A new family of symmetric functions is considered. These functions are analogous to the classical Schur functions, but depend on an integer modulus
p
⩾
2
p \geqslant 2
, as well as on a partition
λ
\lambda
. In the case where
p
p
is prime, certain of these functions are shown to be irreducible characters of the general linear group
G
L
(
n
,
K
)
GL(n,K)
in the natural characteristic
p
p
of the field
K
K
. This dualises a wellknown criterion of G. D. James for such characters to be given by classical Schur functions.