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)