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 $\tspace{p,q}$ refers to the vector space of type $(p,q)$ tensor arrays, i.e. maps $$I^p\times I^q\rightarrow \kfield,$$ where $I$ is some finite list of index labels, and where $\kfield$ 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 $0$ , and the remaining non-zero value is equal to $1$ . For tuples $A\in I^p$ and $B\in I^q$ , we let $$\ca^B_A:I^p\times I^q\rightarrow\kfield,$$ denote the characteristic array defined by

Furthermore the outer multiplication of two characteristic arrays gives a characteristic array of larger valence. In other words, for $$ A_1\in I^{p_1},\; B_1\in I^{q_1},\; A_2\in I^{p_2},\; B_2\in I^{q_2}, $$ we have that $$\ca^{B_1}_{A_1} \ca^{B_2}_{A_2} = \ca^{B_1 B_2}_{A_1 A_2},$$ where the product on the left-hand side is performed by outer multiplication, and where $A_1 A_2$ on the right-hand side refers to the element of $I^{p_1+p_2}$ obtained by concatenating the tuples $A_1$ and $A_2$ , and similarly for $B_1 B_2$ .

In this way we see that the type $(1,0)$ characteristic arrays $\ca_{(i)},\; i\in I$ (the natural basis of $\kfield^I$ ), and the type $(0,1)$ characteristic arrays $\ca^{(i)},\; i\in I$ (the natural basis of $\lp\kfield^I\rp^*$ ) 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 $$\ca_{(i)},\; \ca^{(i)} ,\quad i\in I$$ subject to the commutation relations $$\ca_{(i)} \ca^{(i')} = \ca^{(i')}\ca_{(i)} ,\quad i, i'\in I,$$ add and multiply these symbols using coefficients in $\kfield$ , and use $$\ca^{(i_1 \ldots i_q)}_{(j_1 \ldots j_p)},\quad i_1,\ldots,i_q,j_1,\ldots,j_p\in I$$ as a handy abbreviation for $$\ca^{(i_1)} \ldots \ca^{(i_q)} \ca_{(j_1)} \ldots \ca_{(j_p)}.$$ 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 $\ca$ symbol matching the given index. However, note that in the $\ca$ 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 $I=(1,2)$ . We can now write down a type $(1,0)$ tensor, i.e. a column vector $$u= \begin{pmatrix} u^1 \\ u^2 \end{pmatrix}\in \tspace{1,0} $$ as $$u = u^1 \ca_{(1)} + u^2 \ca_{(2)}.$$ Similarly, a row-vector $$\phi = (\phi_1,\phi_2) \in \tspace{0,1}$$ can be written down as $$\phi = \phi_1 \ca^{(1)} + \phi_2 \ca^{(2)}.$$ In the case of a matrix $$ M = \begin{pmatrix} M\ud{1}{1} & M\ud{2}{1} \\ M\ud{1}{2} & M\ud{2}{2} \end{pmatrix}\in \tspace{1,1} $$ we would write $$ M = M\ud{1}{1}\, \ca^{(1)}_{(1)}+ M\ud{1}{2}\, \ca^{(2)}_{(1)}+ M\ud{2}{1}\, \ca^{(1)}_{(2)}+ M\ud{2}{2}\, \ca^{(2)}_{(2)}. $$