COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

,
I
The computation of irreducible representations
of finite groups of order 2”, n 4 6’
P. G. RUUD and R. KEOWN
1. Introduction. Most textbooks and monographs on group representation
theory include the statement that the construction of the irreducible
representations of a particular group or family of groups is an art rather
than a science. This paper is a contribution to the art in the case of 2-groups
of order 2”, n < 6. The construction is based on the definitions of these
groups given in the monumental work of Hall and Senior [5]. The authors
and their colleagues have computed a representative element from each
one of the classes of equivalent irreducible representations in the case of
each class of isomorphic 2-groups of order 2” (n -= 6) and of numerous
groups of order 64. An omission in the original program, whose correction
is now believed understood, prevented the successful calculation of all
of the representations of the groups of order 64.
This collection of 2-groups contains many abelian and metacyclic
groups for which a general theory of their representations exists. However,
there are many 2-groups in the collection of Hall and Senior for which
such a theory is not currently available. This paper is a description of a
method of computation rather than a theory of the representations of
2-groups. The calculation does not employ trial-and-error or iteration
procedures.
A monomial representation of a finite group G is a matrix representation
T of G such that each matrix T(g), g c G, contains exactly one non-zero
entry in each row and column. An induced monomial representation of G
is any induced representation UG where U is a one-dimensional representation
of some subgroup. It is known, see Curtis and Reiner ([3], pages
314 and 356), that every irreducible K-representation of a finite nilpotent
group G is an induced monomial representation. Every finite 2-group is
known to be nilpotent. The defining relations of Hall and Senior provide
an ascending normal series of the form
(1) cH,c . . . CH, = G, (1.1)
t This work was a collaboration between the authors and R. F. Hansen, E. R.
McCarthy and D. L. Stambaugh.
205