For any linear operator
T acting on
V, there is a (unique) induced operator
K χ (
T) ∈ End(
H d (
G,
χ)) acting on symmetrized decomposable polynomials by
$${K_\chi }\left( T \right)\left( {{f_1} * {f_2} * \cdots * {f_d}} \right) = T{f_1} * T{f_2} * \cdots * T{f_d}.$$
In this paper, we show that the representation
T ?
K χ (
T) of the general linear group
GL(
V) is equivalent to the direct sum of
χ(1) copies of a representation (not necessarily irreducible)
T ?
B χ G (
T).