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Lorenz Haber
Lorenz Haber
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Let $\mathfrak{g}$ be a complex simple Lie algebra. Let $S(\mathfrak{g})$ be the algebra of polynomial functions on $\mathfrak{g}$, viewed as a $\mathfrak{g}$-representation. Are the isotypic components of $S(\mathfrak{g})$ finite-dimensional? In other words, does each finite-dimensional irreducible $\mathfrak{g}$-representation appears with finite multiplicity in $S(\mathfrak{g})$?

This seems to be true for $\mathfrak{sl}_n$, since this is the subject of "plethysm" and there are explicit formulae for these multiplicities. See e.g. this mathoverflow question and the reference therein.

Let $\mathfrak{g}$ be a complex simple Lie algebra. Let $S(\mathfrak{g})$ be the algebra of polynomial functions on $\mathfrak{g}$, viewed as a $\mathfrak{g}$-representation. Are the isotypic components of $S(\mathfrak{g})$ finite-dimensional? In other words, does each finite-dimensional irreducible $\mathfrak{g}$-representation appears with finite multiplicity in $S(\mathfrak{g})$?

This seems to be true for $\mathfrak{sl}_n$, since this is the subject of "plethysm" and there are explicit formulae for these multiplicities. See e.g. this mathoverflow question and the reference therein.

Let $\mathfrak{g}$ be a complex simple Lie algebra. Let $S(\mathfrak{g})$ be the algebra of polynomial functions on $\mathfrak{g}$, viewed as a $\mathfrak{g}$-representation. Are the isotypic components of $S(\mathfrak{g})$ finite-dimensional?

This seems to be true for $\mathfrak{sl}_n$, since this is the subject of "plethysm" and there are explicit formulae for these multiplicities. See e.g. this mathoverflow question and the reference therein.

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Lorenz Haber
Lorenz Haber
  • 443
  • 2
  • 6

Let $\mathfrak{g}$ be a complex simple Lie algebra. Let $S(\mathfrak{g})$ be the algebra of polynomial functions on $\mathfrak{g}$, viewed as a $\mathfrak{g}$-representation. Are the isotypic components of $S(\mathfrak{g})$ finite-dimensional? In other words, does each finite-dimensional irreducible $\mathfrak{g}$-representation appears with finite multiplicity in $S(\mathfrak{g})$?

This seems to be true for $\mathfrak{sl}_n$, since this is the subject of "plethysm" and there are explicit formulae for these multiplicities. See e.g. this mathoverflow question and the reference therein.

Let $\mathfrak{g}$ be a complex simple Lie algebra. Let $S(\mathfrak{g})$ be the algebra of polynomial functions on $\mathfrak{g}$, viewed as a $\mathfrak{g}$-representation. Are the isotypic components of $S(\mathfrak{g})$ finite-dimensional? In other words, does each $\mathfrak{g}$-representation appears with finite multiplicity in $S(\mathfrak{g})$?

This seems to be true for $\mathfrak{sl}_n$, since this is the subject of "plethysm" and there are explicit formulae for these multiplicities. See e.g. this mathoverflow question and the reference therein.

Let $\mathfrak{g}$ be a complex simple Lie algebra. Let $S(\mathfrak{g})$ be the algebra of polynomial functions on $\mathfrak{g}$, viewed as a $\mathfrak{g}$-representation. Are the isotypic components of $S(\mathfrak{g})$ finite-dimensional? In other words, does each finite-dimensional irreducible $\mathfrak{g}$-representation appears with finite multiplicity in $S(\mathfrak{g})$?

This seems to be true for $\mathfrak{sl}_n$, since this is the subject of "plethysm" and there are explicit formulae for these multiplicities. See e.g. this mathoverflow question and the reference therein.

Source Link
Lorenz Haber
Lorenz Haber
  • 443
  • 2
  • 6

Are isotypic components of $S(\mathfrak{g})$ finite-dimensional?

Let $\mathfrak{g}$ be a complex simple Lie algebra. Let $S(\mathfrak{g})$ be the algebra of polynomial functions on $\mathfrak{g}$, viewed as a $\mathfrak{g}$-representation. Are the isotypic components of $S(\mathfrak{g})$ finite-dimensional? In other words, does each $\mathfrak{g}$-representation appears with finite multiplicity in $S(\mathfrak{g})$?

This seems to be true for $\mathfrak{sl}_n$, since this is the subject of "plethysm" and there are explicit formulae for these multiplicities. See e.g. this mathoverflow question and the reference therein.