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 <[email protected]> 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: [email protected]
> > To: [email protected]
> > 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=)
> > ----------------------------------------------------------------------
> > For information about J forums see http://www.jsoftware.com/forums.htm
>
> ----------------------------------------------------------------------
> For information about J forums see http://www.jsoftware.com/forums.htm
>
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