are the eigenvalues of A’ and the<br />
rows of M correspond to the common eigenvectors normalized so that<br />
rn; = 1. If the entries in the column of M corresponding to the eigenvalues<br />
of A’ are all distinct then the whole matrix M can be determined from the<br />
matrix A’ alone. This, however, is not usually the case. An extreme case<br />
occurs when G = Z2XZ2X.. . X22, the direct product of n copies. of the<br />
cyclic group Zz. Here each column of M (except the first) has entries f 1<br />
each sign occurring 2”-l times.<br />
A method is described to overcome the difficulty inherent in multiple<br />
eigenvalues.<br />
The idea of the method is that if a matrix has distinct eigenvalues, then<br />
the eigenvectors are determinate (each to within a scalar multiple).<br />
Let Ui, i=l,2 ,..., r, be indeterminates and consider the matrix<br />
CD = c UiA’ which has eigenvalues c Uimb 1 e s =S r.<br />
I I<br />
By choosing suitable values for the indeterminates we can arrange that the<br />
eigenvalues are distinct; if so, the eigenvectors of @ are just ms, 1 < s =S r.<br />
For computational purposes we replace the indeterminates by random<br />
numbers. We may then associate a probability to the numerical separability<br />
of the eigenvalues.<br />
We require that for each p =l= CJ (= 1, 2, . . . , r) the eigenvalues corresponding<br />
to rnp and rnq should be separable, i.e.<br />
1 C Bimf-C 8imf 1 =- E(t) for all p * fJ = 1, 2, . . . , r,<br />
i<br />
where the 6, are chosen from some suitable normalized distribution and<br />
e(t) is a small number dependent on the accuracy of the computer.<br />
The largest eigenvalue of A’ is 1. We introduce a normalizing factor of<br />
r-l and choose 8i to be the coordinates of a point on an n-dimensional<br />
hyperellipsoid of semi-axes hz:1/2 so that 6i hii are points distributed on<br />
the hypersurface of an n-dimensional sphere.<br />
A detailed error analysis is hindered because of lack of adequate prior<br />
knowledge of the mp.<br />
The numerical method for solving the eigenvalue problem is the accelerated<br />
QR method [3]. The eigenvectors are found by inverse iteration.<br />
Derivation of the algebraic form from the numerical. By normalizing<br />
the solution vectors of di so that the first component is unity, we have the<br />
numerical values of mf. To find the dimension of the representation we<br />
.use the relation, derivable from the row orthogonality relations<br />
By multiplying rnf by d, and dividing by hi we find the numerical characters.<br />
As decimal numbers, these are of little interest; we would prefer them<br />
in an algebraic form.<br />
For a representation of degree d over the complex field and an element<br />
of period p,<br />
x(x) = i 095, OGtiGp-1,<br />
i=l<br />
where w is a primitive pth root of unity. Let xN(x) be a numerical approxi-<br />
mation t0 x(X). We may rearrange the terms so that tl =S tz < . . . =S td.<br />
There are ( d+z- ‘) such sequences. We could generate the sequence sys-<br />
tematically starting at 0, 0, . . . , 0 and ending atp - 1, p- 1, . . . , p - 1, and<br />
examine the value of the cyclic sum each yields. We can improve on this.<br />
The problem may be visualized geometrically in the complex plane as<br />
follows :<br />
Each root of unity may be represented by a unit vector which lies<br />
at an angle which is a multiple of b/p to the horizontal. We form a sum of<br />
these vectors by joining them up, end to end. We seek such a sum starting<br />
from the origin and reaching to x(x). We generate the sequences described<br />
above but check to see whether, after fixing the first s vectors, the distance<br />
from the sum of first s terms to X&Y) is less than d-s; if not, we<br />
alter ts.<br />
Even with the above improvement, the algebraic form of the character<br />
of an element of period p in a representation of degree d such that p, d > 10<br />
would be very time-consuming to determine, and in cases such as presented<br />
by the representations of JI, this is out of the question. The following<br />
fact may be used: among the terms of the sequences computed may be<br />
some whose sum contribution to the total is nil. Each such subsequence<br />
may be decomposed into disjoint subsequences each containing a prime<br />
number of terms. These correspond to regular pi-gons for prime p,. From a<br />
computational viewpoint this implies that, provided u > 0, we can attempt<br />
to fit X&X) with only u = d- i$lcipi (ci 2 0) terms where pi are prime<br />
divisors of p. We now compute the values that c cipi can take.<br />
If p has only one prime factor, the values assumed are multiples of<br />
that factor.<br />
Let pl, p2 be the smallest two distinct prime factors of p. All integers not<br />
less than (pr- 1) (p2- 1) are representable as clpl+c~p~ (cl, cg z= 0). In