Let
V
V
be an
n
n
-dimensional vector space over complex numbers
C
C
. Let
W
W
be the
m
m
th tensor product of
V
V
. If
T
∈
Hom
C
(
V
,
V
)
T \in {\operatorname {Hom} _C}(V,V)
, let
⊗
m
T
∈
Hom
C
(
W
,
W
)
{ \otimes ^m}T \in {\operatorname {Hom} _C}(W,W)
be the
m
m
th tensor product of
T
T
. The homomorphism
T
→
⊗
m
T
T \to { \otimes ^m}T
is a representation of the full linear group
G
L
n
(
C
)
{\text {G}}{{\text {L}}_n}(C)
. If
H
H
is a subgroup of the symmetric group
S
m
{S_m}
, and
χ
\chi
a linear character on
H
H
, let
V
χ
m
(
G
)
V_\chi ^m(G)
be the subspace of
W
W
consisting of all tensors symmetric with respect to
H
H
and
χ
\chi
. Then
V
χ
m
(
H
)
V_\chi ^m(H)
is invariant under
⊗
m
T
{ \otimes ^m}T
. Let
K
(
T
)
K(T)
be the restriction of
⊗
m
T
{ \otimes ^m}T
to
V
χ
m
(
H
)
V_\chi ^m(H)
. For
n
n
large compared with
m
m
and for
H
H
transitive, we determine all cases when the representation
T
→
K
(
T
)
T \to K(T)
is irreducible.