3
$\begingroup$

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?

$\endgroup$
7
$\begingroup$

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 $\mathcal{O}(\lambda_{k_1},\lambda_{k_1 + k_2},\dots,\lambda_{k_1+\dots+k_r})$.

$\endgroup$
1
  • $\begingroup$ This proves my original thought! Thank you very much. $\endgroup$
    – gigi
    Oct 15 '20 at 14:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.