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Directory:Jon Awbrey/Papers/Differential Logic (view source)
Revision as of 19:05, 10 June 2009
, 19:05, 10 June 2009→Note 23: centering + spacing
We've seen a couple of groups, <math>V_4\!</math> and <math>S_3,\!</math> represented in various ways, and we've seen their representations presented in a variety of different manners. Let us look at one other stylistic variant for presenting a representation that is frequently seen, the so-called ''matrix representation'' of a group.
We've seen a couple of groups, <math>V_4\!</math> and <math>S_3,\!</math> represented in various ways, and we've seen their representations presented in a variety of different manners. Let us look at one other stylistic variant for presenting a representation that is frequently seen, the so-called ''matrix representation'' of a group.
Recalling the manner of our acquaintance with the symmetric group <math>S_3,\!</math> we began with the ''bigraph'' (bipartite graph) picture of its natural representation as the set of all permutations or substitutions on the set <math>X = \{ A, B, C \}.\!</math>
Recalling the manner of our acquaintance with the symmetric group <math>S_3,\!</math> we began with the ''bigraph'' (bipartite graph) picture of its natural representation as the set of all permutations or substitutions on the set <math>X = \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}.\!</math>
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="6" width="90%"
|
| align="center" |
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{e}
\operatorname{e}
& = & \operatorname{A}:\operatorname{A}
& = & \mathrm{A}\!:\!\mathrm{A}
& + & \operatorname{B}:\operatorname{B}
& + & \mathrm{B}\!:\!\mathrm{B}
& + & \operatorname{C}:\operatorname{C}
& + & \mathrm{C}\!:\!\mathrm{C}
\\[4pt]
\\[4pt]
\operatorname{f}
\operatorname{f}
& = & \operatorname{A}:\operatorname{C}
& = & \mathrm{A}\!:\!\mathrm{C}
& + & \operatorname{B}:\operatorname{A}
& + & \mathrm{B}\!:\!\mathrm{A}
& + & \operatorname{C}:\operatorname{B}
& + & \mathrm{C}\!:\!\mathrm{B}
\\[4pt]
\\[4pt]
\operatorname{g}
\operatorname{g}
& = & \operatorname{A}:\operatorname{B}
& = & \mathrm{A}\!:\!\mathrm{B}
& + & \operatorname{B}:\operatorname{C}
& + & \mathrm{B}\!:\!\mathrm{C}
& + & \operatorname{C}:\operatorname{A}
& + & \mathrm{C}\!:\!\mathrm{A}
\\[4pt]
\\[4pt]
\operatorname{h}
\operatorname{h}
& = & \operatorname{A}:\operatorname{A}
& = & \mathrm{A}\!:\!\mathrm{A}
& + & \operatorname{B}:\operatorname{C}
& + & \mathrm{B}\!:\!\mathrm{C}
& + & \operatorname{C}:\operatorname{B}
& + & \mathrm{C}\!:\!\mathrm{B}
\\[4pt]
\\[4pt]
\operatorname{i}
\operatorname{i}
& = & \operatorname{A}:\operatorname{C}
& = & \mathrm{A}\!:\!\mathrm{C}
& + & \operatorname{B}:\operatorname{B}
& + & \mathrm{B}\!:\!\mathrm{B}
& + & \operatorname{C}:\operatorname{A}
& + & \mathrm{C}\!:\!\mathrm{A}
\\[4pt]
\\[4pt]
\operatorname{j}
\operatorname{j}
& = & \operatorname{A}:\operatorname{B}
& = & \mathrm{A}\!:\!\mathrm{B}
& + & \operatorname{B}:\operatorname{A}
& + & \mathrm{B}\!:\!\mathrm{A}
& + & \operatorname{C}:\operatorname{C}
& + & \mathrm{C}\!:\!\mathrm{C}
\end{matrix}</math>
\end{matrix}</math>
|}
|}
{| align="center" cellpadding="6" width="90%"
{| align="center" cellpadding="6" width="90%"
|
| align="center" |
<math>\begin{bmatrix}
<math>\begin{bmatrix}
\mathrm{A}\!:\!\mathrm{A} &
\mathrm{A}\!:\!\mathrm{A} &