Anyway, here follow some more remarks. The scenario from above is the periodical remis case. Stalemate is easily contained when some corners are present. Later one has to see how a mate can be modelled (resonance or s.th.).
The idea is that the pieces keep their strategy as long as they don't have to change their plans. It's a deterministic approach.
The question later on arises how to embed this into GL (again involving some multiplier). Perhaps it's again about faces-dim (norming constraints etc. will reduce this of course) over some field (whose elements indicate which piece is standing on the face, or since each piece has got its spin, and besides the pieces can be linked, so perhaps we better won't use a simple field but a double ring or so)? Finite dimensional Lie algebras have to do with this, we only need to construct some cool operators. Look at our group as a solution to some difference equation.
Perhaps furthermore one can break the 'ensemble'-generators into single piece generators (moving one piece and thereby morphing the world for the other piece or so)! When a Chess piece moves, it measures the location of the other piece (and changes its momentum), can we put it this way? Simply let's dump out a second system (this time starting from the other piece's point of view), so again some (a,a), (b,a), (a,b) using a parallel label set (the pieces' numbering interchanged). Then the two generators "a" and "b" will be involutions, jumping from one label set to the other, moving one piece and 'rotating' the other single-piece-label. Well, somehow like this at least...
We want some CSA Hamiltonian (unitary sophisticated representation), make up an partial difference equation for all this and reduce the problem to solving the equation. Can we say that we use a partial difference equation first order in time (as opposed to s.th. like the wave equation which is second order in time) somehow or other using 'complex' differentiation (holomorphic or so), we don't need t=2 but besides looking at dx we also have a fixed relation to dy. And are we giving an adjoint representation of some Lie algebra acting on vector fields or so, that is we have the two generators rot1 and rot2 (acting on the first and second piece respectively), and 'adjoint' means we apply the rotation, then the evolution in time, and then undoing the rotation to arrive at a standardized label. Yes, our procedure has to do with this idea, whereas the combination of rot1 and rot2 is even more intricate (we can't apply rot2 where we would be allowed to apply rot1, because it's the first piece's turn). You must admit, our construction is quite elaborate.
Please keep in mind that the permutation representation will not be the final tool, because the number of labels will grow too much. But imagine we deal with the trivial case of two pieces of type A and increase the board. Whilst alpha-beta search and the permutations will become unhandable, a treatment with number fields will just use bigger fields, and since in general we use funtions and operators on them, the growth of information will be compensated by the use of fields with more elements (the values of the functions) as well as by the fact that the invariants (metrics,
Probably the whole point in the approach to involve group theory will be to find out how to decompose into smaller chunks (see the 4 modes from above). One can choose a connection (along which the holonomy groups are developed) so that the stuff neatly separates (we expect that the movement of the ensemble can be factored out so to say). What does the operator look like meassuring the overall momentum of the ensemble, and does it generate an ideal in the group ring? Just take the subgroup generated by "aa" and "bb" (but thereby pretending that the labels are all having the corresponding orientation which they actually don't have)? Ok, in the GL it's some derivation thing. And does the overall movement couple on the subtleties of the graph, that is to say: is a plain overall movement always possible?
Let's face it, the 2-body case (especially if they are of the same type) is trivial from a Chess player's perspective, and so the underlying mathematics should turn out to be treatable as well. We can already say that (once we use the appropriate representation) a blowing up (like increasing the board size) will not have any effect on the mathematical complexity of our calculations, so we are on the right track. But the technical issues will be understood better and better only after having done a lot of examples.
And due to the situations involving mate approaching (there is an implicit leveling (grading), mate in n steps, then mate in (n-1) and so on), the transitions through the positions of capture (similar to a tunneling spring system) will perhaps have to be taken into account? Or maybe those operations are not invertible anymore, so the algebra (the laws so to say, the equations describing the system regardless of being drawn or won/lost) is more fundamental than the group only dealing with the drawn situations?! And a move of a type D piece, is it invertible or not?!
We have got the impression that certain Brommann algebras are involved here (they still have to be written down to be accessible though).
Perhaps one could model the Go game in the same fashion. Every party has one piece, and (alternating turns) it moves from the location of the previous move to the next place. The several disconnected board areas (former putting of stones will build walls like in the game of Amazons) simply result in intransitive groups (there is no need for an increasing number of generators or s.th.). The pieces don't have any special moving rule (in fact you can place a stone where you want), but in this way our holonomy groups come into play again, this is the goal. And yes, unless in the rare case where we have a Ko situation but no Ko rule is used (or more generally a bigger group has been in Atari, so even with Ko rule retaking would be allowed) and the opponent directly retakes, of course we are not allowed to place another stone into the previous location, so this again can be modelled by constraints which we obey (build the quotient, we are in the cokernel and so on). Somehow model the scoring and we are done, there are classes with specified value under optimal play, invariant while traversing the positions as time evolves. To give more explanation of what we mean, let us give the hint, that for example when a ladder is happening (not as a mistake, but let it be the correct move in some position), then the attacker's subsequent stone puttings are diagonally separated, but that's only the representation, it's the same group element. Just think projectively!