It is known that, given a complex vector space $V$ of dimension $\mathit{dim}(V)=n+1$, all irreducible representations for the group $\mathit{GL}(V)$ are parametrized by Young tableaux $\lambda$. For example for the Young tableaux $\lambda=(d)$ we have that $$S_{\lambda}V=\mathit{Sym}^d(V)$$ and for $\lambda=(1,\dots,1)$ that $$S_{\lambda}V=\bigwedge^k(V)$$ In particular every irreducible $\textit{GL}(V)-$module $S_{\lambda}V$ can be seen as a subvector space of a suitable tensor power $V^{\otimes N_{\lambda}}$.

My question now is the following: is it true that all homogeneous varieties arising as the orbit of a highest weight vector in $S_{\lambda}V$ are only either a Veronese, a Grassmannian or a flag variety?


If $\lambda = (\lambda_1,\dots,\lambda_{n+1})$ is a dominant weight and $$ \lambda_1 = \dots = \lambda_{k_1} > \lambda_{k_1 + 1} = \dots = \lambda_{k_1 + k_2} > \dots > \lambda_{k_1 + \dots + k_r + 1} = \dots = \lambda_{n+1} $$ then the orbit of the highest weight vector in the projective space $\mathbb{P}(S_\lambda V)$ is isomorphic to the flag variety $$ Fl(k_1,k_1+k_2,\dots,k_1+\dots+k_r;n+1) $$ which is embedded by the line bundle $\mathcal{O}(\lambda_{k_1},\lambda_{k_1 + k_2},\dots,\lambda_{k_1+\dots+k_r})$.

  • $\begingroup$ This proves my original thought! Thank you very much. $\endgroup$
    – gigi
    Oct 15 '20 at 14:20

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