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Directory:Jon Awbrey/Papers/Dynamics And Logic (view source)

Revision as of 13:45, 15 June 2009

407 bytes added ,  13:45, 15 June 2009
→‎Note 21: markup
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<pre>
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From the relational representation of <math>\operatorname{Sym} \{ a, b, c \} \cong S_3,</math> one easily derives a ''linear representation'' of the group by viewing each permutation as a linear transformation that maps the elements of a suitable vector space into each other.  Each of these linear transformations is in turn represented by the a 2-dimensional array of coefficients in <math>\mathbb{B},</math> resulting in the following set of matrices for the group:
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From this relational representation of Sym {a, b, c} ~=~ S_3,
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one easily derives a "linear representation", regarding each
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permutation as a linear transformation that maps the elements
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of a suitable vector space into each other, and representing
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each of these linear transformations by means of a matrix,
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resulting in the following set of matrices for the group:
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Table 21.  Matrix Representations of the Permutations in S_3
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<br>
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o---------o---------o---------o---------o---------o---------o
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|        |        |        |        |        |        |
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|    e    |    f    |    g    |    h    |    i    |    j    |
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|        |        |        |        |        |        |
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o=========o=========o=========o=========o=========o=========o
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|        |        |        |        |        |        |
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|  1 0 0  |  0 0 1  |  0 1 0  |  1 0 0  |  0 0 1  |  0 1 0  |
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|  0 1 0  |  1 0 0  |  0 0 1  |  0 0 1  |  0 1 0  |  1 0 0  |
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|  0 0 1  |  0 1 0  |  1 0 0  |  0 1 0  |  1 0 0  |  0 0 1  |
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|        |        |        |        |        |        |
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o---------o---------o---------o---------o---------o---------o
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The key to the mysteries of these matrices is revealed by
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
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observing that their coefficient entries are arrayed and
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|+ <math>\text{Matrix Representations of Permutations in}~ \operatorname{Sym}(3)</math>
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overlayed on a place mat that's marked like so:
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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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1 & 0 & 0
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\\
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0 & 1 & 0
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\\
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0 & 0 & 1
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\end{matrix}</math>
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<math>\begin{matrix}
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0 & 0 & 1
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\\
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1 & 0 & 0
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\\
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0 & 1 & 0
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\end{matrix}</math>
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<math>\begin{matrix}
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0 & 1 & 0
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\\
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0 & 0 & 1
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\\
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1 & 0 & 0
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\end{matrix}</math>
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<math>\begin{matrix}
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1 & 0 & 0
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\\
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0 & 0 & 1
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\\
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0 & 1 & 0
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\end{matrix}</math>
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|
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<math>\begin{matrix}
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0 & 0 & 1
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\\
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0 & 1 & 0
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\\
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1 & 0 & 0
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\end{matrix}</math>
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|
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<math>\begin{matrix}
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0 & 1 & 0
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\\
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1 & 0 & 0
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\\
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0 & 0 & 1
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\end{matrix}</math>
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|}
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o-----o-----o-----o
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<br>
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| a:a | a:b | a:c |
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o-----o-----o-----o
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The key to the mysteries of these matrices is revealed by observing that their coefficient entries are arrayed and overlaid on a place-mat marked like so:
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| b:a | b:b | b:c |
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o-----o-----o-----o
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{| align="center" cellpadding="6" width="90%"
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| c:a | c:b | c:c |
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| align="center" |
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o-----o-----o-----o
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<math>\begin{bmatrix}
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</pre>
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a\!:\!a &
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a\!:\!b &
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a\!:\!c
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\\
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b\!:\!a &
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b\!:\!b &
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b\!:\!c
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\\
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c\!:\!a &
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c\!:\!b &
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c\!:\!c
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\end{bmatrix}</math>
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|}
    
==Note 22==
 
==Note 22==
Jon Awbrey
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