Intern. J. Math. & h. Sci.
Vol. 3 No. 2 (1980) 401-403
401
A NOTE ON THE SUBCLASS ALGEBRA
JOHN KARLOF
Department of Mathematics
University of Nebraska at Omaha
Omaha, Nebraska 68182
U.S.A.
(Received January 16, 1978 and in revised form November 19, 1979)
ABSTRACT.
Each irreducible character of the subclass algebra is paired up with
its irreducible module.
KEY WORDS AND PHRASES. Finite group, irreducible character, subclass algebra.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES:
20C05 primary; 20C15 secondary.
INTRODUCTION.
Let G be a finite group and let H be a subgroup of G.
of G containing g is the set E
B
g
Ex
xEE
g
{hgh-llhEH}
If gG, the subclass
and the subclass sum containing g is
The algebra over the complex numbers, K, generated by these subclass
g
sums is called the subclass algebra (denoted by S) associated with G and H.
Let {MI,
..,Ms }
ible character,
j,
be the irreducible KG-modules with
of G and let {N
I
N
t}
Mj
affording the irreduc-
be the irreducible KH-modules with N
i
402
J. KARLOF
afford n% the irreducible character
i
of H.
{f.}t
I
i=Iveorthogonal idempotents of KH and
{el}ti=l is a set of prim-
Suppose
is the set of primitive central
orthogonal Idempotents of KH where the sets are indexed so that N i
dim
fl (dim Ni)e i
{cij} by Xj
H
i
H
t
E
cij
i=l
E
hell
i(h-1)h
= KHe i and
We define the non-negatlve integers
Oi
In [2], it was demonstrated that the irreducible S-modules are
{eiMj}.
D. Travls [3] has shown that the irreducible characters of S are parameterlzed by
pairs
Xj, #i (cij
# 0)
and are given by
.
[
ij (B)
g
IN
hH
Xj (gh)#i(h-1).
()
Independent of Travis’s work we show that the irreducible character afforded by
eiMj
is
ij"
IE g [x(sg)
LEMMA:
Xi(sB g
PROOF:
Since B
[_
g
IN
7.
hgh- 1
geG
we have
heN
7.
hell
IE g
/sgS
Xi(hsgh -I)
Xi (sB)
g
since hs
[EEl
-IH
sh,
.
h H
Xi (shgh -i)
/heH
Xi(sg)
//
THEOREM:
elmj (cij
Let
# 0).
PROOF:
ij be the character afforded by the
Then
ij is as defined by equation (i).
irreducible S-module
t
By proposition 2 3 of [2]
we have
Mj
S
k=lT.
(dim N
t
7. f.M
k=ikJ
t
Therefore, for seS,
Xj(sfi)
k=iI
(dim
k)kj (sfi)
k) ekMj
A NOTE ON THE SUBCLASS ALGEBRA
(dim
since the trace of the action of sf
action of s on f
Thus
lJ(B g
iMj
#i)@ij (sf i)
i
on
fiMj
is the same as the trace of the
and the trace of the action of sf
di-i Xj (Bgfi)
by
dlm xj(gf i)
i
403
i
on
fkMj
(i # k) is O.
the Lemma
z
//
Acknowledg.nt: We thank
the referee for simplifying the proof of the above
theorem.
REFERENCES
i.
Curtis, C. W. and I. Reiner, Representation Theory of Finite Groups and
Associative Algebr.as New York: Intersclence (1962).
2.
Karlof, J., The Subclass Algebra Associated with a Finite Group and Subgroup,
Trans. A.M.S. 207, (1975) 329-341.
3.
Travis, D., Spherical Functions on Finite Groups, J. of Algebra
65-76.
.29, (1974)