Rational representations of Yangians associated with skew Young diagrams

Abstract

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 $gl_N$.