Schur and Weyl functors
Schur and Weyl functors
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Schur 和 Weyl 函子
DOI：
10.1016/0001-8708(91)90020-8
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发表时间：
1991
影响因子：
1.7
通讯作者：
F. Kouwenhoven
中科院分区：
文献类型：
--
作者：
F. Kouwenhoven
Schur和Weyl函子是一般线性群的表示论中Schur和Weyl模的函子推广，这也解释了它们的名字。这两类函子都定义在任何含1的交换环上，并且它们由Young图参数化。它们在[20，11]中被定义和研究，进一步的工作可以在[2，51]中找到。Schur和Weyl函子是有限生成投射模范畴上的内函子，并且它们是通用定义的，即它们随着基环的变化而交换。Schur函子的特殊情况是对称幂和外幂，实际上它们在某种意义上是极端情况。Weyl函子是Schur函子在自然意义上的对偶，其分幂是对称幂的对偶，外幂是自对偶。Schur和Weyl函子也都是在研究外幂和对称幂时自然产生的。为了说明相关的舒尔和魏尔函子在多线性代数和一般表示理论的群体，我们提到以下：对于每个杨图有一个特殊的普遍定义的自然变换从魏尔到舒尔函子可以表征的签署。如果基环包含有理数，它实际上是等价的。这些变换的象在无限域上构成了有限维向量空间范畴上的不可约多项式内函子的完备的不可冗余系统。当应用于有限维向量空间的自同态幺半群或自同构群的自然表示时，它们分别给出了这个幺半群群的不可约多项式表示的完备集。(For多项式函子和表示的定义见[18，13-J.）而且当基域是有限的时，对于所有Young集的某个子集的自然变换，
The Schur and Weyl functors are the functorial generalisation of the Schur respectively Weyl modules in the representation theory of general linear groups, this also explains their names. Both types of functors are defined over any commutative ring with 1, and they are parametrised by Young diagrams. They were defined and studied in [20, 11, further work can be found in [2, 51.The Schur and Weyl functors are endofunctors on the category of finitely generated projective modules, and they are universally defined, ie, they commute with change of base ring. Special cases of Schur functors are the symmetric and exterior powers, in fact these are the extreme cases in some sense. Weyl functors are the duals of Schur functors in a natural sense, the divided power is the dual of the symmetric power in this sense and the exterior power is self-dual. Both Schur and Weyl functors also arise naturally in the study of exterior and symmetric powers. To illustrate the relevance of Schur and Weyl functors in multilinear algebra and in general representation theory of groups, we mention the following: For each Young diagram there is a special universally defined natural transformation from the Weyl to the Schur functor which can be characterised up to sign. In case the base ring contains the rationals it is in fact an equivalence. The images of these transformations constitute over an infinite field a complete irredundant system of irreducible polynomial endofunctors on the category of finite dimensional vector spaces. And when applied to the natural representation of the endomorphism monoid, or the automorphism group, of a finite dimensional vector space, they give a complete set of irreducible polynomial representations of this monoid respectively group.(For the definition of polynomial functors and representations see [18, 13-J.) Moreover when the base field is finite similar results hold for the natural transformations for a certain subset of the set of all Young