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, 13:20, 15 June 2009→Note 21: markup
==Note 21==
==Note 21==
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 have seen a couple of groups, V_4 and S_3, represented in
Let us look at one other stylistic variant for presenting
a group representation that is often used, the so-called
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>
substitutions acting on a set of letters X = {a, b, c}.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
|+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ a, b, c \}</math>
|- style="background:#f0f0ff"
| width="16%" | <math>\operatorname{e}</math>
| width="16%" | <math>\operatorname{f}</math>
| width="16%" | <math>\operatorname{g}</math>
| width="16%" | <math>\operatorname{h}</math>
| width="16%" | <math>\operatorname{i}</math>
| width="16%" | <math>\operatorname{j}</math>
|-
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
a & b & c
\end{matrix}</math>
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
c & a & b
\end{matrix}</math>
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
b & c & a
\end{matrix}</math>
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
a & c & b
\end{matrix}</math>
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
c & b & a
\end{matrix}</math>
|
<math>\begin{matrix}
a & b & c
\\[3pt]
\downarrow & \downarrow & \downarrow
\\[6pt]
b & a & c
\end{matrix}</math>
|}
<br>
These permutations were then converted to relative form as logical sums of elementary relatives:
{| align="center" cellpadding="10" width="90%"
| align="center" |
<math>\begin{matrix}
\operatorname{e}
& = & a\!:\!a
& + & b\!:\!b
& + & c\!:\!c
\\[4pt]
\operatorname{f}
& = & a\!:\!c
& + & b\!:\!a
& + & c\!:\!b
\\[4pt]
\operatorname{g}
& = & a\!:\!b
& + & b\!:\!c
& + & c\!:\!a
\\[4pt]
\operatorname{h}
& = & a\!:\!a
& + & b\!:\!c
& + & c\!:\!b
\\[4pt]
\operatorname{i}
& = & a\!:\!c
& + & b\!:\!b
& + & c\!:\!a
\\[4pt]
\operatorname{j}
& = & a\!:\!b
& + & b\!:\!a
& + & c\!:\!c
\end{matrix}</math>
|}
<pre>
From this relational representation of Sym {a, b, c} ~=~ S_3,
From this relational representation of Sym {a, b, c} ~=~ S_3,
one easily derives a "linear representation", regarding each
one easily derives a "linear representation", regarding each
