From:

~~~ Subject:

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