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==Note 21==
 
==Note 21==
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<pre>
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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.
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We have seen a couple of groups, V_4 and S_3, represented in
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several different ways, and we have seen each of these types
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of representation presented in several different fashions.
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Let us look at one other stylistic variant for presenting
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a group representation that is often used, the so-called
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"matrix representation" of a group.
     −
Returning to the example of Sym(3), we first encountered
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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>
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this group in concrete form as a set of permutations or
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substitutions acting on a set of letters X = {a, b, c}.
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This set of permutations was displayed in Table 17-a,
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copies of which can be found here:
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http://stderr.org/pipermail/inquiry/2004-May/001419.html
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<br>
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http://forum.wolframscience.com/showthread.php?postid=1321#post1321
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These permutations were then converted to "relative form":
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
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|+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ a, b, c \}</math>
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|- style="background:#f0f0ff"
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| width="16%" | <math>\operatorname{e}</math>
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| width="16%" | <math>\operatorname{f}</math>
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| width="16%" | <math>\operatorname{g}</math>
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| width="16%" | <math>\operatorname{h}</math>
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| width="16%" | <math>\operatorname{i}</math>
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| width="16%" | <math>\operatorname{j}</math>
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|-
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|
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<math>\begin{matrix}
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a & b & c
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\\[3pt]
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\downarrow & \downarrow & \downarrow
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\\[6pt]
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a & b & c
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\end{matrix}</math>
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|
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<math>\begin{matrix}
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a & b & c
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\\[3pt]
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\downarrow & \downarrow & \downarrow
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\\[6pt]
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c & a & b
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\end{matrix}</math>
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|
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<math>\begin{matrix}
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a & b & c
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\\[3pt]
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\downarrow & \downarrow & \downarrow
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\\[6pt]
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b & c & a
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\end{matrix}</math>
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|
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<math>\begin{matrix}
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a & b & c
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\\[3pt]
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\downarrow & \downarrow & \downarrow
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\\[6pt]
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a & c & b
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\end{matrix}</math>
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|
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<math>\begin{matrix}
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a & b & c
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\\[3pt]
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\downarrow & \downarrow & \downarrow
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\\[6pt]
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c & b & a
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\end{matrix}</math>
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|
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<math>\begin{matrix}
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a & b & c
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\\[3pt]
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\downarrow & \downarrow & \downarrow
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\\[6pt]
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b & a & c
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\end{matrix}</math>
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|}
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  e  =  a:a + b:b + c:c
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<br>
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  f  =  a:c + b:a + c:b
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These permutations were then converted to relative form as logical sums of elementary relatives:
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  g = a:b + b:c + c:a
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{| align="center" cellpadding="10" width="90%"
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| align="center" |
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  h = a:a + b:c + c:b
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<math>\begin{matrix}
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\operatorname{e}
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  i = a:c + b:b + c:a
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& = & a\!:\!a
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& + & b\!:\!b
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  j = a:b + b:a + c:c
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& + & c\!:\!c
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\\[4pt]
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\operatorname{f}
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& = & a\!:\!c
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& + & b\!:\!a
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& + & c\!:\!b
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\\[4pt]
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\operatorname{g}
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& = & a\!:\!b
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& + & b\!:\!c
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& + & c\!:\!a
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\\[4pt]
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\operatorname{h}
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& = & a\!:\!a
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& + & b\!:\!c
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& + & c\!:\!b
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\\[4pt]
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\operatorname{i}
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& = & a\!:\!c
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& + & b\!:\!b
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& + & c\!:\!a
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\\[4pt]
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\operatorname{j}
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& = & a\!:\!b
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& + & b\!:\!a
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& + & c\!:\!c
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\end{matrix}</math>
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|}
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<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
Jon Awbrey
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