Invariant theory ofG2 and Spin7
Invariant theory ofG2 and Spin7
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DOI：
10.1007/bf02566782
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发表时间：
1988-12
影响因子：
0.9
通讯作者：
Gerald W. Schwarz
中科院分区：
文献类型：
--
作者：
Gerald W. Schwarz
(0.0)设G是一个半单复代数群. G的不变量理论是一个忠实的表示r以及不变量代数C [nV] c，ne~的生成元和关系，其中nV表示V的n个副本的直和。给定G的不变量理论，人们可以使用符号方法[W]来获得关于G的任何表示的不变量的信息。如果G是经典群SLm，SOm之一，等的标准表示，则经典不变理论（CIT）给出了C [nV] c，ne I%的生成元和关系，即CIT是经典群的不变理论.对于非经典的单连通复代数群，即例外群G2、F4、E6、E7和Ea，以及自旋群Spinm，m-> 7，仍然存在寻找不变理论的问题。第一个要考虑的情况是（32）和Spin 7（也表示为1113），它们分别具有忠实的不可约7维和8维表示。本文建立了G2和B3的不变量理论。我们对G2的结果在IS 4中公布]。
(0.0) Let G be a semisimple complex algebraic group. An invariant theory for G is a faithful representation r together with generators and relations for the algebras of invariants C [nV] c, ne~, where nV denotes the direct sum of n copies of V. Given an invariant theory for G, one can use the symbolic method [W] to garner information about the invariants of any representation of G.If G is one of the classical groups SLm, SOm, etc. with its standard representation on V= C m, then classical invariant theory (CIT) tells us generators and relations for C [nV] c, ne I% ie CIT is an invariant theory for the classical groups. There remains the problem of finding an invariant theory for the non-classical simple connected complex algebraic groups, ie for the exceptional groups G2, F4, E6, E7 and Ea, and the spin groups Spinm, m-> 7. The first cases to consider are (32 and Spin7 (also denoted 1113) which have faithful irreducible 7-dimensional and 8-dimensional representations, respectively. In this paper we establish an invariant theory for G2 and B3. Our results for G2 were announced in IS4].