Line 4,048: |
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| These days one is much more likely to encounter the natural representation of <math>S_3\!</math> in the form of a ''linear representation'', that is, as a family of linear transformations that map the elements of a suitable vector space into each other, all of which would in turn usually be represented by a set of matrices like these: | | These days one is much more likely to encounter the natural representation of <math>S_3\!</math> in the form of a ''linear representation'', that is, as a family of linear transformations that map the elements of a suitable vector space into each other, all of which would in turn usually be represented by a set of matrices like these: |
| | | |
− | {| align="center" cellpadding="6" width="90%" | + | <br> |
− | | align="center" |
| + | |
− | <pre> | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" |
− | Table 2. Matrix Representations of Permutations in Sym(3)
| + | |+ <math>\text{Matrix Representations of Permutations in}~ \operatorname{Sym}(3)</math> |
− | o---------o---------o---------o---------o---------o---------o
| + | |- style="background:#f0f0ff" |
− | | | | | | | | | + | | width="16%" | <math>\operatorname{e}</math> |
− | | e | f | g | h | i | j | | + | | width="16%" | <math>\operatorname{f}</math> |
− | | | | | | | | | + | | width="16%" | <math>\operatorname{g}</math> |
− | o=========o=========o=========o=========o=========o=========o
| + | | width="16%" | <math>\operatorname{h}</math> |
− | | | | | | | |
| + | | width="16%" | <math>\operatorname{i}</math> |
− | | 1 0 0 | 0 0 1 | 0 1 0 | 1 0 0 | 0 0 1 | 0 1 0 |
| + | | width="16%" | <math>\operatorname{j}</math> |
− | | 0 1 0 | 1 0 0 | 0 0 1 | 0 0 1 | 0 1 0 | 1 0 0 |
| + | |- |
− | | 0 0 1 | 0 1 0 | 1 0 0 | 0 1 0 | 1 0 0 | 0 0 1 |
| + | | |
− | | | | | | | |
| + | <math>\begin{matrix} |
− | o---------o---------o---------o---------o---------o---------o
| + | 1 & 0 & 0 |
− | </pre> | + | \\ |
| + | 0 & 1 & 0 |
| + | \\ |
| + | 0 & 0 & 1 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0 & 0 & 1 |
| + | \\ |
| + | 1 & 0 & 0 |
| + | \\ |
| + | 0 & 1 & 0 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0 & 1 & 0 |
| + | \\ |
| + | 0 & 0 & 1 |
| + | \\ |
| + | 1 & 0 & 0 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 1 & 0 & 0 |
| + | \\ |
| + | 0 & 0 & 1 |
| + | \\ |
| + | 0 & 1 & 0 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0 & 0 & 1 |
| + | \\ |
| + | 0 & 1 & 0 |
| + | \\ |
| + | 1 & 0 & 0 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0 & 1 & 0 |
| + | \\ |
| + | 1 & 0 & 0 |
| + | \\ |
| + | 0 & 0 & 1 |
| + | \end{matrix}</math> |
| |} | | |} |
| + | |
| + | <br> |
| | | |
| The key to the mysteries of these matrices is revealed by noting that their coefficient entries are arrayed and overlaid on a place-mat marked like so: | | The key to the mysteries of these matrices is revealed by noting that their coefficient entries are arrayed and overlaid on a place-mat marked like so: |