From BRYAN@wvnvm.wvnet.edu Thu May 25 12:01:36 1995 Return-Path: Received: from LCS.MIT.EDU (mintaka.lcs.mit.edu) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA19301; Thu, 25 May 95 12:01:36 EDT Received: from wvnvm.wvnet.edu by MINTAKA.LCS.MIT.EDU id aa25656; 25 May 95 11:55 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 8346; Thu, 25 May 95 11:51:27 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 7593; Thu, 25 May 1995 11:51:00 -0400 Message-Id: Date: Thu, 25 May 1995 11:50:55 -0400 (EDT) From: "Jerry Bryan" To: "Dan Hoey" , "Cube Lovers List" Subject: Re: M-conjugacy vs. C-Conjugacy in the Slice group In-Reply-To: Message of 05/23/95 at 13:11:27 from hoey@AIC.NRL.Navy.Mil I said: >> In fact, I have now verified with a quick search program that >> all M-conjugates in the slice group are also C-conjugates. Hence, >> there are 50 C-conjugate classes in slice, just as there are >> 50 M-conjugate classes. >> In retrospect, I don't think the search program was necessary.... On 05/23/95 at 13:11:27 hoey@AIC.NRL.Navy.Mil said: >and (Jerry) continues with an argument that did not convince me, but the >following does: I think I can both greatly simplify and greatly strengthen the argument that did not convince Dan. My argument is based on the idea (copied from _Symmetry and Local Maxima_) that M-conjugation can be viewed as a permutation on Q, the set of twelve quarter turns. Call the six clockwise quarter turns Q+ and the six counter-clockwise quarter turns Q-. We can observe that the 24 rotations in M all map Q+ to Q+ and map Q- to Q-, and that the 24 reflections in M all map Q+ to Q- and map Q- to Q+. We also note that in particular, the central inversion v is a reflection. Suppose X and Y are M-conjugates in with Y=m'Xm for some fixed m in M. Write X as pairs of quarter turns (each pair is a slice), and write Y as pairs of quarter turns which are respective M-conjugates (via the fixed permutation m) of the quarter turns in X. If the respective quarter turns have been mapped Q+ to Q+ and Q- to Q-, then m is a rotation and we are done. Otherwise, commute the halves of each slice in Y. We first note that so commuting is the identity on Y. We also note that so commuting is equivalent to performing the permutation operation v on Q, and is therefore equivalent to performing v-conjugation on Y. (In passing, we see that this effectively proves Dan's first point, namely that X=v'Xv for all X in . Given that, I would shorten the rest of Dan's argument by saying Y=m'Xm=v'('m'Xm)v=v'm'Xmv, and noting that either m or mv is a rotation). But having started with the "commuting the halves of slices" argument, I would continue as follows. Having commuted the halves of the slices, we still have an M-conjugate (and still the same M-conjugate) because commuting is equivalent to v-conjugation, v is in M, and v-conjugation is the identity in . Finally, having commuted the halves of the slices, we are now mapping Q+ to Q+ and Q- to Q-, so we have a rotation. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU