2003/03/31 by Maxim Nazarov
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Finite Group Theory Research #math.CO #math.QA #math.RT #msc:17B37 #msc:20C30 #msc:22E46 #msc:81R50
paper · pdf · doi:10.1007/s00209-003-0619-7
published as Math. Z. 247 (2004), 21-63 · 44 pages, final version
arxiv created 2003/08/16 · openalex publication_date 2004/04/07 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/29
Let GLM be general linear Lie group over the complex field. The irreducible rational representations of the group GLM are labeled by pairs of partitions μ and μ such that the total number of non-zero parts of μ and μ does not exceed M. Let U be the representation of GLM corresponding to such a pair. Regard the direct product GLN× GLM as a subgroup of GLN+M. Let V be the irreducible rational representation of the group GLN+M corresponding to a pair of partitions λ and λ. Consider the vector space W=HomGM(U,V). It comes with a natural action of the group GLN. Let n be sum of parts of λ less the sum of parts of μ. Let n be sum of parts of λ less the sum of parts of μ. For any choice of two standard Young tableaux of skew shapes λ/μ and λ/μ respectively, we realize W as a subspace in the tensor product of n copies of the defining N-dimensional representation of GLN, and of 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 GLN, 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 λ. 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 glN.