The present entry employs the terminology and notation defined and described in the entry on tensor arrays. To keep things reasonably self-contained we mention that the symbol refers to the vector space of type tensor arrays, i.e. maps
where is some finite list of index labels, and where is a field.
We say that a tensor array is a characteristic array, a.k.a. a basic tensor, if all but one of its values are , and the remaining non-zero value is equal to . For tuples and , we let
denote the characteristic array defined by
The type characteristic arrays form a natural basis for .
Furthermore the outer multiplication of two characteristic arrays gives a characteristic array of larger valence. In other words, for
we have that
where the product on the left-hand side is performed by outer multiplication, and where on the right-hand side refers to the element of obtained by concatenating the tuples and , and similarly for .
In this way we see that the type characteristic arrays (the natural basis of ), and the type characteristic arrays (the natural basis of ) generate the tensor array algebra relative to the outer multiplication operation.
The just-mentioned fact gives us an alternate way of writing and thinking about tensor arrays. We introduce the basic symbols
subject to the commutation relations
add and multiply these symbols using coefficients in , and use
as a handy abbreviation for
We then interpret the resulting expressions as tensor arrays in the obvious fashion: the values of the tensor array are just the coefficients of the symbol matching the given index. However, note that in the symbols, the covariant data is written as a superscript, and the contravariant data as a subscript. This is done to facilitate the Einstein summation convention.
By way of illustration, suppose that . We can now write down a type tensor, i.e. a column vector
Similarly, a row-vector
can be written down as
In the case of a matrix
we would write
|Date of creation||2013-03-22 12:40:37|
|Last modified on||2013-03-22 12:40:37|
|Last modified by||rmilson (146)|