Symmetry Classes Of Tensors And Induced Modules

This dissertation focuses on symmetry classes of tensors and induced modules. It consists of three parts. It begins with an introduction of the background and the main ideas of the present research, and a sketch of the main results of the dissertation. The first chapter is concerned with symmetry classes of tensors and symmetry classes of functions: For the former we discuss the ranks and the existence of orthogonal bases; for the latter we study the decompositions of functions of several variables and the corresponding symmetry classes of functions. The second chapter is devoted to a theorem of Fong's type for group-graded algebras of p-solvable groups.In Chapter 1, we first consider the tensor product of a free module of finite rank over any integral domain, and the multilinear maps which are symmetric with respect to a function on a permutation group, which acts on the tensor product as permutation operators, and establish relationships between the symmetric property of the multilinear maps and the symmetry operators. In Section 3 where it is assumed that the integral domain has characteristic zero and any finitely generated projective module over the domain is free, a formula for the ranks of symmetry classes of tensors is obtained, and a criterion for the non-triviality of symmetry classes of tensors is provided. The next section is devoted to the symmetry classes of tensors associated with the direct product of permutation groups, and we thus obtain some information of the symmetry classes of tensors from each permutation group. As applications of the above researches, in Section 5 we derive the information on the symmetry classes of tensors in the usual ordinary case and the modular case; the results cover and extend many known facts. In Section 6, we then concentrate our study on two special types of groups and prove that the corresponding symmetry classes of tensors in the complex case have no orthogonal bases. Finally, we consider the functions of several variables valued in an integral domain of characteristic zero such that the corresponding group rings have block decompositions, and then we describe their symmetric properties and determine all the symmetry classes of functions associated with a given permutation group.In Chapter 2, assuming that p is a prime number and G is a p-solvable finite group, we consider a G-graded algebra A over a complete discrete valuation ringwith an algebraically closed residue field of characteristic p, and we prove that any primitive idempotent of A is conjugate to an idempotent of the H-part Ah of A where if is a Hall p'-subgroup of G; on the other hand, we show a necessary and sufficient condition for a primitive idempotent of Ah being still primitive in A.