代数共变导数曲率张量的生成元和Young对称化子
Generators of algebraic covariant derivative curvature tensors and Young symmetrizers
摘要：
我们证明了代数共变导数曲率张量空间R'是由Young对称张量积W*U或U*W生成的，其中W和U是2阶和3阶的共变张量，它们的对称类是不可约的，并由以下几对划分刻画：{（2），（3）}，{（2），（2 1）}或{（1 1），（2 1）}.划分（2）、（3）和（11）中的每一个都精确地描述一个对称类，而划分（21）表征不可约对称类的无限集S。这个集合S恰好包含一个对称类S_0，其元素U不能充当张量R '的生成元。来自S\{S_0}的所有其他对称类的张量U可以用作张量R '的生成元。我们研究的基础是S. A.富灵河C.金湾G. Wybourne和C. J. Cummins关于生成代数共变导数曲率张量的对称类的Young对称化子。此外，我们适用于理想和幂等元群环C[Sr]，Littlewood-Richardson规则和离散傅里叶变换的对称群Sr。对于某些符号计算，我们使用的Mathematica软件包Ricci和PERMS。
英文摘要：
We show that the space of algebraic covariant derivative curvature tensors R' is generated by Young symmetrized tensor products W*U or U*W, where W and U are covariant tensors of order 2 and 3 whose symmetry classes are irreducible and characterized by the following pairs of partitions: {(2),(3)}, {(2),(2 1)} or {(1 1),(2 1)}. Each of the partitions (2), (3) and (1 1) describes exactly one symmetry class, whereas the partition (2 1) characterizes an infinite set S of irreducible symmetry classes. This set S contains exactly one symmetry class S_0 whose elements U can not play the role of generators of tensors R'. The tensors U of all other symmetry classes from S\{S_0} can be used as generators for tensors R'. Foundation of our investigations is a theorem of S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins about a Young symmetrizer that generates the symmetry class of algebraic covariant derivative curvature tensors. Furthermore we apply ideals and idempotents in group rings C[Sr], the Littlewood-Richardson rule and discrete Fourier transforms for symmetric groups Sr. For certain symbolic calculations we used the Mathematica packages Ricci and PERMS.
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