From cube-lovers-errors@mc.lcs.mit.edu Tue Apr 27 18:00:36 1999 Return-Path: Received: from sun28.aic.nrl.navy.mil (sun28.aic.nrl.navy.mil [132.250.84.38]) by mc.lcs.mit.edu (8.9.1a/8.9.1-mod) with SMTP id SAA28553 for ; Tue, 27 Apr 1999 18:00:35 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Date: Tue, 27 Apr 1999 15:07:29 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Conjugation done right [Re: Inventing your own techniques] To: Cube Lovers Message-Id: On Thu, 08 Apr 1999 19:45:22 -0400 (EDT) Dan Hoey wrote: > Jerry Bryan wrote: > > > A process of the form Y'XY is called a conjugate, and in particular is called > > the conjugate of X by Y. Note that YXY' is also a conjugate, and in particular > > is called the conjugate of X by Y'. This can be a little confusing because a > > few books (incorrectly in my opinion) call YXY' the conjugate of X by Y. ... and Dan wrote: > > I tried to explain this a while ago, but I guess it didn't quite work. > One form of conjugate is right, and the other form is wrong, but just > which is right depends on how you write function composition. My apologies for leaving out a couple of things in my note about conjugation and commutators. But if I hadn't left them out, I doubt that we would have received Dan's very nice little message about rightward and leftward composition. I should have mentioned the rightward vs. leftward composition issue. As Dan points out, it is certainly the case that Y'XY can be correct in one book while YXY' can be correct in another. However, what I *meant* to say was that I had seen books in which (unless I was missing something obvious) the author's definition of conjugation did not correspond to the author's composition direction. All I was really trying to say was that irrespective of the author's chosen direction, YXY' and Y'XY are both conjugates. It's just that in one case, you have the conjugate of X by Y and in the other you have the conjugate of X by Y', and which is which depends on the right vs. left system the author is using. I cited the reason "X shifted by Y" as a reason for preferring the conjugate of X by Y to be Y'XY in the Cube-Lovers system. I should also have mentioned the homomorphism vs. antihomomorhism issue. There are two reasons for preferring Y'XY to YXY' for the conjugate of X by Y in the Cube-Lovers system, and regrettably I only mentioned one of them. Homomorphism is the other. Dan covered the homomorphism issue extremely well, so I would like to make some additional comments about the "X shifted by Y" interpretation of conjugation. Let us suppose that we have a maneuver A = L2 F2 L2 U L' R B R2 D2 R2 B' L R' U' which flips the uf cubie and the df cubie while leaving the rest of the cube unchanged (the rest of the cubies are said to be fixed by A). The uf and df cubies are the edge cubies which are in the middle of the top row and the middle of the bottom row of the front face, respectively. (The given maneuver for A is minimal in the face turn metric, but the exact maneuver doesn't matter for our purposes.) Suppose that instead we want to flip the ul cubie and the df cubie. A maneuver which will do so is U'AU. The U' move brings the ul cubie into the uf cubie's place while leaving the df cubie where it was. The A maneuver flips the *contents* of the uf cubicle which is now the ul cubie and flips the df cubie as usual. The U moves returns the (now flipped) ul cubie to the ul cubicle. We might write the actions of the A maneuver as follows: A: uf -> fu (flips the uf cubie) df -> fd (flips the df cubie) We might write the actions of the U and U' moves on the edge cubies as follows: U: uf -> ul U': uf -> ur ul -> ub ur -> ub ub -> ur ub -> ul ur -> uf ul -> uf Hence, if we just consider the AU part of U'AU, we have that the uf cubie goes to fu which in turn goes to lu. (If U performs uf -> ul, then it equivalently performs fu -> lu). So the uf cubie is carried to the ul cubicle and flipped to be lu. This is the general idea of what we want (to flip the cubie which is in the ul cubicle), but it is the wrong cubie in the ul cubicle. So preceding AU by U' "cancels" the movement of the cubie (and also the movement of all the other cubies) and retains only the flip of the cube. The net result is that the cubie which is flipped is shifted from being the uf cubie to being the ul cubie, as desired. A is shifted by U, which is what we wanted. It should be clear that UAU' flips the ur and the df cubies. UAU' flips the ur rather than the uf cubie, which we can describe as saying that A has been shifted by U'. Essentially any two edge cubies can be flipped by variations on this basic theme. Our next example will involve whole cube moves. We denote the whole cube move of grasping the right face and turning the whole cube clockwise and counterclockwise by c_R and c_R', respectively. So c_R' would bring the bu and fu cubies into the uf and df cubicles (respectively), A would flip them, and c_R would return them to their original locations. The net result is that the maneuver (c_R' A c_R) flips the bu and the fu cubies. It is standard on Cube-Lovers to denote the 24 rotations of the cube by C, and we might write a C-conjugate as c'Ac where c is some fixed but arbitrary element of C. c_R and c_R' are just two particular elements of C. Working with a real cube, you probably wouldn't even think about C-conjugation in this particular context -- you would just do it. That is, if your hands knew how to perform the A maneuver to flip the uf and df cubies, and if you needed to flip two edge cubies which were on opposite sides of the same face, you would just rotate the whole cube in space to bring the two cubies which needed to be flipped into the uf and df locations and then you would perform the A maneuver -- simpler to do than to describe. It is more common on Cube-Lovers to talk about M-conjugation than to talk about C-conjugation, where M is the group of 24 rotations and 24 reflections of the cube. C is a subgroup of M. So c_R and c_R' just as well elements of M as they are of C, and our (c_R' A c_R) maneuver is a good example of M-conjugation. M-conjugation lets us deal with reflections in addition to rotations, which in effect means it let's us treat clockwise and counterclockwise moves as equivalent when appropriate for symmetry purposes. But when we are dealing with whole cube rotations of a real cube, we are just dealing with C-conjugation. In the case of our A maneuver, C-conjugation means that c'Ac lets us flip any two cubies anywhere on the cube which are opposite edge cubies on the same face of the cube. Finally, whole cube rotations are a convenient way to apply the maneuver A to any face of a real cube. But mathematically, we really do not have to perform whole cube rotations. We can use C-conjugation (and more generally, M-conjugation) to apply the "same" maneuver to a different face. Consider again (c_R' A c_R). If we write out A, we get c_R' (L2 F2 L2 U L' R B R2 D2 R2 B' L R' U') c_R But the maneuver (c_R c_R') is equal to the identity, so we can insert it between each face move thusly. c_R' L2 (c_R c_R') F2 (c_R c_R') L2 (c_R c_R') U (c_R c_R') etc. Now, we can re-associate thusly so that we have the c_R-conjugate of each face turn. (c_R' L2 c_R) (c_R' F2 c_R) (c_R' L2 c_R) (c_R' U c_R) etc. Finally, if we actually perform the calculations, we discover that conjugation by c_R leaves L, L', L2, R, R', and R2 alone; it takes F, F', and F2 to U, U', and U2, respectively; it takes U, U', and U2 to B, B', and B2, respectively;, it takes B, B', and B2 to D, D', and D2, respectively; and it takes D, D', and D2 to F, F', and F2, respectively. Hence, conjugation by r_C gives us a maneuver to flip the bu and fu cubies thusly. A = L2 F2 L2 U L' R B R2 D2 R2 B' L R' U' (flip uf and df) c_R' A C_R = L2 U2 L2 B L' R D R2 F2 R2 D' L R' B' (flip bu and fu) ---------------------------------------- Jerry Bryan jbryan@pstcc.cc.tn.us