COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

210 P. G. Ruud and R. Keown
3. The calculation of the irreducible representations of 290~~s. This
section discusses a systematic method of calculation of the irreducible
representations of 2-groups. Some basic definitions and results are given
to make the material more intelligible to the non-specialist.
Let t be an r-dimensional matrix representation of the subgroup H of
index k in the finite group G. Denote by [glH,. . . , gkfl the collection of
all distinct left cosets of H in G. The function T which makes correspond
to each x of G the block matrix
t’(g;1xg1) . . . t’(g;lXgk)
T(x) = . . . 3 (3.1)
ugklxgl) . . * t’(gklXgk) 1
where each t’(gilxgJ is an r X r matrix, 1 s i, j < k,
with
t’(g;lXgj) = 0, g; lxgj t H,
and
t’(g?lxgj) = t(g;lxgj), g;lXgj E H,
is an (rk X rk)-dimensional representation of G which is said to be induced
by the representation t of H. Only under special circumstances is T an
irreducible representation of G even though t is an irreducible representation
of H. Given a representation T of G, there exists a representation
t of H whose values are given by
t(h) = T(h), h E H. (3.2)
The representation t is said to be subduced by the representation T. As in
the case of induced representations, the subduced representation t of
H need not be irreducible even though the subducing representation T
of G is.
The following remarks are valid in the case where H is a normal subgroup
of G, but not always in the general case. Let t be a representation
of Hand g any element of G. The mapping tg defined by
t,(h) = wlk), h E H, (3.3)
is a representation of H which is said to be a conjugate of t. The representation
tg is said to be obtained through conjugation of t by g. The representation
t may or may not be equivalent to its conjugate tg. The set K
of all g such that tg is equivalent to t is a group called the little group of t.
A representation t which is equivalent to each of its conjugates is said to
be self-conjugate.
Let t be a representation of H subduced by the irreducible representation
T of G. The representation t is the direct sum
t = nltl-i- . . . inktk (3.4)
Irreducible representations of finite groups 211
of irreducible representations of H. The set [ti,, . . ., tij] of irreducible
representations which appear with non-zero coefficients in (3.4) is called the
orbit of T. The number of elements in an orbit is called its order. According
to a result of Clifford, the elements of an orbit are mutually conjugate
and every conjugate appears. Furthermore, each conjugate occurs the
same number of times in equation (3.4) which can be written
t = n(ti,/ . . . -i-t,). (3.5)
The number n is referred to as the multiplicity of the orbit. It is known that
every irreducible representation t of H belongs to the orbit of at least
one irreducible representation T of G.
Two irreducible representations T and T’ of G are said to be associates
if their orbits have an irreducible representation t of H in common, which
implies that their orbits coincide. An irreducible representation T of G
is said to be self-associate if its orbit is disjoint from that of every other
non-equivalent irreducible representation of G. When His a subgroup of
index two in G, the orbit of an irreducible representation T of G is either
a single, self-conjugate irreducible representation t or a pair, t and t’, of
mutually conjugate irreducible representations of H. In either case, the
multiplicity of the orbit is always one. In the second case, the representation
R of G induced by t is an irreducible representation of G equivalent
to T. In the first, the representation R induced by t is equivalent to the direct
sum, Tf T’, where T’ is the only other associate of T.
These results suggest that the irreducible representations of 2-groups
can be computed by induction if a convenient algorithm can be developed
for reducing the induced representations arising from self-conjugate representations
of a subgroup of index two. The remainder of this section is
devoted to the development of such an algorithm and the description of a
practical scheme of induction. We begin with an observation concerning
the one-dimensional representations of any 2-group given by the Hall-
Senior defining relations.
THEOREM. Let [bl,. . . , b,] be the Hall-Senior generators of a 2-group
G of order 2”, 1 ==n < 6. Let C be the set [cl,. . ., cd] of those generators,
defined inductively, which either appear as a commutator in the defining
relations or as one factor of a commutator given by the defining relations
in which the other factors are already in C. Each set K of complex numbers
[kl, . . ., k,] which satisfy the defining relations, ki having the value 1 for
each element ci of C, determines a one-dimensional representation of G
which is specified by its values on the generators, namely,
TK(bi) = ki, 1 G i < n. (3.6)
Each one-dimensional, irreducible representation T of G corresponds to
exactly one such K. The derived group G’ of G is generated by the elements
of the set C. Each maximal subgroup M of G is the kernel of a representation