On Infinitesimal Schur Algebras
On Infinitesimal Schur Algebras
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DOI：
10.1112/plms/s3-72.3.588
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发表时间：
1996-05
影响因子：
1.8
通讯作者：
S. Doty;D. Nakano;Karl M. Peters
中科院分区：
文献类型：
--
作者：
S. Doty;D. Nakano;Karl M. Peters
一般线性群pg =GLn（在无限域上，在定义特征中）的表示理论的一种方法依赖于乘法monoidM=Mnofn×nmatrices的有理表示与GLn的多项式表示之间的等价性。这个等价将G的多项式表示理论简化为某些有限维代数（n,d）的（普通）Schur代数的表示理论。素数特征的另一种标准方法是通过表示与第一个Frobenius核grofg和一个极大环grofg相关的群方案grt来研究gln的表示。这两种方法使我们为单形格式MrD构造了一个模范畴，它结合了这两种理论的各个方面。我们得到了grt与若干子代数（n,d）rofS（n,d）的多项式表示理论。这些子代数是无限小舒尔代数。
One approach to the representation theory of the general linear groupG=GLn(over an infinite field, in the defining characteristic) depends on the equivalence between rational representations of the multiplicative monoidM=Mnofn×nmatrices and polynomial representations ofGLn. This equivalence reduces polynomial representation theory of G to representation theory of certain finite‐dimensional algebrasS(n,d), the (ordinary) Schur algebras. Another standard approach in prime characteristic is to study the representations ofGLnthrough representations of the group schemesGrTassociated to therth Frobenius kernelGrofGand a maximal torusTofG. These two methods have led us to construct a module category for a monoid scheme,MrD, which combines facets of both theories. The monoidMrDbears the same relation toGrTas does M to G. We are led to a polynomial representation theory ofGrTand to certain associated subalgebrasS(n,d)rofS(n,d). These subalgebras are the infinitesimal Schur algebras.