94 hence C * a@ = C atma, t and so x = a,m,. John McKay Construction of characters of a Jinite group 95 We have the following situation. The entries in the ith column of M are the eigenvalues of A’ and the rows of M correspond to the common eigenvectors normalized so that rn; = 1. If the entries in the column of M corresponding to the eigenvalues of A’ are all distinct then the whole matrix M can be determined from the matrix A’ alone. This, however, is not usually the case. An extreme case occurs when G = Z2XZ2X.. . X22, the direct product of n copies. of the cyclic group Zz. Here each column of M (except the first) has entries f 1 each sign occurring 2”-l times. A method is described to overcome the difficulty inherent in multiple eigenvalues. The idea of the method is that if a matrix has distinct eigenvalues, then the eigenvectors are determinate (each to within a scalar multiple). Let Ui, i=l,2 ,..., r, be indeterminates and consider the matrix CD = c UiA’ which has eigenvalues c Uimb 1 e s =S r. I I By choosing suitable values for the indeterminates we can arrange that the eigenvalues are distinct; if so, the eigenvectors of @ are just ms, 1 < s =S r. For computational purposes we replace the indeterminates by random numbers. We may then associate a probability to the numerical separability of the eigenvalues. We require that for each p =l= CJ (= 1, 2, . . . , r) the eigenvalues corresponding to rnp and rnq should be separable, i.e. 1 C Bimf-C 8imf 1 =- E(t) for all p * fJ = 1, 2, . . . , r, i where the 6, are chosen from some suitable normalized distribution and e(t) is a small number dependent on the accuracy of the computer. The largest eigenvalue of A’ is 1. We introduce a normalizing factor of r-l and choose 8i to be the coordinates of a point on an n-dimensional hyperellipsoid of semi-axes hz:1/2 so that 6i hii are points distributed on the hypersurface of an n-dimensional sphere. A detailed error analysis is hindered because of lack of adequate prior knowledge of the mp. The numerical method for solving the eigenvalue problem is the accelerated QR method [3]. The eigenvectors are found by inverse iteration. Derivation of the algebraic form from the numerical. By normalizing the solution vectors of di so that the first component is unity, we have the numerical values of mf. To find the dimension of the representation we .use the relation, derivable from the row orthogonality relations By multiplying rnf by d, and dividing by hi we find the numerical characters. As decimal numbers, these are of little interest; we would prefer them in an algebraic form. For a representation of degree d over the complex field and an element of period p, x(x) = i 095, OGtiGp-1, i=l where w is a primitive pth root of unity. Let xN(x) be a numerical approxi- mation t0 x(X). We may rearrange the terms so that tl =S tz < . . . =S td. There are ( d+z- ‘) such sequences. We could generate the sequence sys- tematically starting at 0, 0, . . . , 0 and ending atp - 1, p- 1, . . . , p - 1, and examine the value of the cyclic sum each yields. We can improve on this. The problem may be visualized geometrically in the complex plane as follows : Each root of unity may be represented by a unit vector which lies at an angle which is a multiple of b/p to the horizontal. We form a sum of these vectors by joining them up, end to end. We seek such a sum starting from the origin and reaching to x(x). We generate the sequences described above but check to see whether, after fixing the first s vectors, the distance from the sum of first s terms to X&Y) is less than d-s; if not, we alter ts. Even with the above improvement, the algebraic form of the character of an element of period p in a representation of degree d such that p, d > 10 would be very time-consuming to determine, and in cases such as presented by the representations of JI, this is out of the question. The following fact may be used: among the terms of the sequences computed may be some whose sum contribution to the total is nil. Each such subsequence may be decomposed into disjoint subsequences each containing a prime number of terms. These correspond to regular pi-gons for prime p,. From a computational viewpoint this implies that, provided u > 0, we can attempt to fit X&X) with only u = d- i$lcipi (ci 2 0) terms where pi are prime divisors of p. We now compute the values that c cipi can take. If p has only one prime factor, the values assumed are multiples of that factor. Let pl, p2 be the smallest two distinct prime factors of p. All integers not less than (pr- 1) (p2- 1) are representable as clpl+c~p~ (cl, cg z= 0). In
96 John McKay Construction of characters of a jinite group 97 cases when (pi- 1) (pZ-- 1) Z- d, values not greater than u are computed directly. All integer valued characters are extracted before attempting to match terms because certain values, e.g. - 1, are time-consuming to fit. The above discussion has not taken into account that only an approximation to the numerical value of x(x) is the starting point. If two distinct values of sums of roots of unity differ by less than E in modulus, where e is the accuracy of computation of the value of x&x), then the results of the above algorithm will not necessarily be correct. A lower bound is required for the non-zero values of P(m)= ~~W~~-~cO~q O
Change language