COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

118 J. S. Frame
pair of characters of degrees fi, fi or f3, f4 such that
120(56)(63)/fifi is a square, and fi+f2 = 119, (4.1)
135(64)(7O)/f3f, is a square, and f3+f4 = 134.
The unique positive integral solutions of these equations are
fi, fi = 35 and 84 in 120,, (4.2)
fs, f4 = 50 and 84 in 135,. (4.3)
It is easily verified that 120, contains the character 35, already computed,
so
84, = 120,- lx-35,, (4.4)
50, = 135,- 120,+ 35,. (4.5)
It can be checked that 84, is also a constituent of the character 960,
induced by the symmetric subgroup S = &. Hence the character 960,- lx-
84, of degree 875 splits into two irreducible constituents whose degrees
divide 213 35 52 7. The splitting into 175, and 700, is versed when it is
found that this character has a constituent in common with the Murnaghan
character [6] of degree 1386 for the orthogonal group Os, which splits as
follows into three known constituents and one new constituent 700,:
[6] = 700,+567,+84,+35,. (4.6)
The other constituent 175, of 960, is
175, = 960,-lx-84,-700,. (4.7)
To check this character 175,, we note that it splits in H into the sum of
the two characters 105b+70i. Thus its values for the 46 classes of A in H
can be computed directly from these subgroup characters without evaluating
[6]. Both degrees 175 and 700 are divisible by 52 and 7, so the characters
175, and 700, both vanish for the 11 classes of elements whose orders
are divisible by 5 or 7. Thus it may be easier to determine 175, first for
most classes, and then find 700, from (4.7) rather than from (4.6).
Having the additional characters 84,, 50,, 175, and 700,, we can now
compute several more characters of types x and y quite simply as follows:
1400, = 28,(50,)
(4.8)
1050, = 35,(50,) - 700,
1575, = 28,(84,)-567,-210, = 8,(560,)-35,(84,-lx)
1344, = 35,(84,)-8,(112,)-700,
2100, = 28,(210,-84,)-84,- 1344,
2268, = 35,(210,-84,)- 567,- 1575,
525, = 84:“1-567,- 1050,- 1344,
700,, = 5Og’l- 525
4200, = 28,( 175,) y700x,.
The characters of the Weyl group Es 119
The symbol 700,, denotes a second character of type x and degree 700,
distinct from 700,. Other new characters are defined by
972, = 50,(84,) - 50, - 84, - 1050, - 1344, - 700,, (4.9)
4096, = (35x- 1,)(300,-28,)-840,-700,- 1344,-2268x.
The character 972, of degree 2235 is of highest type modulo 3, so it must
vanish in all 3-singular classes.
By decomposing Kronecker products involving characters of relatively
small degree such as 70,, 50,, 84,, 28, and 35, we can now solve for ten
more of the 13 self-associated characters of type y as follows :
1134, = 70,(28x) - 70, - (28, + 28;) - (350, + 350;) (4.10)
1680, = 70,(35,)-70,-(350,+350;)
= 70,(84x)-(2100,+2100~)
168, = SOP1 - 50, - 84,- 972
420, = 70,(70,+lx)-28x(28~+28~)-(840,+840~)-1134Y-168,
3150, = 28,(168,)- 1134,-420,
4200, = 35,( 168,) - (840,f 840;)
2688, = 28,(420, - 168,) - 168, - 4200,
2100, = 50x(168,- lx-I:)-(700,+700;,)-(972,+972;)- 168,-2688,
1400, = 50,(70,) - 2100,
4536, = 28,(525,)-300,-700,- 1400,-2268,-4096,- 1400,.
5. Kronecker products with the character 8,. Products of the character
8, with any constituent of an even (odd) Kronecker power will split into
constituents of powers of the opposite parity. By splitting products of 8,
with the even-power constituents of types x and y already found we can
complete the list of irreducible characters of types z and w as follows.
We start with a second character 1400, of type z and degree 1400, not to
be confused with 1400, already obtained.
CPA 9
1400,, = 8,(175,) (5.1)
4200, = 8,(700,x- 175,)
400, = 8,(50x)
3240, = 8,(700, - 50,) - 1400, - 560,
4536, = 8,(972,j - 3240,
2400, = 56,(84,) - 1296, - 1008,
3360, = 8,(1400,-525,-50,)-3240,
2800, = 56,(50x)
4096, = 8,( 1575,) - 56,(84x) - 560, - 3240,
5600, = 8,(2268,-525,)- 1008,-3240,-2800,
448, = 8,(1344,- 1575,)f 1296,- 1400,+2400,