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Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra, and $V$ an irreducible finite-dimensional $\mathfrak{g}$-module. Then $\mathfrak{g}$ also acts on the symmetric algebra $S(V)$.

Is there a description of the invariants $S(V)^\mathfrak{g}$?

If $V$ is the standard module of the classical algebras, then this reduces to the fundamental theorems of invariant theory. Is there something in the literature of this more general kind? Is this known, at least for $\mathfrak{sl}_2$?

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    $\begingroup$ Even for $\mathfrak{g} = \mathfrak{sl}_2(\mathbb{C})$ and $V$ irreducible, I think in general it is not obvious what is the dimension of $S^k(V)^\mathfrak{g}$, which equals the number of times the $1$-dimensional irreducible occurs as a composition factor. Of course in a given small-dimensional case you can compute the character and figure this out. See eg OEIS: here and sequences linked there. Also here $\endgroup$ – spin Sep 17 at 9:52
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    $\begingroup$ You can replace $\mathfrak{g}$ with a compact Lie group and use the compact form of Molien's theorem: en.wikipedia.org/wiki/Molien_series $\endgroup$ – Qiaochu Yuan Sep 17 at 20:58
  • $\begingroup$ I am linking this answer here because it was very helpful to me: mathoverflow.net/questions/299900/… $\endgroup$ – Rafael Mrđen Nov 28 at 14:29
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The question subsumes all of 19th Century invariant theory, so I don't think there is much chance of a really explicit answer.

For example, take $\mathfrak{g} = \mathfrak{sl}_d(\mathbb{C})$ and let $V$ be the $\mathfrak{sl}_d(\mathbb{C})$-module obtained from the polynomial representation $\nabla^\lambda(\mathbb{C}^d)$ of $\mathrm{GL}_d(\mathbb{C})$ canonically labelled by the partition $\lambda$ of $n$ with at most $n-1$ parts. (So $\lambda$ is a dominant integral weight.) The invariants for $\mathfrak{sl}_d(\mathbb{C})$ in the degree $r$ component of $S(V)$ lift to elements of $\mathrm{Sym}^r \nabla^\lambda(\mathbb{C}^d)$ on which $\mathrm{GL}_d(\mathbb{C})$ acts as a power of the determinant: since det has polynomial degree $d$, the relevant power is $m = r|\lambda|/d$. These elements span a subspace of $\mathrm{Sym}^r \nabla^\lambda(\mathbb{C}^d)$ isomorphic to $\nabla^{(m^d)}(\mathbb{C}^d)$. Thus finding the dimension of the invariant spaces (as $d$ varies) is equivalent to finding the multiplicities of polynomial representations labelled by rectangular partitions in $\mathrm{Sym}^r \nabla^\lambda(\mathbb{C}^d)$.

This can be restated in the language of Schur functions as asking for $\langle s_{(r)} \circ s_\lambda, s_{(m^d)} \rangle$, where $\circ$ is the plethystic product. Decomposing plethysms into Schur functions is a notorious open problem in algebraic combinatorics. As far as I know, it is not made easier by restricting to constituents labelled by rectangular partitions.

This paper https://arxiv.org/pdf/0807.0430.pdf has a formula for the dimensions in the special case when $\lambda = (n)$ has one part, i.e. it finds the dimension of the invariant spaces of $\mathfrak{sl}_d(\mathbb{C})$ on the space $\mathrm{Sym}^n \mathbb{C}^d$ of $n$-ary forms in $d$ variables. The formula looks somewhat intractable to me.

For $\mathfrak{sl}_2(\mathbb{C})$ the simple modules are the symmetric powers of the natural module $E \cong \mathbb{C}^2$ and much more is known. Thinking of $\mathrm{Sym}^2 E$ as the space of quadratic forms, the invariant algebra $S(\mathrm{Sym}^2 E)^{\mathfrak{sl}_2(\mathbb{C})}$ is generated by the discriminant in degree $2$. Hence for $\mathrm{Sym}^2 E$ there is a unique invariant (up to scalars) in each even degree.

Still for $\mathfrak{sl}_2(\mathbb{C})$, but for all $r$ and $\ell$, the dimension of the degree $r$ component of $S(\mathrm{Sym}^\ell \!E)^{\mathfrak{sl}_2(\mathbb{C})}$ is the number of partitions of $\ell r/2$ that fit in an $\ell \times r$ box, minus the number of partitions of $\ell r/2-1$ in the same box. This is the Cayley–Sylvester formula: see e.g. Lecture XVII in Hilbert Theory of algebraic invariants. The dimensions are easily computed but do not grow in a regular way when $r \ge 3$. The symmetry swapping $r$ and $\ell$ is known as Hermite reciprocity. There is some more recent work on generalizations of the Cayley–Sylvester formula, see e.g. papers by King, Manivel and (please excuse the self-publicity) Paget and Wildon.

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  • $\begingroup$ Thanks! I hoped that at least for $\mathfrak{sl_2}$ one can write down algebraically independent generators for any $Sym^k E$, but now I see that I was way too optimistic. $\endgroup$ – Rafael Mrđen Sep 20 at 7:05
  • $\begingroup$ @Mark Wildon Mentioned you as somewhat intractable formula nevertheless allowed some explicit calculations, see arxiv.org/pdf/1007.1064.pdf and arxiv.org/pdf/0911.5717 $\endgroup$ – Leox Nov 3 at 10:02
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Please, check the work of Stephen Yau published in memoirs of AMS, i believe you will find a partial answer to your questions. best Jorge

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  • $\begingroup$ Please could you give a more precise reference. $\endgroup$ – Mark Wildon Sep 30 at 17:14
  • $\begingroup$ I think jorge meant: Stephen S.-T. Yau "Classification of Jacobian ideals invariant by sl(2,C) actions", Mem. Amer. Math. Soc. 72 (1988), no. 384, iv+180 pp. It is related to the classical invariant theory of binary forms. $\endgroup$ – Abdelmalek Abdesselam Sep 30 at 18:21
  • $\begingroup$ jorge, this is not a proper format for an MO answer. First, references to the literature need to be precise in order to be helpful; if you can point to specific results or to page numbers in the article, that's even better. But mostly what question-posters want to get is a to-the-point, informative, and informed answer from someone with expertise, not some vague hand-waving. Please see Mark's answer above for an idea of a proper response. $\endgroup$ – Todd Trimble Sep 30 at 19:17

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