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Let $\rho\colon \GL(g,\C)\to \GL(V)$ be a finite-dimensional irreducible representation Then there exists a unique $1$-dimensional subspace $\langle v_0 \rangle$ of $V$ and a unique $g$-tuple of integers $(\lambda_1 \geqslant\lambda_2 \geqslant \ldots \geqslant \lambda_g)$ such that $\rho(\mathrm{diag}(a_1,\ldots,a_g))\,v_0=\prod_{i=1}^{g}a_i^{\lambda_i}\, v_0$. The $g$-tuple of integers $(\lambda_1,\lambda_2, \ldots, \lambda_g)$ is called ** the highest weight** of $\rho$.

For example the $1$-dimensional representation of $\GL(g,\C)$ given by the determinant has highest weight $(1,\ldots,1)$ while the standard $g$-dimensional representation has highest weight $(1,0,\ldots,0)$.

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  • Review status: beta
  • Last edited by Fabien Cléry on 2023-11-17 14:31:09
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