Induced class function
From Groupprops
Definition
Suppose is a group and
is a subgroup of finite index in
. Suppose
is a class function on
(i.e., a function on
that is constant on each conjugacy class of
). Then, the induced class function on
, denoted
is defined by the following summation over a left transversal
of
in
:
where is
if
and
otherwise.
Note that this is well-defined (independent of _ precisely because
is a class function on
, so replacing
by
, gives an element conjugate via
and hence the same value of
.
Facts
- Character of induced representation equals induced class function from character: If
is the character of a linear representation, then
is the character of the induced representation from
to
. it is often called the induced character from
.
- Similarly, if
is a virtual character, the induced class function is also a virtual character.