Date: Fri, 9 Oct 1998 10:02:00 -0500 (CDT)
To: Dave Rusin
From: bcolletti@mail.utexas.edu (Bruce W. Colletti)
Subject: Re: Transversal of Orbits
Dave
Thank you for the message and for withholding my intended posting.
>You'll have to specify the context -- what kind of groups? of actions? --
>if you want useful response. Presumably you expect X/G to be finite. If
>G is too, you "just" loop over this:
> pick x in X ; replace X by X - { g*x | g in G }
The acting groups are subgroups of S(n), the symmetric group on n-letters,
and the sets acted upon are conjugacy classes in S(n). While your algorithm
above certainly works, I'm wondering if one can avoid that set difference.
There are ways to build transversals of cosets but I can't find anything
similar for orbital transversals.
Should I repost a modified message or should I send it to you for review
first? Perhaps the answer is that there is no such algorithm and if so, I'd
rather not pose the question to our broader community.
Thanks again for being an alert moderator and saving me from embarassment.
Bruce
Bruce W. Colletti
PO BOX 7022
Austin TX 78713
512-832-5347
bcolletti@mail.utexas.edu
bcolletti@compuserve.com (permanent)
==============================================================================
Date: Sat, 10 Oct 1998 11:45:15 -0500 (CDT)
To: Dave Rusin
From: bcolletti@mail.utexas.edu (Bruce W. Colletti)
Subject: Re: Transversal of Orbits
>If I understand your situation now, you have a subgroup H of S(n) acting
>on a conjugacy class C of S(n) and you want to decompose it into orbits.
>As a S(n)-set, C is the same as the coset space S(n)/C(x) (where C(x)
>is the centralizer in S(n) of a fiduciary element x in that conjugacy
class).
>Thus what you are asking for is a set of double coset representatives in
>H \ S(n) / C(x). Have I got that right?
Dear Dave
Thank you for the reply. It was incredibly useful and I've worked out its
logic (thus my delayed reply). However, I may misunderstand your notation H
\ S(n) / C(x), as found below. Here, conjugation is x^t = inverse(t)*x*t.
A GAP program shows that if T is a transversal on the double cosets
{H*s*C(x): s in S(n)}, then X = {x^t: t in T} isn't a transversal on the
orbits. However, X is an orbital transversal if the double cosets are
instead {C(x)*s*H: s in S(n)}. Does your notation H \ S(n) / C(x)
represent these latter double cosets? My "proof" suggests yes but I feel it
prudent to ask.
The result which you cited is useful but I find it nowhere in my algebra
texts. Some weeks ago I posed the original question to algebraists here but
they didn't know (undoubtedly I explained it poorly, since I am not a
mathematician). Thus, a bibliographic citation would be welcome but only if
you have it at hand.
Instinct says it's best not to repost any messages on this matter to
sci.math.research. If you feel otherwise, please say so.
Thank you again for opening the door which I had sought in vain. I
appreciate the courtesy which you extended. It has definitely brought peace
of mind. We'll now put this question to rest.
Bruce
Bruce W. Colletti
PO BOX 7022
Austin TX 78713
512-832-5347
bcolletti@mail.utexas.edu
bcolletti@compuserve.com (permanent)