Question

As in Exercise 2, let $\pi$ be an irreducible representation of $\mathrm{sl}(3 ; \mathrm{C})$ and let $\pi^$ be the dual representation to $\pi$. Show that if $\pi$ has highest weight ( $m_1, m_2$ ), $\pi^$ has highest weight ( $m_2, m_1$ ). Hint: Establish this first in the cases $\left(m_1, m_2\right)=(1,0)$ and $\left(m_1, m_2\right)=$ $(0,1)$.

   As in Exercise 2, let $\pi$ be an irreducible representation of $\mathrm{sl}(3 ; \mathrm{C})$ and let $\pi^*$ be the dual representation to $\pi$. Show that if $\pi$ has highest weight ( $m_1, m_2$ ), $\pi^*$ has highest weight ( $m_2, m_1$ ).
Hint: Establish this first in the cases $\left(m_1, m_2\right)=(1,0)$ and $\left(m_1, m_2\right)=$ $(0,1)$.

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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 3

Instant Answer

Step 1

If $\pi$ is a representation of $\mathfrak{sl}(3,\mathbb{C})$ on a vector space $V$, then $\pi^*$ is the representation on the dual space $V^*$ defined by: $(\pi^*(X)f)(v) = -f(\pi(X)v)$ for all $X \in \mathfrak{sl}(3,\mathbb{C})$, $f \in V^*$, and $v \in V$. Show more…

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As in Exercise 2, let $\pi$ be an irreducible representation of $\mathrm{sl}(3 ; \mathrm{C})$ and let $\pi^*$ be the dual representation to $\pi$. Show that if $\pi$ has highest weight ( $m_1, m_2$ ), $\pi^*$ has highest weight ( $m_2, m_1$ ). Hint: Establish this first in the cases $\left(m_1, m_2\right)=(1,0)$ and $\left(m_1, m_2\right)=$ $(0,1)$.

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