## enigma-era representation theory, constant spaces and orthogonal duals

enigma-era math is famous for the work of the Polish cryptographers who exploited the notion of conjugacy classes (that relate to the character functions of permutation functions).

In Turing’s only known math-centric treat of enigma-era theories about crypto, in his On Permutations manuscript,  we see him also focusing on represenation theory of the symmetric group.

When turing modeled U+x.U-y

we now see that his whole argument is about swapping the order or rotors (by the action of the representation of the input group member – a cycle from the upright, recall).

What is more, we finally have a solid metal model for why, in sub-representation theory, one wises to constrain the +x and –y (etc) so they are 0 – before and after the permuting effect of the input cycle. We see how his H is what these days might be called Vper in contrast to V and Vconst.

The constraining of the representation of the input group “cycle” to the Vper impacts the degree of the resulting Cayley graph -reminding use somewhat of those NKD codes of the form (n, n-1, n-1). But, since we are in a q-ary world already with enigma wheels (which makes q=26) we should not be thinking in binary (i.e. q=2) but in terms of Reed Solomon (where each of the sequence of rotors, above, is the nth degree of the reed solomon world polynomial).

With this perspective we see how rotors, with suitable feedback, can be seen in the that reed solomon coding sense. They are calculating in a particular field (or ring); in which “rotors” can divide “rotors” to form residue classes – in analogy to polynomial fields.

We have to recall that the unitary condition means that distances measured after the transform are the same as the distance between the terms used in the domain applied to the transform. And we do recall that Turing’s main argument hinged on such numbers – formed by considering the pairwise distances between the terms in the image space, figured after computing the rod specified by the power term (conjugating the original upright’s alphabet, upon rotating it relative to the diagonal by +x, or –y, etc).

Now it was always a little confusing what Turing meant about his use of alpha term, and the associated 1-dimensional argument. But that becomes a little clearer if we think in terms of “diagonalizing a matrix, having computed eigenvalues and their multiplicities”. If we now take Vconst as a (rotationally “scaled” (by alpha degrees) Sn-invariant subspace (just as is Vper) then we have another clear analogy to coding theory, Vper is the space orthogonal  to Vconst.

Take a circle’ and take the line from origin to the rightmost point. Now rotate that line a bit anticlockwise thereby “scaling it” by “alpha” degrees. Next, create cones of future and historical tree development at right angle to that (rotated, translated, scaled) line…. and this is our coding space – with bi-infinite sequences of historical and future branching events and a p-adic distance measures between points on the cones’ leading edges.

I’ve actually purchase a e-copy of a good book that discusses the background theory to qudits in the context of expander graphs – all expressed in terms of group theory and permutation representation of symmetric groups (without mentioning enigma and rotors, of course). The text hardly says eigenvalue (or makes one do boring linear algebra fiddling) unless the eigenvalues means something of relevance either to diagonalizing matrices to form quantum gates or noting how relations between eigenvalues in a spectral basis relate to the conditions of expander graph families.

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