Question

Let $\pi$ be an irreducible finite-dimensional representation of $\mathrm{sl}(3 ; \mathbb{C})$ acting on a space $V$ and let $\pi^$ be the dual representation to $\pi$, acting $\mathrm{cu} V^$, as defined in Section 4.7. Show that the weights of $\pi^*$ are the negatives of the weights of $\pi$. Hint: Choose a basis for $V$ in which both $\pi\left(H_1\right)$ and $\pi\left(H_2\right)$ are diagonal.

   Let $\pi$ be an irreducible finite-dimensional representation of $\mathrm{sl}(3 ; \mathbb{C})$ acting on a space $V$ and let $\pi^*$ be the dual representation to $\pi$, acting $\mathrm{cu} V^*$, as defined in Section 4.7. Show that the weights of $\pi^*$ are the negatives of the weights of $\pi$.
Hint: Choose a basis for $V$ in which both $\pi\left(H_1\right)$ and $\pi\left(H_2\right)$ are diagonal.

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Lie Groups, Lie Algebras, and Representations: An Elementary Understanding

Lie Groups, Lie Algebras, and Representations: An Elementary Understanding

Brian C. Hall 1st Edition

Chapter 5, Problem 2

Instant Answer

Step 1

If $\pi$ is a representation of a Lie algebra $\mathfrak{g}$ on a vector space $V$, then the dual representation $\pi^*$ on $V^*$ is defined by: $$\pi^*(X)(f)(v) = -f(\pi(X)(v))$$ for all $X \in \mathfrak{g}$, $f \in V^*$, and $v \in V$. Show more…

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Let $\pi$ be an irreducible finite-dimensional representation of $\mathrm{sl}(3 ; \mathbb{C})$ acting on a space $V$ and let $\pi^*$ be the dual representation to $\pi$, acting $\mathrm{cu} V^*$, as defined in Section 4.7. Show that the weights of $\pi^*$ are the negatives of the weights of $\pi$. Hint: Choose a basis for $V$ in which both $\pi\left(H_1\right)$ and $\pi\left(H_2\right)$ are diagonal.

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