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===Note 21===
===Note 21===
We've seen a couple of groups, ''V''<sub>4</sub> and ''S''<sub>3</sub>, 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, V_4 and S_3, 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 S_3,
Recalling the manner of our acquaintance with the symmetric group ''S''<sub>3</sub>, we began with the "bigraph" (bipartite graph) picture of its natural representation as the set of all permutations or substitutions on the set ''X'' = {''A'', ''B'', ''C''}.
we began with the "bigraph" (bipartite graph) picture of its natural
representation as the set of all permutations or substitutions on
the set X = {A, B, C}.
<pre>
Table 1. Permutations or Substitutions in Sym {A, B, C}
Table 1. Permutations or Substitutions in Sym {A, B, C}
o---------o---------o---------o---------o---------o---------o
o---------o---------o---------o---------o---------o---------o
| | | | | | |
| | | | | | |
o---------o---------o---------o---------o---------o---------o
o---------o---------o---------o---------o---------o---------o
</pre>
Then we rewrote these permutations -- since they are
Then we rewrote these permutations — since they are functions ''f'' : ''X'' → ''X'' they can also be recognized as 2-adic relations ''f'' ⊆ ''X'' × ''X'' — in "relative form", in effect, in the manner to which Peirce would have made us accustomed had he been given a relative half-a-chance:
functions f : X -> X they can also be recognized as
2-adic relations f c X x X -- in "relative form",
in effect, in the manner to which Peirce would
have made us accustomed had he been given
a relative half-a-chance:
<pre>
e = A:A + B:B + C:C
e = A:A + B:B + C:C
j = A:B + B:A + C:C
j = A:B + B:A + C:C
</pre>
These days one is much more likely to encounter the natural representation
These days one is much more likely to encounter the natural representation of ''S''<sub>3</sub> in the form of a "linear representation", that is, as a family of linear transformations that map the elements of a suitable vector space into each other, all of which would in turn usually be represented by a set of matrices like these:
of S_3 in the form of a "linear representation", that is, as a family of
linear transformations that map the elements of a suitable vector space
into each other, all of which would in turn usually be represented by
a set of matrices like these:
<pre>
Table 2. Matrix Representations of the Permutations in Sym(3)
Table 2. Matrix Representations of the Permutations in Sym(3)
o---------o---------o---------o---------o---------o---------o
o---------o---------o---------o---------o---------o---------o
| | | | | | |
| | | | | | |
o---------o---------o---------o---------o---------o---------o
o---------o---------o---------o---------o---------o---------o
</pre>
The key to the mysteries of these matrices is revealed by noting that their
The key to the mysteries of these matrices is revealed by noting that their coefficient entries are arrayed and overlayed on a place mat marked like so:
coefficient entries are arrayed and overlayed on a place mat marked like so:
<pre>
[ A:A A:B A:C |
[ A:A A:B A:C |
| B:A B:B B:C |
| B:A B:B B:C |
| C:A C:B C:C ]
| C:A C:B C:C ]
</pre>
Of course, the place-settings of convenience at different symposia may vary.
Of course, the place-settings of convenience at different symposia may vary.
==Differential Logic : Series B==
==Differential Logic : Series B==