Rational representations of Yangians associated with skew Young diagrams
Rational representations of Yangians associated with skew Young diagrams
复制标题
与偏杨图相关的杨吉亚有理表示
DOI：
10.1007/s00209-003-0619-7
复制
发表时间：
2002
影响因子：
0.8
通讯作者：
M. Nazarov
中科院分区：
文献类型：
--
作者：
M. Nazarov
摘要：考虑复域上一般线性群GLM。群GLM的不可约有理数表示可以用划分对和来标记，使得μ和的非零部分的总数不超过M。令EQ4是对应于这样的对的不可约表示。把直积看成GLN+M的一个子群。取GLN+M的任意不可约有理表示。向量空间带有GLN群的自然作用。输入n=。对于任意一对标准的斜形状的Young tableaux，我们分别给出了GLN的定义表示的n个副本和逆表示（）* 的n个副本的张量积中的子空间的实现。这个子空间被确定为Wn n n上某个线性算子的像。我们通过一个显式的乘法公式引入这个算子。当M=0且是GLN的不可约表示时，我们将已知的实现恢复为中所有无迹张量空间中的某个子空间。然后，该算子可以被视为杨氏对称化子的有理模拟，对应于形状为λ的表Ω。即使M=0，我们的公式也是新的。我们的结果是李代数Yangian表示理论的应用。特别地，是上的代数的某些表示之间的交织算子。我们还介绍了一个合理的Yangian表示的概念。作为的表象，的象是有理的，不可约的。
Abstract.Consider the general linear group GLM over the complex field. The irreducible rational representations of the group GLM can be labeled by the pairs of partitions and such that the total number of non-zero parts of μ and does not exceed M. Let EQ4 be the irreducible representation corresponding to such a pair. Regard the direct product as a subgroup of GLN+M . Take any irreducible rational representation of GLN+M. The vector space comes with a natural action of the group GLN. Put n=. For any pair of standard Young tableaux of skew shapes respectively, we give a realization of as a subspace in the tensor product of n copies of defining representation of GLN, and of ñ copies of the contragredient representation ()*. This subspace is determined as the image of a certain linear operator on Wnñn. We introduce this operator by an explicit multiplicative formula. When M=0 and is an irreducible representation of GLN, we recover the known realization of as a certain subspace in the space of all traceless tensors in . Then the operator may be regarded as the rational analogue of the Young symmetrizer, corresponding to the tableau Ω of shape λ . Even when M=0, our formula for is new. Our results are applications of the representation theory of the Yangian of the Lie algebra . In particular, is an intertwining operator between certain representations of the algebra on . We also introduce the notion of a rational representation of the Yangian . As a representation of , the image of is rational and irreducible.