8.2 Basis representation of r-vectors

中文

Let $\left\{{\mathbf{e}}_i\right\}$ be any basis of $V$ with dual basis $\{{\epsilon }^j\}$ , then setting ${\omega }^1={\epsilon }^{i_1},{\omega }^1={\epsilon }^{i_1},...,{\omega }^r={\epsilon }^{i_r}$ in Eq. (8.2) results in an equation for the components of any tensor $T$ of type ($r$, 0)

$$(\mathcal{A}T{)}^{i_1{i}_2\dots i_r}=T^{[i_1{i}_2\dots i_r]}\equiv \frac{1}{r!}\sum _{\sigma }(-1{)}^{\sigma }{T}^{i_{\sigma (1)}{i}_{\sigma (2)}\dots i_{\sigma (r)}}.$$

From the properties of the antisymmetrization operator, the square bracketing of any set of indices satisfies

$$\begin{array}{l}{T}^{[i_1{i}_2\dots i_r]}=(-1{)}^{\pi }{T}^{[i_{\pi (1)}{i}_{\pi (2)}\dots i_{\pi (r)}]},\text{ for any permutation }\pi \\ T^{[[i_1{i}_2\dots i_r]]}=T^{[i_1{i}_2\dots i_r]}.\end{array}$$

If $A$ is an $r$-vector then $AA=A$ , or in components,

$$A^{i_1{i}_2\dots i_r}=A^{[i_1{i}_2\dots i_r]}.$$

Similar statements apply to tensors of covariant type, for example

$$\begin{array}{rl}& T_{[ij]}=\frac{1}{2}(T_{ij}-T_{ji}),\\ & T_{[ijk]}=\frac{1}{6}(T_{ijk}+T_{jki}+T_{kij}-T_{ikj}-T_{jik}-T_{kji}).\end{array}$$

Theorem 8.1 can be expressed in components as

$$T^{\left[\left[i_1{i}_2\dots i_r\right]S^{j_{r+1}{j}_{r+2}\dots j_{r+s}}\right]}=T^{[i_1{i}_2\dots i_r}{S}^{j_{r+1}{j}_{r+2}\dots j_{r+s}}],$$

or, with a slight generalization, square brackets occurring anywhere within square brackets may always be eliminated,

$$T^{[i_1\dots [i_k\dots i_{k+l}]\dots i_r]}=T^{[i_1\dots i_k\dots i_{k+l}\dots i_r]}.$$

By the antisymmetry of its components every $r$-vector $A$ can be written

$$\begin{array}{c}A=A^{i_1{i}_2\dots i_r}{e}_{i_1}\otimes e_{i_2}\otimes \cdots \otimes e_{i_r}\\ =A^{i_1{i}_2\dots i_r}{e}_{i_1{i}_2\dots i_r}\end{array}\text{\hspace{1em}(8.3)}$$

where

$$e_{i_1{i}_2\dots i_r}=\frac{1}{r!}\sum _{\sigma }(-1{)}^{\sigma }{e}_{i_{\sigma (1)}}\otimes e_{i_{\sigma (2)a}}\otimes \cdots \otimes e_{i_{\sigma (r)a}}.\text{\hspace{1em}(8.4)}$$

For example

$$\begin{array}{c}{e}_{12}=\frac{1}{2}(e_1\otimes e_2-e_2\otimes e_1),\\ e_{123}=\frac{1}{6}(e_1\otimes e_2\otimes e_3-e_1\otimes e_3\otimes e_2+e_2\otimes e_3\otimes e_1\\ -e_2\otimes e_1\otimes e_3+e_3\otimes e_1\otimes e_2-e_3\otimes e_2\otimes e_1),\text{etc.}\end{array}$$

The $r$-vectors $e_{i_1\dots i_r}$ have the property

$$e_{i_1\dots i_r}=\{\begin{array}{ll}0& \text{ if any pair of indices are equal, }\\ (-1{)}^{\pi }{e}_{i_{\pi (1)}\dots i_{\pi (r)}}& \text{for any permutation }\pi \text{ of }1,2,\dots ,r.\end{array}\text{\hspace{1em}(8.5)}$$

Hence the expansion Eq. (8.3) can be reduced to one in which every term has $i_1<i_2<\cdots <i_r$

$$A=r!\underset{i_1<i_2<\cdots <i_r}{\underbrace{\sum \cdots \sum }}{A}^{i_1{i}_2\dots i_r}{e}_{i_1{i}_2\dots i_r}.\text{\hspace{1em}(8.6)}$$

Hence ${\Lambda }^r(V)$ is spanned by the set

$$E_r=\{e_{i_1{i}_2\dots i_r}\mid i_1<i_2<\cdots <i_r\}.$$

Furthermore this set is linearly independent, for if there were a linear relation

$$0=\underset{i_1<i_2<\cdots <i_r}{\underbrace{\sum \cdots \sum }}{B}^{i_1{i}_2\dots i_r}{e}_{i_1{i}_2\dots i_r},$$

application of this multilinear function to arguments ${\epsilon }^{j_1},{\epsilon }^{j_2},\dots ,{\epsilon }^{j_r}$ with $j_1<j_2\cdots <j_r$ gives

$$0=\underset{i_1<i_2<\cdots <i_r}{\underbrace{\sum \cdots \sum }}{B}^{i_1{i}_2\dots i_r}{\delta }_{i_1}^{j_1}{\delta }_{i_2}^{j_2}\dots {\delta }_{i_r}^{j_r}=B^{j_1{j}_2\dots j_r}.$$

Hence $E_r$ forms a basis of ${\Lambda }^r(V)$

The dimension of the vector space ${\Lambda }^r(V)$ is the number of subsets $\{i_1<i_2<\cdots <i_r\}$ occurring in the first $n$ integers $\{1,2,\dots ,n\}$

$$\dim {\Lambda }^r(V)=\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\frac{n!}{r!(n-r)!}.$$

In particular $\dim$ ${\Lambda }^n(V)=1$ , while $\dim$ ${\Lambda }^{(n+k)}(V)=0$ for all $k>0$ . The latter follows from the fact that if $r>n$ then all $r$-vectors $e_{i_1{i}_2\dots i_r}$ vanish, since some pair of indices must be equal.

An analogous argument shows that the set of $r$-forms

$${\mathcal{E}}_r=\{{\epsilon }^{i_1\dots i_r}|i_i<\cdots <i_r\},\text{\hspace{1em}(8.7)}$$

where

$${\epsilon }^{i_1\dots i_r}=\frac{1}{r!}\sum _{\pi }(-1{)}^{\pi }{\epsilon }^{i_{\pi (1)}}\otimes {\epsilon }^{i_{\pi (2)}}\otimes \cdots \otimes {\epsilon }^{i_{\pi (r)}},\text{\hspace{1em}(8.8)}$$

is a basis of ${\Lambda }^{\ast r}(V)$ and the dimension of the space of $r$-forms is also ${}^n$ .