COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

108
C. Brott and J. Neubiiser
elements of G in q, are computed and printed. The programme stops as
soon as r irreducible representations have been found. This often happens
long before all triplets N, U, V have been investigated.
4.7. In debugging the programme we have used the special version (2.1 d)
of the programme @ to generate a matrix group from the images of the
generators of G under an irreducible representation T. This must be isomorphic
to G/ker T.
5. The numerical part of the programme.
5.1. The part of the programme described so far finds only the monomial
irreducible representations and characters. For low orders there are very
few groups having non-monomial irreducible representations. Up to order
96 there are one of order 24, four of order 48, one of order 60 and two of
order 72. No method is included yet in our programme for the determination
of non-monomial irreducible representations, but at least an attempt is
made to complete the character table by a more numerical part of the programme.
For its description we define:
Then from [2], p. 235,
we have
I
WsJ)W(i) = C CijkW,Q, lei,j,s==r,
k=l
kg1 (Ciik -&&y))w~) = 0 ,
(5.1.1)
(5.1.2)
(5.1.3)
i.e. for each s the r values WV), 1 4 I 4 r, belonging to the sth character satisfy
the ra equations
k$l(Cijk-SikXj)Xk = 0, 1 =S i, j e r. (5.1.4)
To solve this system we choose a fixed j = jo and consider only the r equations
k~l(c&&-djJ&jo)x~ = 0, 1 4 i < r. (5.1.5)
Then for each s, 1~ s 4 r, the vector (IV?), . . . , WY)) is an eigenvector of the
matrix (cildr) belonging to the eigenvalue wji’. If for some j,-, this matrix has
r different eigenvalues, the eigenvectors are essentially uniquely determined
and hence must coincide up to a factor with the vectors (WY), . . . , w$@),
1 4 s 4 r. This factor is calculated from
h
From (21, 31.18,
Group characters and representations 109
(5.1.6)
we obtain the dimensions dl, . . . , d, and hence the values #).
In our programme we use the characters already obtained by the process
of induction to reduce the task of finding the eigenvalues of the (cijk) as
roots of the characteristic polynomials.
5.2. For the numerical programme we first compute the structure constants
cijke Let X1, . . . , x, be representatives of the r classes of elements conjugate
in G. Then for all k = l( 1)r and all b E Ci the number of solutions of
bx = xk in Cj is counted.
For each matrix (cu.& 2 ==j--r, its characteristic polynomial is computed
by the Hessenberg procedure [12]. Zeros of this polynomial belonging to
known characters are split off and the zeros of the remaining polynomial are
computed by the Bairstow procedure. For a simple root of this polynomial
a character is found as an eigenvector.
The numerical method described above does not work if there is a nonmonomial
character x(“) such that for each j there exists s’ + s with
$I = $‘I. This case has not yet been covered in our programme, but we
intend to replace the numerical part by a method proposed by John D.
Dixon+ which makes use of the fact that the vectors w@), l< s < r, are essentially
UniqUdy determined as common eigenvectors of all matrices (cijk),
1