Some Introductory Algebra

In this chapter, we summarize some basic theory concerning permutations, multilinear algebra, set partitions, and diagrams for usage throughout the later part of the book. We start with the notion of permutations and then discuss the tensor product (T-product), vec operator, and commutation matrices. A transformation of multiple T-products of vectors into a given order is also considered. In the section on symmetrization and multilinear algebra, we introduce the symmetrizer in the framework of multilinear algebra of Cartesian tensors. In connection with the symmetric subspace of tensors we construct linear operators in terms of matrices for the elimination of identical entries from a tensor as well as the duplication, triplication, quadruplication, etc. of tensors of distinct entries. The Partitions and Diagrams section includes the inclusive and exclusive method of extending partitions to derive all partitions of a finite set. The use of Bell polynomials and Bell numbers for obtaining the partitions is outlined. Partitions with lattice structure are considered mainly to discuss indecomposable partitions and diagrams. A discussion of the particular cases of diagrams, such as closed diagrams without loops and closed diagrams with arms and no loops, concludes this chapter.