Thanks,
I'm worried your lcosets and rcosets are actually the same verb, given that
the arguments are swapped.
The idea is that left multiplying (C.) everything by v4 and right multiplying
everything by v4 will give individually different result, because A4 is not
commutative, but will partition A4 into the same sets.
Does not the "~" swap a4 and v4 in rcosets, so that it is effectively the same
as lcosets?
If so, this is not what I intended.
a4 lcosets v4 should be the results of doing " for all a in a4, v in v4,
a*v", where * can be considered as C.
whereas,a4 rcosets v4 should be the results of doing the same but v*a, instead
of a*v.
Both your lcosets and rcosets verbs seem to do the same, v C. a.
Unless, that is, my reading of your verbs is wrong.
Regards.
> From: rauldmil...@gmail.com
> Date: Sat, 12 Jul 2014 22:41:38 -0400
> To: programm...@jsoftware.com
> Subject: Re: [Jprogramming] I just don't C. it.
>
> I've reduced your notation slightly - playing with alternate expressions
> helps me understand.
> Sym=: A.&i.~ !
> Alt=: ({~ 1 I.@:= C.!.2)@:Sym
> rcosets=: ~.@:(/:~"2)@:(C."_ 1~)
> a4=: Alt 4
> v4=: 0 7 16 23 A.i.4
> lcosets=: ~.@:(/:~"2)@:(C."_ 1)
>
> (a4 rcosets v4) -: (v4 lcosets a4)
> 1
>
> Note that I played with sub expressions (and sometimes looked at result
> shapes instead of results) while playing with this.
>
> It seems obvious now that I think I have things straight, but originally it
> was not. But let's see if you agree.
>
> Thanks,
>
> --
> Raul
>
>
>
> On Sat, Jul 12, 2014 at 9:16 PM, Jon Hough <jgho...@outlook.com> wrote:
>
> > Thanks for the reply, but my problem still persists.
> > Please consider the following verbs:
> >
> > Sym =: (i. @: !) A. i.
> >
> >
> >
> >
> > Alt =: ( (I. @: (1&=) @: (C.!.2)) { ] ) @: Sym
> >
> >
> >
> >
> > a4 =. Alt 4
> >
> >
> > a4 is the alternating group on 4 letters.
> > Now I'll sloppily create v4
> >
> >
> >
> >
> > v4 =. 4 4 $ 0 1 2 3 1 0 3 2 2 3 0 1 3 2 1 0
> >
> >
> > v4 (klien four group) is a normal subgroup of a4. This means the left
> > cosets and right cosets of v4 in a4 should be the same.
> >
> >
> > i.e.
> >
> >
> > if I multiply ( use C.) all elements of a4 with v4, and then nub out
> > duplicates, a4 will be partitioned into 3 sets of size 4.
> >
> >
> > here is the right coset verb:
> >
> >
> > rcosets =: ~.@:(/:~"2)@:( ] C."_ 1 [ ) NB. a4 should be on left, v4 on
> > right.
> >
> >
> > Yes the verb is weird, I swap x and y when using C. So we have v4 C."(_ 1)
> > a4 , and then nub duplicates. I get
> >
> >
> > a4 rcosets v4
> >
> >
> >
> >
> > 0 1 2 3
> > 1 0 3 2
> > 2 3 0 1
> > 3 2 1 0
> >
> > 0 2 3 1
> > 1 3 2 0
> > 2 0 1 3
> > 3 1 0 2
> >
> >
> > 0 3 1 2
> > 1 2 0 3
> > 2 1 3 0
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >
> > 3 0 2 1
> >
> >
> > i.e. I get 3 sets of 4 permutations. Exactly what I wanted.
> > What I cannot do, is find the equivalent verb for getting left cosets.
> > i.e.
> > I need a verb that does:
> > a4 C."(r1 r2) v4 NB. what ranks r1, r2?
> > and then nub the resulting duplicate sets ~.@:(/:~"2
> > And the answer SHOULD be the same as the right cosets.
> > So, if I try
> >
> > lcosets =: ~.@:(/:~"2) @: ( C."(1 _) ) NB. superfluous brackets, I know
> >
> >
> > The reuslt of
> >
> >
> > a4 lcosets v4
> >
> >
> > is simply
> >
> >
> > 0 1 2 31 0 3 22 3 0 1
> >
> >
> >
> >
> > 3 2 1 0
> >
> >
> > i.e. it's just v4 again. This makes no sense to me.
> >
> >
> >
> >
> > > Date: Sat, 12 Jul 2014 07:24:32 -0400
> > > From: schott.br...@gmail.com
> > > To: programm...@jsoftware.com
> > > Subject: Re: [Jprogramming] I just don't C. it.
> > >
> > > The explanation may be the rank of C, as you ask.
> > >
> > > C. b. 0
> > > 1 1 _
> > > 0 2 3 1 C."1 v
> > > 0 2 3 1
> > > 1 3 2 0
> > > 2 0 1 3
> > > 3 1 0 2
> > >
> > >
> > >
> > >
> > > --
> > > (B=)
> > > ----------------------------------------------------------------------
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> >
> > ----------------------------------------------------------------------
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> >
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