Here FFT is the standard abbreviation for First Fundamental Theorem of classical invariant theory. If one has a group inclusion $H\subset G$ (and I suspect also more generally a morphism $H\rightarrow G$), I see two ways of relating the FFT for $H$ to that of $G$. This is somewhat similar to induction and restriction for representations.
One may deduce the FFT for $H$ (the principal group) from that of $G$ (the comprehensive group) and this is called the adjunction method. See this MO answer on the FFT for $F_4$ for an (admittedly rather vague) account of how to proceed. The terminology principal/comprehensive group and adjunction is from Felix Klein's Erlangen Program which I understand as the study of relations between FFTs for different groups (see the link to the Erlangan Program as well as to other related references in the MO answer for $F_4$ mentioned above).
One also has what one could call the recombination method which deduces the FFT for $G$ from that of smaller group $H$. Indeed, by restriction, an invariant of $G$ is an invariant of $H$ and by the FFT for the latter it can be written as a linear combination of assemblies (tensor contractions) of elementary pieces which are $H$-invariant. The latter need not be $G$-invariant and so this first step must be followed by a second one that recombines the $H$-elementary pieces into $G$-elementary pieces. For a detailed example of this second step see my two answers to this MO question, where $H=SL(d)$ and $G=GL(d)$. In these answers, this second step uses the identity ${\rm det}(AB)={\rm det}(A){\rm det}(B)$ where $A$ is seen as a bunch of row vectors and $B$ as another bunch of column vectors. Taking the numerical coefficients of this polynomial identity in the coordinates of the $2d$ vectors present results in the needed identity which replaces two Levi-Civita tensors (the $SL$-elementary pieces) by $d$ Kronecker deltas (the $GL$-elementary pieces). One can do a similar recombination for $H=SL(2)$ and $G=SL(d)$ and the embedding given by the $(d-1)$-th symmetric power. Namely $\left(\begin{array}{c} d\\ 2\end{array}\right)$ $SL(2)$-pieces may be recombined into a single $SL(d)$-piece and this is nothing but the factorized formula for the Vandermonde determinant. Something similar may be done for $SL(k)$ instead of $SL(2)$ but it seems like a difficult question related to generalizations of Pascal's Theorem. This kind of relation between FFTs for different $SL$'s is what Klein calls the Hesse transfer principle in his Erlangen Program.
My motivation: One of my medium term projects is to write a book on classical invariant theory and classical elimination theory. It would have to include proofs of FFTs for various groups and it would be esthetically pleasing to do so by proving the FFT for the "mother of all groups" and deduce the other ones from it.
My questions:
1) What is this mother of all groups? It is not clear to me which is more fundamental $SL$ or $GL$. This is indeed a chicken and egg type question, so I am asking for some insights or philosophical reasons to view one as more fundamental than the other. Of course $SL\subset GL$ leans towards ${\rm Mother}=GL$. On the other hand, I know nice formulas for $SU(N)$ integration but not for $U(N)$ integration (see this MO question).
2) What literature is there on the study of these relationships between FFTs? I am particularly interested in examples that are exotic (like in the $F_4$ MO answer I mentioned) or extreme (like $G=GL(n)$ and $H=S_n$ is finite).
3) Can one deduce the Fundamental Theorem of symmetric functions theory from the FFT of $GL$? This is just a specific instance of the last question that I am curious about.
