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Guessing the right candidates for Step 2) looks hard to me. Knowing beforehand that some multiplicities ${\rm mult}_{\lambda}(I[\overline{G\cdot F_1}]_d)$ are nonzero would definitely help. Although, one could procrastinate and defer the proof of nonidentical vanishing of the concomitant to Step 3) which should show more than that anyway. If one has such right candidates, showing they vanish on $F_1$ may be easy by arguments one could call Pauli's exclusion principle (contracting symmetrizations with antisymmetrizations), high chromatic number property, or simply `lack of space'.

PS: I should add that my pessimism is specific to the Valiant Hypothesis which is the `Riemann Hypothesis' in the field. Of course, one should not throw the baby with the bath water and denigrate GCT because it so far failed to prove this conjecture. There are plenty of more approachable problems in this area where progress has been made and more progress is expected. See in particular the above-mentioned article by Grochow and review by Landsberg.

Guessing the right candidates for Step 2) looks hard to me. Knowing beforehand that some multiplicities ${\rm mult}_{\lambda}(I[\overline{G\cdot F_1}]_d)$ are nonzero would definitely help. Although, one could procrastinate and defer the proof of nonidentical vanishing of the concomitant to Step 3) which should show more than that anyway. If one has such right candidates, showing they vanish on $F_1$ may be easy by arguments one could call Pauli's exclusion principle (contracting symmetrizations with antisymmetrizations), high chromatic number property, or simply "lack of space".

PS: I should add that my pessimism is specific to the Valiant Hypothesis which is the "Riemann Hypothesis" in the field. Of course, one should not throw the baby with the bath water and denigrate GCT because it so far failed to prove this conjecture. There are plenty of more approachable problems in this area where progress has been made and more progress is expected. See in particular the above-mentioned article by Grochow and review by Landsberg.

Let $X=(X_{ij})_{1\le i,j\le n}$ be a generic $n\times n$ matrix and $F_1(X)={\rm det}(X)$ the degree $n$ homogeneous polynomial given by the determinant. Let $$ F_2(X)=(X_{nn})^{n-m}\times {\rm perm}\left[(X_{ij})_{1\le i,j\le m}\right] $$ which takes the permanent of an $m\times m$ submatrix and multiplies by one's favorite linear form in order to make another homogeneous polynomial of degree $n$ (one could also use the entry $X_{11}$ instead of $X_{nn}$). This modification is called padding. Then define the number $$ c(m)=\min\{\ n\ |\ n\ge m\ \ {\rm and}\ \ \overline{G\cdot F_2}\subset \overline{G\cdot F_1}\ \} $$ where $G$ is $GL(n^2)$ acting on the affine space of dimension $n^2$ where $X$ lives and $\overline{G\cdot F_i}$ are Zariski closures of orbits. The big conjecture in the area or Valiant's Hypothesis (a complex version of ${\rm P}\neq{\rm NP}$) is that $c(m)$ grows faster than any polynomial in $m$.

I think the approach by Mulmuley is to try prove the existence of such multiplicity obstructions by leveraging all the tools available from representation theory for the computation of these multiplicities. Personally, I have never been a fan of this approach. Having studied 19th century invariant theory in some depth, it seems more natural to me to approach the orbit separation problem using the explicit tools from that era. This article by Grochow seems to also point in a similar direction (I suspect the third article mentioned by Joseph is in the same vein). In classical language (see Turnbull or Littlewood), one has to explicitly construct a mixed concomitant which vanishes on $F_1$ but not on $F_2$. One also has to do this infinitely often (in $m$) in order to establish the superpolynomial growth property. Such a concomitant is the same as a specific $G$-equivariant map from your favorite model for the irreducible representation $\lambda$ to the polynomial algebra in the $n^2$ variables $X$ (Grochow calls that a separating module). Invariant theorists from the 19th century had two methods for generating such objects: elimination theory and diagrammatic algebra.

Let $X=(X_{ij})_{1\le i,j\le n}$ be a generic $n\times n$ matrix and $F_1(X)={\rm det}(X)$ the degree $n$ homogeneous polynomial given by the determinant. Let $$ F_2(X)=(X_{nn})^{n-m}\times {\rm perm}\left[(X_{ij})_{1\le i,j\le m}\right] $$ which takes the permanent of an $m\times m$ submatrix and multiplies by one's favorite linear form in order to make another homogeneous polynomial of degree $n$ (one could also use the entry $X_{11}$ instead of $X_{nn}$). This modification is called padding. Then define the number $$ c(m)=\min\{\ n\ |\ n\ge m\ \ {\rm and}\ \ \overline{G\cdot F_2}\subset \overline{G\cdot F_1}\ \} $$ where $G$ is $GL(n^2)$ acting on the space $V$ of dimension $n^2$ where $X$ lives and therefore also on the space of degree $n$ polynomial functions of $X$. The $\overline{G\cdot F_i}$ are Zariski closures of orbits. The big conjecture in the area or Valiant's Hypothesis (a complex version of ${\rm P}\neq{\rm NP}$) is that $c(m)$ grows faster than any polynomial in $m$.

I think the approach by Mulmuley is to try prove the existence of such multiplicity obstructions by leveraging all the tools available from representation theory for the computation of these multiplicities. Personally, I have never been a fan of this approach. Having studied 19th century invariant theory in some depth, it seems more natural to me to approach the orbit separation problem using the explicit tools from that era. This article by Grochow seems to also point in a similar direction (I suspect the third article mentioned by Joseph is in the same vein). In classical language (see Turnbull or Littlewood), one has to explicitly construct a mixed concomitant which vanishes on $F_1$ but not on $F_2$. One also has to do this infinitely often (in $m$) in order to establish the superpolynomial growth property. Such a (degree $d$) concomitant is the same as a specific $G$-equivariant map from your favorite model for the irreducible representation $\lambda$ to ${\rm Sym}^d({\rm Sym}^n(V))$ (Grochow calls that a separating module). Invariant theorists from the 19th century had two methods for generating such objects: elimination theory and diagrammatic algebra.

I think the approach by Mulmuley is to try prove the existence of such multiplicity obstructions by leveraging all the tools available from representation theory for the computation of these multiplicities. Personally, I have never been a fan of this approach. Having studied 19th century invariant theory in some depth, it seems more natural to me to approach the orbit separation problem using the explicit tools from that era. This article by Gorchow seems to also point in a similar direction (I suspect the third article mentioned by Joseph is in the same vein). In classical language (see Turnbull or Littlewood), one has to explicitly construct a mixed concomitant which vanishes on $F_1$ but not on $F_2$. One also has to do this infinitely often (in $m$) in order to establish the superpolynomial growth property. Such a concomitant is the same as a specific $G$-equivariant map from your favorite model for the irreducible representation $\lambda$ to the polynomial algebra in the $n^2$ variables $X$ (Grochow calls that a separating module). Invariant theorists from the 19th century had two methods for generating such objects: elimination theory and diagrammatic algebra.

I think the approach by Mulmuley is to try prove the existence of such multiplicity obstructions by leveraging all the tools available from representation theory for the computation of these multiplicities. Personally, I have never been a fan of this approach. Having studied 19th century invariant theory in some depth, it seems more natural to me to approach the orbit separation problem using the explicit tools from that era. This article by Grochow seems to also point in a similar direction (I suspect the third article mentioned by Joseph is in the same vein). In classical language (see Turnbull or Littlewood), one has to explicitly construct a mixed concomitant which vanishes on $F_1$ but not on $F_2$. One also has to do this infinitely often (in $m$) in order to establish the superpolynomial growth property. Such a concomitant is the same as a specific $G$-equivariant map from your favorite model for the irreducible representation $\lambda$ to the polynomial algebra in the $n^2$ variables $X$ (Grochow calls that a separating module). Invariant theorists from the 19th century had two methods for generating such objects: elimination theory and diagrammatic algebra.