Galois groups and genes

In an article discussing a TGD inspired model for possible variations of Geff (see this) , I ended up with an old idea that subgroups of Galois group could be analogous to conserved genes in that they could be conserved in number theoretic evolution. In small variations such as above variation Galois subgroups as genes would change only a little bit. For instance, the dimension of Galois subgroup would change.

The analogy between subgoups of Galois groups and genes goes also in other direction. I have proposed long time ago that genes (or maybe even DNA codons) could be labelled by heff/h=n . This would mean that genes (or even codons) are labelled by a Galois group of Galois extension (see this) of rationals with dimension n defining the number of sheets of space-time surface as covering space. This could give a concrete dynamical and geometric meaning for the notin of gene and it might be possible some day to understand why given gene correlates with particular function. This is of course one of the big problems of biology.

One should have some kind of procedure giving rise to hierarchies of Galois groups assignable to genes. One would also like to assign to letter, codon and gene and extension of rationals and its Galois group. The natural starting point would be a sequence of so called intermediate Galois extensions EH leading from rationals or some extension K of rationals to the final extension E. Galois extension has the property that if a polynomial with coefficients in K has single root in E, also other roots are in E meaning that the polynomial with coefficients K factorizes into a product of linear polynomials. For Galois extensions the defining polynomials are irreducible so that they do not reduce to a product of polynomials.

Any sub-group H⊂ Gal(E/K)) leaves the intermediate extension EH invariant in element-wise manner as a sub-field of E (see this). Any subgroup H⊂ Gal(E/K)) defines an intermediate extension EH and subgroup H1⊂ H2⊂... define a hierarchy of extensions EH1>EH2>EH3... with decreasing dimension. The subgroups H are normal - in other words Gal(E) leaves them invariant and Gal(E)/H is group. The order |H| is the dimension of E as an extension of EH. This is a highly non-trivial piece of information. The dimension of E factorizes to a product ∏i |Hi| of dimensions for a sequence of groups Hi.

Could a sequence of DNA letters/codons somehow define a sequence of extensions? Could one assign to a given letter/codon a definite group Hi so that a sequence of letters/codons would correspond a product of some kind for these groups or should one be satisfied only with the assignment of a standard kind of extension to a letter/codon?

Irreducible polynomials define Galois extensions and one should understand what happens to an irreducible polynomial of an extension EH in a further extension to E. The degree of EH increases by a factor, which is dimension of E/EH and also the dimension of H. Is there a standard manner to construct irreducible extensions of this kind?

1. What comes into mathematically uneducated mind of physicist is the functional decomposition Pm+n(x)= Pm(Pn(x)) of polynomials assignable to sub-units (letters/codons/genes) with coefficients in K for a algebraic counterpart for the product of sub-units. Pm(Pn(x)) would be a polynomial of degree n+m in K and polynomial of degree m in EH and one could assign to a given gene a fixed polynomial obtained as an iterated function composition. Intuitively it seems clear that in the generic case Pm(Pn(x)) does not decompose to a product of lower order polynomials. One could use also polynomials assignable to codons or letters as basic units. Also polynomials of genes could be fused in the same manner.
2. If this indeed gives a Galois extension, the dimension m of the intermediate extension should be same as the order of its Galois group. Composition would be non-commutative but associative as the physical picture demands. The longer the gene, the higher the algebraic complexity would be. Could functional decomposition define the rule for who extensions and Galois groups correspond to genes? Very naively, functional decomposition in mathematical sense would correspond to composition of functions in biological sense.
3. This picture would conform with M8-M4× CP2 correspondence (see this) in which the construction of space-time surface at level of M8 reduces to the construction of zero loci of polynomials of octonions, with rational coefficients. DNA letters, codons, and genes would correspond to polynomials of this kind.
Could one say anything about the Galois groups of DNA letters?
1. Since n=heff/h serves as a kind of quantum IQ, and since molecular structures consisting of large number of particles are very complex, one could argue that n for DNA or its dark variant realized as dark proton sequences can be rather large and depend on the evolutionary level of organism and even the type of cell (neuron viz. soma cell). On the other, hand one could argue that in some sense DNA, which is often thought as information processor, could be analogous to an integrable quantum field theory and be solvable in some sense. Notice also that one can start from a background defined by given extension K of rationals and consider polynomials with coefficients in K. Under some conditions situation could be like that for rationals.
2. The simplest guess would be that the 4 DNA letters correspond to 4 non-trivial finite groups with smaller possible orders: the cyclic groups Z2,Z3 with orders 2 and 3 plus 2 finite groups of order 4 (see the table of finite groups in this). The groups of order 4 are cyclic group Z4=Z2× Z2 and Klein group Z2⊕ Z2 acting as a symmetry group of rectangle that is not square - its elements have square equal to unit element. All these 4 groups are Abelian.
3. On the other hand, polynomial equations of degree not larger than 4 can be solved exactly in the sense that one can write their roots in terms of radicals. Could there exist some kind of connection between the number 4 of DNA letters and 4 polynomials of degree less than 5 for whose roots one can write closed expressions in terms of radicals as Galois found? Could the polynomials obtained by a a repeated functional composition of the polynomials of DNA letters also have this solvability property?

This could be the case! Galois theory states that the roots of polynomial are solvable in terms of radicals if and only if the Galois group is solvable meaning that it can be constructed from abelian groups using Abelian extensions (see this).

Solvability translates to a statement that the group allows so called sub-normal series 1<G0<G1 ...<Gk=G such that Gj-1 is normal subgroup of Gj and Gj/Gj-1 is an abelian group: it is essential that the series extends to G. An equivalent condition is that the derived series is G→ G(1) → G(2) → ...→ 1 in which j+1:th group is commutator group of Gj: the essential point is that the series ends to trivial group.

If one constructs the iterated polynomials by using only the 4 polynomials with Abelian Galois groups, the intuition of physicist suggests that the solvability condition is guaranteed!

4. Wikipedia article also informs that for finite groups solvable group is a group whose composition series has only factors which are cyclic groups of prime order. Abelian groups are trivially solvable, nilpotent groups are solvable, and p-groups (having order, which is power prime) are solvable and all finite p-groups are nilpotent. This might relate to the importance of primes and their powers in TGD.

Every group with order less than 60 elements is solvable. Fourth order polynomials can have at most S4 with 24 elements as Galois groups and are thus solvable. Fifth order polynomial can have the smallest non-solvable group, which is alternating group A5 with 60 elements as Galois group and in this case is not solvable. Sn is not solvable for n>4 and by the finding that Sn as Galois group is favored by its special properties (see this). It would seem that solvable polynomials are exceptions.

A5 acts as the group of icosahedral orientation preserving isometries (rotations). Icosahedron and tetrahedron glued to it along one triangular face play a key role in TGD inspired model of bio-harmony and of genetic code (see this and this). The gluing of tetrahedron increases the number of codons from 60 to 64. The gluing of tetrahedron to icosahedron also reduces the order of isometry group to the rotations leaving the common face fixed and makes it solvable: could this explain why the ugly looking gluing of tetrahedron to icosahedron is needed? Could the smallest solvable groups and smallest non-solvable group be crucial for understanding the number theory of the genetic code.

An interesting question inspired by M8-H-duality (see this) is whether the solvability could be posed on octonionic polynomials as a condition guaranteeing that TGD is integrable theory in number theoretical sense or perhaps following from the conditions posed on the octonionic polynomials. Space-time surfaces in M8 would correspond to zero loci of real/imaginary parts (in quaternionic sense) for octonionic polynomials obtained from rational polynomials by analytic continuation. Could solvability relate to the condition guaranteeing M8 duality boiling down to the condition that the tangent spaces of space-time surface are labelled by points of CP2. This requires that tangent or normal space is associative (quaternionic) and that it contains fixed complex sub-space of octonions or perhaps more generally, there exists an integrable distribution of complex subspaces of octonions defining an analog of string world sheet.