TITLE:
Character Products and Q-polynomial Group Association Schemes
ABSTARACT:
We study a finite group having a faithful
character whose square has small number of irreducible characters as
constituents.
Let Irr(G) be the set of absolutely irreducible ordinary characters of
a finite group G. For each f in Irr(G), let f = f if
f is real valued and f= f+f' otherwise,
where f' denotes the complex conjugate of f. Let
RIrr(G) = {f | f is in Irr(G)}. For g in RIrr(G), let
g2 = b*1 + a*g + h
such that h is a character of G which does not contain g
nor the principal character 1 as a constituent. We study the case
when h is a scalar multiple of a sum of the characters in RIrr(G),
which are in a single orbit with respect to the action of the Galois
group Gal(Q/Q(g)). Here Q denotes the
algebraic closure of Q in C and Q(g) is the field
generated by the values of g.
As an application,
we give a classification of Q-polynomial group association schemes.
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