# Varieties in the $\mathit{GL}(V)$-module $S_{\lambda}V$

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?

## 1 Answer

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

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