To summarize, what i learned from Red Ed's
classes for symmetry techniques:
Only the 26 transformations (or equivalents to them) with >0 invariant grids are possible candidates for grid automorphisms (but there are grids with more than one of these automorphisms).
The most interesting for ST are those, where one or more cells are fixed by the transformation/automorphism. When you know that a cell is fixed, it only can hold a number, that maps to itself, therefore all other candidates (with a cyclic mapping) can be eliminated.
The ones with fixed cells we already know now are (i added the class number):
37 diagonal symmetry (9 cells fixed on the diagonal)
86 90 degree symmetry (1 fixed in the center)
79 180 degree rotational symmetry (1 fixed in the center)
The "new" one is represented by class 134. Fixed are 3 minicolums in the 3 boxes of a band (or 3 minirows in the 3 boxes of a stack). The transformation can be described as: swap 2 bands and in each stack 2 columns. Let me call it "sticks" symmetry here.
E.g. exchanging bands 2 and 3 and columns 2 and 3, 5 and 6, 8 and 9 is a transformation, that maps
r123c1, r123c4 and r123c7 to themselves
r123c2 to r123c3 and vice versa, same for r123c4/r123c5 and r123c8/r123c9
r456c1 to r789c1 and vice versa, same for r456c4/r789c4 and r456c7/r789c7
r456c2 to r789c3 and vice versa, same for r456c3/r789c2, r456c5/r789c6, r456c6/r789c5, r456c8/r789c9, r456c9/r789c8
There are
no other transformations under the 26
with a fixed cell.
Already known are
7 chunk symmetry (e.g. one stack maps to the next)
28 box symmetry (box1->b6->b8, b2->b9->b4 and b3->b5->b7)
AFAIK there are no really ST's for them, but each time, when a "normal" technique allows you to assign/eliminate 1 number, you can do the same for 2 more (or generally for as much as the cycle length is for this cell, e.g 4 for the non center cells with 90 degree symmetry). But i am not sure, if not some Emerald techniques (symm. UR, turbot etc.) can be applied also here. Any samples?
Not for all of the 26 classes there can be found sudoku grids with such an automorphism. I only looked at class 40 closer, where it was rather easy to see, that there cant be such a grid.
samples of ravel's symmetry (or is this gurth's "tenfold symmetry"?), the first three are minimal. ST and singles.
