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Date: Sun, 6 Nov 94 23:29 PST
From: Martin.Schoenert@math.rwth-aachen.de
To: cube-lovers@life.ai.mit.edu
Cc: CRSO.Cube@canrem.com
In-Reply-To: Mark Longridge's message of Sat, 5 Nov 1994 22:16:00 -0500 <60.846.5834.0C1BB9FA@canrem.com>
Subject: Re: Shifty Invariance

Mark writes in his e-mail message of 1994/11/05

    After a bit of computer cubing I found:

    p183 6 Twist      R1 U3 R2 U3 R1 D3 U3 R1 U3 R3 D2 R3 U3 R1 D3 U3
                      (18 q  or  16 h  moves)

     This requires using the larger group of <U1, R1, D1>, although I
    expected a 16 turn process. Note the fact this larger group has face
    index 3 (rather than 2). But now the process is NOT shift invariant
    and we see the route itself can determine whether it will be
    shift invariant!

    I welcome any mathematical explanation!

As I tried to explain in my first e-mail message, a shift invariant
process is a process in a subgroup X of G corresponding to an element
x in the centre *of this subgroup*.

The ``odd'' element is an element in the centre of the subgroup < U, R >.
Thus any process effecting this element written in U and R is a shift
invariant process.  UR11 is one such process.

However, the ``odd'' element does not lie in the centre of the subgroup
< U, R, D > (in fact this subgroup has trivial centre).  Thus a process
effecting this element *involving D*, will *not* be shift invariant.

Some shift invariant processes are in fact in the centre of multiple
subgroups.  For example the square elements, except for the ``diagonal
square'' element, have this property.

For such elements one has some choice which generators to use.  For
example the ``single square'' elements (U2 R2)^3 lies in the centre of
< U2, R2 > and < U2, D, R2, L > (and all subgroups inbetween), so every
process effecting this element involving any subset of U2, D, D2, R2, L,
and L2, will be a shift invariant process.

For the ``odd'' element, one has now choice.  It lies in the centre of
< U, R >, but not in the centre of any larger group.  Thus a shift
invariant process effecting the ``odd'' element must involve U and R,
and cannot involve more generators.

Martin.

-- .- .-. - .. -.  .-.. --- ...- . ...  .- -. -. .. -.- .-
Martin Sch"onert,   Martin.Schoenert@Math.RWTH-Aachen.DE,   +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany