Changes

==Note 18==

==Note 18==

By way of collecting a short-term pay-off for all the work that we did on the regular representations of the Klein 4-group <math>V_4,\!</math> let us write out as quickly as possible in ''relative form'' a minimal budget of representations for the symmetric group on three letters, <math>\operatorname{Sym}(3).</math>  After doing the usual bit of compare and contrast among the various representations, we will have enough concrete material beneath our abstract belts to tackle a few of the presently obscured details of Peirce's early "Algebra + Logic" papers.

By way of collecting a short-term pay-off for all the work that we

did on the regular representations of the Klein 4-group V_4, let us

write out as quickly as possible in "relative form" a minimal budget

of representations for the symmetric group on three letters, Sym(3).

After doing the usual bit of compare and contrast among the various

representations, we will have enough concrete material beneath our

abstract belts to tackle a few of the presently obscured details

of Peirce's early "Algebra + Logic" papers.

Writing the permutations or substitutions of Sym {a, b, c}

Writing the permutations or substitutions of <math>\operatorname{Sym} \{ a, b, c \}</math> in relative form generates what is generally thought of as a ''natural representation'' of <math>S_3.\!</math>

in relative form generates what is generally thought of as

a "natural representation" of S_3.

  e = a:a + b:b + c:c

\operatorname{e}

& = & a\!:\!a

& + & c\!:\!b

& = & a\!:\!c

& + & b\!:\!b

& = & a\!:\!b

& + & b\!:\!a

& + & c\!:\!c

I have without stopping to think about it written out this natural representation of <math>S_3\!</math> in the style that comes most naturally to me, to wit, the "right" way, whereby an ordered pair configured as <math>x\!:\!y</math> constitutes the turning of <math>x\!</math> into <math>y.\!</math> It is possible that the next time we check in with CSP we will have to adjust our sense of direction, but that will be an easy enough bridge to cross when we come to it.

 

 

 

 

 

I have without stopping to think about it written out this natural

representation of S_3 in the style that comes most naturally to me,

to wit, the "right" way, whereby an ordered pair configured as x:y

constitutes the turning of x into y.  It is possible that the next

time we check in with CSP that we will have to adjust our sense of

direction, but that will be an easy enough bridge to cross when we

come to it.

==Note 19==

==Note 19==