COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

130 J. S. Frame
Counting the associates of characters already computed, the list of 112
characters of the largest of the five Weyl groups GO, F4, ES, ET, and EB is
now complete. In Table 1 we list symbols for the 67 classes of A, together
with the orders of centralizers of an element, the class numeral used by
Hamill [7], and Edge [2], and the characters of this class for the permutation
representations induced by the subgroups H of index 120, A4 of
index 135, and S of index 960. In F the centralizer orders must be doubled
for classes of type a or c, and each odd cycle symbol k or E replaced by
iz or k to obtain the additional 45 classes of F. As explained above in
(1.3) the information for the complete 112 x 112 character table is conveyed
by four square blocks of dimensions 40 for [Xob, Y,], 27 for [X,,] which
include the 67 characters of A, and 25 for [Z,, W,], 20 for [Z,] which
include the characters of faithful representations of F.
Reference should also be made to Dye’s papers (10, 111, which appeared
after this paper was submitted.
REFERENCES
1. H. S. M. COXETER: Regular Polytopes (Macmillan, 1963).
2. W. L. EDGE: An orthogonal group of order 213-35.52.7. Annali di Matematica (4)
61 (1963), l-96.
3. J. S. FRAME: The degrees of the irreducible representation s of simply transitive
permutation groups. Duke Math. Journal 3 (1937), 8-17.
4. J. S. FRAME: The classes and representations of the groups of 27 lines and 28 bitangents.
Annali di Matematicu (4) 32 (1951), 83-169.
5. J. S. FRAME: An irreducibIe representation extracted from two permutation groups.
Annals of Math. 55 (1952), 85-100.
6. J. S. FRAME: The constructive reduction of finite group representations. Proc. of
Symposia in Pure Math. (Amer. Math. Sot.) 6 (1962), 89-99.
7. C. M. HAMILL: A collineation group of order 213-35.52-7. Proc. London Math. Sot.
(3) 3 (1953), 54-79.
8. T. KONDO:~ The characters of the Weyl group of type F4. J. Fat. Sci. Univ. Tokyo
1 (1965), 145-153.
9. F. D. MURNAGHAN: The Orthogonal and Symplectic Groups. Comm. of the Dublin
Inst. for Adv. Study, Ser. A, No. 13 (Dublin, 1958).
10. R. H. DYE: The simple group FH (8,2) of order 2i2 35 52 7 and the associated
geometry of triality. Proc. London Math. Sot. (3) 18 (1968), 521-562.
11. R. H. DYE: The characters of a collineation group in seven dimensions. J. London
Math. Sot. 44 (1969), 169-174.
On some applications of group-theoretical
programmes to the derivation of the crystal classes of R,
R. Bii~ow AND J. NEUB~~SER
1. In mathematical crystallography symmetry properties of crystals are
described by group-theoretical means [l, 71. One considers groups of
motions fixing a (point-)lattice. These groups can therefore be represented
as groups of linear or affine transformations over the ring Z of integers.
For such groups certain equivalence relations are introduced.
In particular two subgroups 8 and Q of GL,(Z) are called geometrically
equivalent, if there exists an integral nonsingular matrix X, such that
x-wx = sj. If, moreover, such X can be found with det X = f 1 (i.e.
with X-l integral, too), @ and 8 are called arithmetically equivalent.
The equivalence classes are called geometrical and arithmetical crystal
classes respectively.
The lists of both geometrical and arithmetical crystal classes for dimensions
n = 1,2,3 have been known for some time. In 1951 A. C. Hurley
[5] published a list of the geometrical crystal classes for n = 4, which has
since been slightly corrected [6].
2. In 1965 E. C. Dade [2] gave a complete list of representatives of the
“maximal” arithmetical crystal classes, a problem which had also been
considered by C. Hermann [4]. Dade’s list consists of 9 groups:
grow
Q,
cu,
sx, c3 cu,
sx3 @ ccl,
%
Sx,'Z)
PY38 CUl
PY4
sxz@2
order
1152
384
96
96
240
288
96
240
144
All crystal classes can be found by classifying the subgroups of these nine
groups. Obviously subgroups conjugate in one of these groups are arithmetically
and hence geometrically equivalent. Therefore it suffices to take
131