COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

110 C. Brott and J. Neubiiser
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The characters of the Weyl group E,
J. S. FRAME
1. Introduction. The group F of order 192*10! = 2143V7 = 696,729,600
whose 112 absolutely irreducible characters (all rational) are described in
this paper is isomorphic to the Weyl group ES. The group F itself is described
by Coxeter [l] as the &dimensional group 3[4,2*11 of symmetries of
Gosset’s semi-regular polytope 421, and it is the largest of the irreducible
finite groups generated by reflections. Its factor group A = F/C with respect
to its center C = {I, -I} is the orthogonal group of half the order
investigated by Hamill [7] as a collineation group and by Edge [2] as the
group A of automorphisms of the non-singular quadric consisting of 135
points of a finite projective space [7]. The simple group denoted FH(8,2)
by Dickson is a subgroup A+ of index 2 in A = F/C.
The 8-dimensional orthogonal representation of F, called 8, below,
contains a monomial subgroup M of order 2’8 !, consisting of the products
of 8 ! permutation matrices by 2’ involutory diagonal matrices of determinant
1. There are 64 right cosets of M in the double coset MRM of M generated
by the involution R:
1 1 1 1
14-E - E
R= where E = + 1 1 1 1 =E2
(1.1)
[ - E 14-E1 1 1 1 1
i 1 1 1 1I
Here 14 is the 4X4 identity matrix. Each matrix in MRM has 7 entries
f l/4, and one entry & 314 in each row and column, and the product of the
entries in any row or column is negative. The remaining 70 right cosets of
M in F lie in the double coset MQM where
Q = RR’ = [‘7E” -9 [‘-I I-;] = [‘-T I-“zE] (1.2)
For each row or column of a matrix in MQM there are four entries 0 and
four entries f l/2 and the product of the four signs has a common value for
the eight rows, and a common value for the eight columns.
If the signs in row 8 and then in column 8 of all the 8 ! permutation matrices
of Mare changed, the resulting group isomorphic with Ss can be combined
with the matrix R of (1. l), acting in the role of the transposition (8 9),
111