  Let $GL_M$ be general linear Lie group over the complex field. The
irreducible rational representations of the group $GL_M$ are labeled by pairs
of partitions $\mu$ and $\tilde\mu$ such that the total number of non-zero
parts of $\mu$ and $\tilde{\mu}$ does not exceed $M$. Let $U$ be the
representation of $GL_M$ corresponding to such a pair. Regard the direct
product $GL_N\times GL_M$ as a subgroup of $GL_{N+M}$. Let $V$ be the
irreducible rational representation of the group $GL_{N+M}$ corresponding to a
pair of partitions $\lambda$ and $\tilde{\lambda}$. Consider the vector space
$W=Hom_{G_M}(U,V)$. It comes with a natural action of the group $GL_N$. Let $n$
be sum of parts of $\lambda$ less the sum of parts of $\mu$. Let $\tilde{n}$ be
sum of parts of $\tilde{\lambda}$ less the sum of parts of $\tilde{\mu}$. For
any choice of two standard Young tableaux of skew shapes $\lambda/\mu$ and
$\tilde{\lambda}/\tilde{\mu}$ respectively, we realize $W$ as a subspace in the
tensor product of $n$ copies of the defining $N$-dimensional representation of
$GL_N$, and of $\tilde{n}$ copies of the contragredient representation. This
subspace is determined as the image of a certain linear operator $F$ in the
tensor product, given by explicit multiplicative formula. When M=0 and $W=V$ is
an irreducible representation of $GL_N$, we recover the classical realization
of $V$ as a subspace in the space of all traceless tensors. Then the operator
$F$ can be regarded as the rational analogue of the Young symmetrizer,
corresponding to the chosen standard tableau of shape $\lambda$. Even in the
special case M=0, our formula for the operator $F$ is new. Our results are
applications of representation theory of the Yangian of the Lie algebra $gl_N$.
