| Abstract: | Let $G_M$ be either the orthogonal group $O_M$ or thesymplectic group $Sp_M$ over the complex field; in the latter casethe non-negative integer $M$ has to be even. Classically, theirreducible polynomial representations of the group $G_M$ arelabeled by partitions $mu=(mu_{1},mu_{2},,ldots)$such that $mu^{prime}_1+mu^{prime}_2le M$ in the case $G_M=O_M$, or$2mu^{prime}_1le M$ in the case $G_M=Sp_M$. Here$mu^{prime}=(mu^{prime}_{1},mu^{prime}_{2},,ldots)$ is the partitionconjugate to $mu$. Let $W_mu$ be the irreducible polynomialrepresentation of the group $G_M$ corresponding to $mu$. Regard $G_Ntimes G_M$ as a subgroup of $G_{N+M}$.Then take any irreducible polynomial representation$W_lambda$ of the group $G_{N+M}$.The vector space$W_{lambda}(mu)={rm Hom}_{,G_M}( W_mu, W_lambda)$comes with a natural action of the group $G_N$.Put $n=lambda_1-mu_1+lambda_2-mu_2+ldots,$.In this article, for any standard Young tableau $varOmega$ ofskew shape $lm$ we give a realization of $W_{lambda}(mu)$as a subspace in the $n$-fold tensor product$(mathbb{C}^N)^{bigotimes n}$, compatible with the action of the group $G_N$.This subspace is determined as the image of a certain linear operator$F_varOmega (M)$ on $(mathbb{C}^N)^{bigotimes n}$, given by an explicit formula. When $M=0$ and $W_{lambda}(mu)=W_lambda$ is an irreducible representation ofthe group $G_N$, we recover the classical realization of $W_lambda$as a subspace in the space of all traceless tensors in $(mathbb{C}^N)^{bigotimes n}$.Then the operator $F_varOmega(0)$ may be regarded as the analoguefor $G_N$ of the Young symmetrizer, corresponding to thestandard tableau $varOmega$ of shape $lambda$.This symmetrizer is a certain linear operator on$CNn$$(mathbb{C}^N)^{bigotimes n} $ with the image equivalent to the irreduciblepolynomial representation of the complex general linear group$GL_N$, corresponding to the partition $lambda$. Even in the case$M=0$, our formula for the operator $F_varOmega(M)$ is new. Our results are applications of the representationtheory of the twisted Yangian, corresponding to thesubgroup $G_N$ of $GL_N$. This twisted Yangianis a certain one-sided coideal subalgebra of the Yangian correspondingto $GL_N$. In particular, $F_varOmega(M)$ is an intertwiningoperator between certain representations of the twisted Yangianin $(mathbb{C}^N)^{bigotimes n}$. |
