[Inquiry] Re: Dynamics And Logic

[Jon Awbrey]

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DAL.  Note 21

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We have seen a couple of groups, V_4 and S_3, represented in
several different ways, and we have seen each of these types
of representation presented in several different fashions.
Let us look at one other stylistic variant for presenting
a group representation that is often used, the so-called
"matrix representation" of a group.

Returning to the example of Sym(3), we first encountered
this group in concrete form as a set of permutations or
substitutions acting on a set of letters X = {a, b, c}.
This set of permutations was displayed in Table 17-a,
copies of which can be found here:

http://stderr.org/pipermail/inquiry/2004-May/001419.html
http://forum.wolframscience.com/showthread.php?postid=1321#post1321

These permutations were then converted to "relative form":

   e  =  a:a + b:b + c:c

   f  =  a:c + b:a + c:b

   g  =  a:b + b:c + c:a

   h  =  a:a + b:c + c:b

   i  =  a:c + b:b + c:a

   j  =  a:b + b:a + c:c

>From this relational representation of Sym {a, b, c} ~=~ S_3,
one easily derives a "linear representation", regarding each
permutation as a linear transformation that maps the elements
of a suitable vector space into each other, and representing
each of these linear transformations by means of a matrix,
resulting in the following set of matrices for the group:

Table 21.  Matrix Representations of the Permutations in S_3
o---------o---------o---------o---------o---------o---------o
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` `e` ` | ` `f` ` | ` `g` ` | ` `h` ` | ` `i` ` | ` `j` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
o=-=-=-=-=o=-=-=-=-=o=-=-=-=-=o=-=-=-=-=o=-=-=-=-=o=-=-=-=-=o
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| `1 0 0` | `0 0 1` | `0 1 0` | `1 0 0` | `0 0 1` | `0 1 0` |
| `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` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
o---------o---------o---------o---------o---------o---------o

The key to the mysteries of these matrices is revealed by
observing that their coefficient entries are arrayed and
overlayed on a place mat that's marked like so:

   o-----o-----o-----o
   | a:a | a:b | a:c |
   o-----o-----o-----o
   | b:a | b:b | b:c |
   o-----o-----o-----o
   | c:a | c:b | c:c |
   o-----o-----o-----o

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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