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One sees that given the $SL(2,\mathbb{C})$ action on $\mathbb{C}^5$, thought of as the space of polynomials of the form,

$$a_0 x^4 + 4a_1 x^3 y + 6a_2x^2y^2 + 4a_3xy^3 + a_4 y^4$$

the ring of invariants is generated by the following functions,

$$g_2(a) = a_0a_4 - 4a_1 a_3 + 3a_2^2$$

and

$$g_3(a) = a_0a_2a_4 - a_0a_3^2 - a_1^2a_4 + 2a_1a_2a_3 - a_2^3$$

But if these $g_2$ and $g_3$ satisfy the discriminant $=0$ condition then there are inequivalent $SL(2,\mathbb{C})$ polynomials which map to the same $(g_2,g_3)$ point.

But if I look at say Theorem 5.9 in the book by Mukai then I get to see that the closure equivalence equivalent classes of orbits of the action of a linearly reductive group like $SL(2,\mathbb{C})$ on $\mathbb{C}^5$ are in bijective correspondence to the the points of $\mathbb{C}^5\/\/SL(2,\mathbb{C})$ \mathbb{C}^5//SL(2,\mathbb{C})$(which is defined as the spectrum of the invariant polynomials under$SL(2,\mathbb{C})$) Also look at the theorem at the end of page 11 of this paper. In the above paper "//" is defined as identifying points in the affine variety if one lies in the closure of the orbit through the other. • Are these two notions of "//" equivalent? If yes, how? • In the light of the above two theorems, can one say that the$SU(2)$invariant polynomials among binary homogeneous quartics are in bijection with those closure equivalent classes of orbits of$SU(2)^{\mathbb{C}} = SL(2,\mathbb{C})$which are labeled by the pairs of invariants$(g_2,g_3)$such that$g_2^3 - 27 g_3^3 g_3^2 \neq 0$? • Hence if I am interested in only closure equivalent orbits can I just forget those pairs of values of the invariants which lie on the discriminant$0$curve? • Conversely given a$(g_2,g_3)$for which the above discriminant condition is satisfied can one write down the family of$SU(2)$invariant polynomials explicitly? • I would anyway like to know how to distinguish the orbits corresponding to the discriminant$0$condition. In light of the various extremely helpful and references that have come up, I realize that there there is a notion of a "discriminant" for homogeneous polynomials of degree$d$in$n$variables. (call this space of polynomials as$P(n,d)$) This discriminant is in some sense a "homogeneous invariant" and for the$n=2$case in which I am interested in, it generates the subalgebra of the coordinate ring over of these polynomials which is invariant under this group action (call that$\mathbb{C}[P(n=2,d=4)]^{SL(2,\mathbb{C})}$). (this is lucky!) I guess the discriminant in this case has to be a polynomial in polynomials of homogeneous degree$4$in$2$variables. The above I guess implies that$\mathbb{C}[P(n=2,d=4)]^{SL(2,\mathbb{C})}$is generated by just one such polynomial in polynomials. I would like to know how is a discriminant" defined for such polynomials. (searching and asking around I am only getting definitions for the single variable case) I want to know if knowing the generator of$\mathbb{C}[P(n=2,d=4)]^{SL(2,\mathbb{C})}$tells me about the initial objective of knowing$P(n=2,d=4)\text{ }mod\text{ }SL(2,\mathbb{C})$I wonder if this unique generator of the invariant subalgebra is related to the null-cone that was pointed out by Bart in his comment. Also I would like to be pointed out if there is any mistake in what I said above! I had recently tried asking a similar question here. But I think I could not precisely convey what I was looking for. Let me here try to give a specific situation that I need to understand coming from certain other considerations in Superconformal Quantum Field Theories. I can think of$\mathbb{C}^5$as being the space of all homogeneous degree$4$polynomials in$2$variables. On this space$SL(2,\mathbb{C})$has the standard action. I want to know what is the most explicit (or the best!) way to describe the quotient space thus obtained. I want to understand how do the orbits look like. [EDIT: I was initially asking if there exists fixed subspaces etc but then from the comments I realized that I was missing the elementary fact that it is an irreducible representation! Hence nothing like this can exist.] I tried something naive. I wrote down the most general element of$SL(2,\mathbb{C})$using its canonical polar decomposition and then acted it on the most general homogeneous degree$4$polynomial in$2$variables and tried to see how the coefficients change. Unfortunately the equations are very complicated and I didn't see any hope of me being able to solve them to find the fixed points. Apart from this specific example I would also like to know of references to simpler examples than this where a similar question is asked and answered. 7 Major changes and refinements of the question Having seen some more One sees that given the$SL(2,\mathbb{C})$action on the space of polynomials of the references pooled inform, $$a_0 x^4 + 4a_1 x^3 y + 6a_2x^2y^2 + 4a_3xy^3 + a_4 y^4$$ the ring of invariants is generated by the following functions, $$g_2(a) = a_0a_4 - 4a_1 a_3 + 3a_2^2$$ and $$g_3(a) = a_0a_2a_4 - a_0a_3^2 - a_1^2a_4 + 2a_1a_2a_3 - a_2^3$$ But if these$g_2$and$g_3$satisfy the discriminant$=0$condition then there are inequivalent$SL(2,\mathbb{C})$polynomials which map to the same$(g_2,g_3)$point. But if I put look at say Theorem 5.9 in some more specific queries the book by Mukai then I get to see that the closure equivalence classes of orbits of a linearly reductive group like$SL(2,\mathbb{C})$are in bijective correspondence to the the points of$\mathbb{C}^5\/\/SL(2,\mathbb{C})$(which is defined as comments addressed the spectrum of the invariant polynomials under$SL(2,\mathbb{C})$) Also look at the theorem at the end of page 11 of this paper. In the above paper "//" is defined as identifying points in the affine variety if one lies in the closure of the orbit through the other. • Are these two notions of "//" equivalent? If yes, how? • In the light of the above two theorems, can one say that the$SU(2)$invariant polynomials among binary homogeneous quartics are in bijection with those closure equivalent classes of orbits of$SU(2)^{\mathbb{C}} = SL(2,\mathbb{C})$which are labeled by the pairs of invariants$(g_2,g_3)$such that$g_2^3 - 27 g_3^3 \neq 0$? • Hence if I am interested in only closure equivalent orbits can I just forget those pairs of values of the invariants which lie on the discriminant$0$curve? • Conversely given a$(g_2,g_3)$for which the above discriminant condition is satisfied can one write down the family of$SU(2)$invariant polynomials explicitly? • I would anyway like to Jose and Abdelmalek know how to distinguish the orbits corresponding to the discriminant$0$condition. • 6 added 154 characters in body Having seen some more of the references pooled in, I put in some more specific queries as comments addressed to Jose and Abdelmalek In light of the various extremely helpful and references that have come up, I realize that there there is a notion of a "discriminant" for homogeneous polynomials of degree$d$in$n$variables. (call this space of polynomials as$P(n,d)$) This discriminant is in some sense a "homogeneous invariant" and for the$n=2$case in which I am interested in, it generates the subalgebra of the coordinate ring over of these polynomials which is invariant under this group action (call that$\mathbb{C}[P(n=2,d=4)]^{SL(2,\mathbb{C})}$). (this is lucky!) I guess the discriminant in this case has to be a polynomial in polynomials of homogeneous degree$4$in$2$variables. The above I guess implies that$\mathbb{C}[P(n=2,d=4)]^{SL(2,\mathbb{C})}$is generated by just one such polynomial in polynomials. I would like to know how is a discriminant" defined for such polynomials. (searching and asking around I am only getting definitions for the single variable case) I want to know if knowing the generator of$\mathbb{C}[P(n=2,d=4)]^{SL(2,\mathbb{C})}$tells me about the initial objective of knowing$P(n=2,d=4)\text{ }mod\text{ }SL(2,\mathbb{C})$I wonder if this unique generator of the invariant subalgebra is related to the null-cone that was pointed out by Bart in his comment. Also I would like to be pointed out if there is any mistake in what I said above! I had recently tried asking a similar question here. But I think I could not precisely convey what I was looking for. Let me here try to give a specific situation that I need to understand coming from certain other considerations in Superconformal Quantum Field Theories. I can think of$\mathbb{C}^5$as being the space of all homogeneous degree$4$polynomials in$2$variables. On this space$SL(2,\mathbb{C})$has the standard action. I want to know what is the most explicit (or the best!) way to describe the quotient space thus obtained. I want to understand how do the orbits look like. [EDIT: I was initially asking if there exists fixed subspaces etc but then from the comments I realized that I was missing the elementary fact that it is an irreducible representation! Hence nothing like this can exist.] I tried something naive. I wrote down the most general element of$SL(2,\mathbb{C})$using its canonical polar decomposition and then acted it on the most general homogeneous degree$4$polynomial in$2\$ variables and tried to see how the coefficients change. Unfortunately the equations are very complicated and I didn't see any hope of me being able to solve them to find the fixed points.

Apart from this specific example I would also like to know of references to simpler examples than this where a similar question is asked and answered.

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