The Complexity of Group Algebra Computations

In this paper we study the computation of a set of bilinear forms associated with a finite group. In the case of a cyclic group these bilinear forms represent the well-knowrl circular convolution of two sequences and can be evaluated efficiently using the fast finite Fourier transform. WC shall use a similar technique in the case of a general finite group and will apply it to calculate gror;p algebra products. Let G den&e a finite group of order y1 and let A denote an assot:iative algebra defined over the complex field C. T!le group algebra AG consists of the set of formal sums C g&+g where the coeficients ug belong to A. and in which addition, scalar multiplication and multiplication of elements are defined in the following natural way: c agg-t c b,g= c be+h-k, (1) c agg c b,,h = c c,g, where c, = c a,b,-1,. f3EG h EG gEG XEG (2) (3) In these formulae ag, bg E A a The formula for the sequence {c,}~~~ in (3) is called the convolution of sequences {6Zg}pEG and {b&G. We are concerned with eficient computations within AG. Of the three oplera-tions above (I) and (2) can clearly be accomplished in time O(n) whereas (3) apparently requires time O(n'). e therefore kVestigate faster ways of corny;. :.3ng the convolution. Our techniqu are likely to have mos ication in gro putatiorls where a large number cf group al ra elements ar computed from an initial small number of elemen such as those defined by the c,: g a set of bilinear forms 205