COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

112 J. S. Frame The characters of the WeyI group Es 113
to produce a subgroup S of F of index 1920 isomorphic to the symmetric
group S9.
We denote the 112 irreducible characters of F by their degrees marked
with a subscript X, y, z, or w to indicate one of four types. The symbols 700,
and 700,, denote two different characters of degree 700 and type X. The 67
characters of F that are irreducible characters of F/C include 27 associated
pairs classified as type x, and 13 self-associated characters classified as
type y. The remaining 45 faithful characters of F include 20 associated pairs
classified as type z and 5 self-associated characters of type w. Pairs of associated
characters are equal for elements of the even subgroups F+ of For A+
of A, and have opposite signs in the odd coset of F+ of A+, while selfassociated
characters vanish in these odd cosets.
We also classify the classes into four types a, b, c, d as follows. Of the 40
classes of elements of A in A+, the 25 classes C, of type a split into pairs
of classes C, and - C, in Fin which an element of I;+ is not conjugate to
its negative, whereas 15 classes C, of type b do not split in F. Of the remaining
27 classes of A in the odd coset of A +, 20 classes C, split into pairs C,
and - C, in F with elements not conjugate to their negatives, and 7 classes
C, of A do not split in F. This classification enables us to partition the
112 x 112 character table as follows.
Character
blocks of A
ixdi-xd 0 / 7
Class Character blocks of F
No. type No.
/
c. 1 25 x, x, Y, Z, Z, W,
I
1 -c. /I 25 1 -x2 / x. Y, I-z. -z, -w.
(1.3)
k, 20 x, I -x,
c,/ 71~~ i-x, li -i? i? ii /
27 + 27+13=67 112 = 27 + 27 + 13 + 20 + 20 + 5
All the 112 x 112 entries of the character table may thus be displayed in
four square blocks totaling 402 + 272 + 252 + 202 = 3354 entries, of which
the first two blocks describe characters of type x or y of A in its even and
odd classes, and the last two describe the faithful characters of F of types
z and w in its even and odd classes. To check orthogonality by rows in
these subtables all products involving x’s or Z’s must be doubled in
forming scalar products.
All the faithful irreducible characters of F of type z and w are found
among the irreducible constituents of the odd Kronecker powers of the
fundamental character S,, whereas the characters of F/C of types x and
y are constituents of even Kronecker powers. To each partition (A) of
m corresponds a Schur character {A} which is irreducible for the general
linear group GL (8, C) containing a Murnaghan character [ill which is
irreducible for the infinite g-dimensional real orthogonal group G [9]. For
m = 1, 2, 3, 4, all these characters except [4] are also irreducible for the
finite subgroup F of G, but for m w 4 many are reducible for F and must
be split by other means.
2. The classes. The class symbol 1” 26 3”. . . , where 8 = Q. + 2bf 3y + . . . ,
is commonly used to describe a class of permutations having a l-cycles,
,J 2-cycles, y 3-cycles, etc. We extend it to other classes of the monomial
group M by denoting by it or k-l(2k) a k-cycle with an odd number of
minus signs, whose eigenvalues are those (2k)th roots of unity which are
not kth roots of unity. Similarly the symbol 1” 28 3Y. . . , where 8 = CI +2g+
3y+. . ., in which one or more of the exponents is negative, denotes a
class of matrices in 8, whose eigenvalues consist of a l’s, plus /I pairs 1
and - 1, plus y complete sets of -cube - roots of unity, etc., and a symbol K
is equivalent to k-l(2.k). Thus 10/Z, or 2 4-1(10)-1(20), denotes a class
of elements of order 20 whose eigenvalues are the eight primitive 20th
roots of unity that are not 10th roots nor 4th roots of unity.
For a matrix in the representation 8, some power of which is an involution
of trace 0 and type 24, these symbols do not specify the class uniquely.
One class, denoted 240, contains diagonal monomial matrices of type l4 i4
and also permutation matrices of M with four 2-cycles and no negative
signs. These each commute with 21333 elements of A, and correspond
to class 3 (called 1-124v) in Ho. This class is not represented in the symmetric
subgroup &. Another class, denoted 24u or Z 23, contains elements
of type 1 24 in &, but these are represented in M by four 2-cycles one of
which has both its signs changed. Type l2 T222 of M also contributes to
class 24u in A. Each element commutes with 2113 elements of A. The letters
v or u follow the class symbols for matrices some power of which is
in class 2% or 2%. We also use 2: and u to distinguish the class 224~ that
contains permutations in M from the class 224u which represents permutations
of Ss whose image in M has two minus signs in one 2-cycle. Thus the
classes containing the squares, cubes, or other powers of any element of
F can be read directly from the class symbol.
We obtain the 67 classes of A = F/C directly by using the classes of two
important subgroups of index 120 and 135 and then finding 8-dimensional
matrices that represent the missing nine classes, rather than by using the
geometrical arguments of Hamill [7] and Edge [2]. Of the 67 classes
of A, 25 even classes (type a) and 20 odd classes (type c) split in F to produce
two classes each, whereas 15 even classes (type b) and 7 odd classes
(type d) represent single classes of F.
(1) A subgroup H of index 120 in A is isomorphic to the Weyl group of
type E7, which is the direct product of its center C = {I, -Z} and the group