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→‎Note 19: markup
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==Note 19==
 
==Note 19==
    +
To construct the regular representations of <math>S_3,\!</math> we begin with the data of its operation table:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 
<pre>
 
<pre>
−
To construct the regular representations of S_3,
+
Symmetric Group S_3
−
we pick up from the data of its operation table,
+
o-------------------------------------------------o
−
DAL 17, Table 17-b, at either one of these sites:
+
|                                                |
−
 
+
|                        ^                        |
−
http://stderr.org/pipermail/inquiry/2004-May/001419.html
+
|                    e / \ e                    |
−
http://forum.wolframscience.com/showthread.php?postid=1321#post1321
+
|                      /  \                      |
−
 
+
|                    /  e  \                    |
−
Just by way of staying clear about what we are doing,
+
|                  f / \  / \ f                  |
−
let's return to the recipe that we worked out before:
+
|                  /  \ /  \                  |
−
 
+
|                  /  f  \  f  \                  |
−
It is part of the definition of a group that the 3-adic
+
|              g / \  / \  / \ g              |
−
relation L c G^3 is actually a function L : G x G -> G.
+
|                /  \ /  \ /  \                |
−
It is from this functional perspective that we can see
+
|              /  g  \  g  \  g  \              |
−
an easy way to derive the two regular representations.
+
|            h / \  / \  / \  / \ h            |
−
 
+
|            /  \ /  \ /  \ /  \            |
−
Since we have a function of the type L : G x G -> G,
+
|            /  h  \  e  \  e  \  h  \            |
−
we can define a couple of substitution operators:
+
|        i / \  / \  / \  / \  / \ i        |
−
 
+
|          /  \ /  \ /  \ /  \ /  \          |
−
1. Sub(x, <_, y>) puts any specified x into
+
|        /  i  \  i  \  f  \  j  \  i  \        |
−
    the empty slot of the rheme <_, y>, with
+
|      j / \  / \  / \  / \  / \  / \ j      |
−
    the effect of producing the saturated
+
|      /  \ /  \ /  \ /  \ /  \ /  \      |
−
    rheme <x, y> that evaluates to xy.
+
|      (  j  \  j  \  j  \  i  \  h  \  j  )      |
 +
|      \  / \  / \  / \  / \  / \  /       |
 +
|        \ /   \ /   \ /   \ /  \ /  \ /        |
 +
|        \  h  \  h  \  e  \  j  \  i  /        |
 +
|          \  / \  / \  / \  / \  /          |
 +
|          \ /  \ /  \ /  \ /  \ /          |
 +
|            \  i  \  g  \  f  \  h  /            |
 +
|            \  / \  / \  / \  /            |
 +
|              \ /  \ /  \ /  \ /              |
 +
|              \  f  \  e  \  g  /              |
 +
|                \  / \  / \  /                |
 +
|                \ /  \ /  \ /                |
 +
|                  \  g  \  f  /                  |
 +
|                  \  / \  /                  |
 +
|                    \ /  \ /                    |
 +
|                    \ e  /                    |
 +
|                      \  /                      |
 +
|                      \ /                      |
 +
|                        v                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
</pre>
 +
|}
   −
2.  Sub(x, <y, _>) puts any specified x into
+
Just by way of staying clear about what we are doing, let's return to the recipe that we worked out before:
−
    the empty slot of the rheme <y, _>, with
  −
    the effect of producing the saturated
  −
    rheme <y, x> that evaluates to yx.
     −
In (1), we consider the effects of each x in its
+
It is part of the definition of a group that the 3-adic relation <math>L \subseteq G^3</math> is actually a function <math>L : G \times G \to G.</math> It is from this functional perspective that we can see an easy way to derive the two regular representations.
−
practical bearing on contexts of the form <_, y>,
  −
as y ranges over G, and the effects are such that
  −
x takes <_, y> into xy, for y in G, all of which
  −
is summarily notated as x = {<y : xy> : y in G}.
  −
The pairs <y : xy> can be found by picking an x
  −
from the left margin of the group operation table
  −
and considering its effects on each y in turn as
  −
these run along the right margin. This produces
  −
the regular ante-representation of S_3, like so:
     −
  e  =  e:e  +  f:f  +  g:g  +  h:h  +  i:i  +  j:j
+
Since we have a function of the type <math>L : G \times G \to G,</math> we can define a couple of substitution operators:
   −
  f  =   e:f  +  f:g  +  g:e  +  h:j  +  i:h  +  j:i
+
{| align="center" cellpadding="6" width="90%"
 +
| valign="top" | 1.
 +
| <math>\operatorname{Sub}(x, (\underline{~~}, y))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(\underline{~~}, y),</math> with the effect of producing the saturated rheme <math>(x, y)\!</math> that evaluates to <math>xy.\!</math>
 +
|-
 +
| valign="top" | 2.
 +
| <math>\operatorname{Sub}(x, (y, \underline{~~}))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(y, \underline{~~}),</math> with the effect of producing the saturated rheme <math>(y, x)\!</math> that evaluates to <math>yx.\!</math>
 +
|}
   −
  g  =   e:g +  f:e  +  g:f  +  h:i  +  i:j  + j:h
+
In (1) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(\underline{~~}, y)</math> into <math>xy,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : xy) ~|~ y \in G \}.</math> The pairs <math>(y : xy)\!</math> can be found by picking an <math>x\!</math> from the left margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the right margin. This produces the ''regular ante-representation'' of <math>S_3,\!</math> like so:
   −
  h   =   e:h + f:i + g:j + h:e + i:f + j:g
+
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{array}{*{13}{c}}
 +
\operatorname{e}
 +
& = & \operatorname{e}\!:\!\operatorname{e}
 +
& + & \operatorname{f}\!:\!\operatorname{f}
 +
& + & \operatorname{g}\!:\!\operatorname{g}
 +
& + & \operatorname{h}\!:\!\operatorname{h}
 +
& + & \operatorname{i}\!:\!\operatorname{i}
 +
& + & \operatorname{j}\!:\!\operatorname{j}
 +
\\[4pt]
 +
\operatorname{f}
 +
& = & \operatorname{e}\!:\!\operatorname{f}
 +
& + & \operatorname{f}\!:\!\operatorname{g}
 +
& + & \operatorname{g}\!:\!\operatorname{e}
 +
& + & \operatorname{h}\!:\!\operatorname{j}
 +
& + & \operatorname{i}\!:\!\operatorname{h}
 +
& + & \operatorname{j}\!:\!\operatorname{i}
 +
\\[4pt]
 +
\operatorname{g}
 +
& = & \operatorname{e}\!:\!\operatorname{g}
 +
& + & \operatorname{f}\!:\!\operatorname{e}
 +
& + & \operatorname{g}\!:\!\operatorname{f}
 +
& + & \operatorname{h}\!:\!\operatorname{i}
 +
& + & \operatorname{i}\!:\!\operatorname{j}
 +
& + & \operatorname{j}\!:\!\operatorname{h}
 +
\\[4pt]
 +
\operatorname{h}
 +
& = & \operatorname{e}\!:\!\operatorname{h}
 +
& + & \operatorname{f}\!:\!\operatorname{i}
 +
& + & \operatorname{g}\!:\!\operatorname{j}
 +
& + & \operatorname{h}\!:\!\operatorname{e}
 +
& + & \operatorname{i}\!:\!\operatorname{f}
 +
& + & \operatorname{j}\!:\!\operatorname{g}
 +
\\[4pt]
 +
\operatorname{i}
 +
& = & \operatorname{e}\!:\!\operatorname{i}
 +
& + & \operatorname{f}\!:\!\operatorname{j}
 +
& + & \operatorname{g}\!:\!\operatorname{h}
 +
& + & \operatorname{h}\!:\!\operatorname{g}
 +
& + & \operatorname{i}\!:\!\operatorname{e}
 +
& + & \operatorname{j}\!:\!\operatorname{f}
 +
\\[4pt]
 +
\operatorname{j}
 +
& = & \operatorname{e}\!:\!\operatorname{j}
 +
& + & \operatorname{f}\!:\!\operatorname{h}
 +
& + & \operatorname{g}\!:\!\operatorname{i}
 +
& + & \operatorname{h}\!:\!\operatorname{f}
 +
& + & \operatorname{i}\!:\!\operatorname{g}
 +
& + & \operatorname{j}\!:\!\operatorname{e}
 +
\end{array}</math>
 +
|}
   −
  i  =   e:i +  f:j  +  g:h  +  h:g  +  i:e  + j:f
+
In (2) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(y, \underline{~~}),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(y, \underline{~~})</math> into <math>yx,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : yx) ~|~ y \in G \}.</math> The pairs <math>(y : yx)\!</math> can be found by picking an <math>x\!</math> on the right margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the left margin. This produces the ''regular post-representation'' of <math>S_3,\!</math> like so:
   −
  j   =   e:j + f:h + g:i + h:f + i:g + j:e
+
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{array}{*{13}{c}}
 +
\operatorname{e}
 +
& = & \operatorname{e}\!:\!\operatorname{e}
 +
& + & \operatorname{f}\!:\!\operatorname{f}
 +
& + & \operatorname{g}\!:\!\operatorname{g}
 +
& + & \operatorname{h}\!:\!\operatorname{h}
 +
& + & \operatorname{i}\!:\!\operatorname{i}
 +
& + & \operatorname{j}\!:\!\operatorname{j}
 +
\\[4pt]
 +
\operatorname{f}
 +
& = & \operatorname{e}\!:\!\operatorname{f}
 +
& + & \operatorname{f}\!:\!\operatorname{g}
 +
& + & \operatorname{g}\!:\!\operatorname{e}
 +
& + & \operatorname{h}\!:\!\operatorname{i}
 +
& + & \operatorname{i}\!:\!\operatorname{j}
 +
& + & \operatorname{j}\!:\!\operatorname{h}
 +
\\[4pt]
 +
\operatorname{g}
 +
& = & \operatorname{e}\!:\!\operatorname{g}
 +
& + & \operatorname{f}\!:\!\operatorname{e}
 +
& + & \operatorname{g}\!:\!\operatorname{f}
 +
& + & \operatorname{h}\!:\!\operatorname{j}
 +
& + & \operatorname{i}\!:\!\operatorname{h}
 +
& + & \operatorname{j}\!:\!\operatorname{i}
 +
\\[4pt]
 +
\operatorname{h}
 +
& = & \operatorname{e}\!:\!\operatorname{h}
 +
& + & \operatorname{f}\!:\!\operatorname{j}
 +
& + & \operatorname{g}\!:\!\operatorname{i}
 +
& + & \operatorname{h}\!:\!\operatorname{e}
 +
& + & \operatorname{i}\!:\!\operatorname{g}
 +
& + & \operatorname{j}\!:\!\operatorname{f}
 +
\\[4pt]
 +
\operatorname{i}
 +
& = & \operatorname{e}\!:\!\operatorname{i}
 +
& + & \operatorname{f}\!:\!\operatorname{h}
 +
& + & \operatorname{g}\!:\!\operatorname{j}
 +
& + & \operatorname{h}\!:\!\operatorname{f}
 +
& + & \operatorname{i}\!:\!\operatorname{e}
 +
& + & \operatorname{j}\!:\!\operatorname{g}
 +
\\[4pt]
 +
\operatorname{j}
 +
& = & \operatorname{e}\!:\!\operatorname{j}
 +
& + & \operatorname{f}\!:\!\operatorname{i}
 +
& + & \operatorname{g}\!:\!\operatorname{h}
 +
& + & \operatorname{h}\!:\!\operatorname{g}
 +
& + & \operatorname{i}\!:\!\operatorname{f}
 +
& + & \operatorname{j}\!:\!\operatorname{e}
 +
\end{array}</math>
 +
|}
   −
In (2), we consider the effects of each x in its
+
If the ante-rep looks different from the post-rep, it is just as it should be, as <math>S_3\!</math> is non-abelian (non-commutative), and so the two representations differ in the details of their practical effects, though, of course, being representations of the same abstract group, they must be isomorphic.
−
practical bearing on contexts of the form <y, _>,
  −
as y ranges over G, and the effects are such that
  −
x takes <y, _> into yx, for y in G, all of which
  −
is summarily notated as x = {<y : yx> : y in G}.
  −
The pairs <y : yx> can be found by picking an x
  −
on the right margin of the group operation table
  −
and considering its effects on each y in turn as
  −
these run along the left margin.  This generates
  −
the regular post-representation of S_3, like so:
  −
 
  −
  e  =  e:e  +  f:f  +  g:g  +  h:h  +  i:i  +  j:j
  −
 
  −
  f  =  e:f  +  f:g  +  g:e  +  h:i  +  i:j  +  j:h
  −
 
  −
  g  =  e:g  +  f:e  +  g:f  +  h:j  +  i:h  +  j:i
  −
 
  −
  h  =  e:h  +  f:j  +  g:i  +  h:e  +  i:g  +  j:f
  −
 
  −
  i  =  e:i  +  f:h  +  g:j  +  h:f  +  i:e  +  j:g
  −
 
  −
  j  =  e:j  +  f:i  +  g:h  +  h:g  +  i:f  +  j:e
  −
 
  −
If the ante-rep looks different from the post-rep,
  −
it is just as it should be, as S_3 is non-abelian
  −
(non-commutative), and so the two representations
  −
differ in the details of their practical effects,
  −
though, of course, being representations of the
  −
same abstract group, they must be isomorphic.
  −
</pre>
      
==Note 20==
 
==Note 20==
Jon Awbrey
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