B.5 Chapter 9

B.5 Chapter 9

Concept links · terms present in this machine draft; source roles are unverified: vector space · eigenvalue · irreducible representation

Problem 1:

Consider the action of $SU(2)$ on the tensor product $V^1\otimes V^1$ of two spin representations. According to the Clebsch-Gordan decomposition, this breaks up into irreducibles as $V^0\oplus V^2$

1. Show that

$$\frac{1}{\sqrt{2}}\left(\left(\genfrac{}{}{0ex}{}{1}{0}\right)\otimes \left(\genfrac{}{}{0ex}{}{0}{1}\right)-\left(\genfrac{}{}{0ex}{}{0}{1}\right)\otimes \left(\genfrac{}{}{0ex}{}{1}{0}\right)\right)$$

is a basis of the $V^0$ component of the tensor product, by computing first the action of $SU(2)$ on this vector, and then the action of $\mathfrak{su}(2)$ on the vector $(i.e.$ , compute the action of ${\pi }^{\prime }(X)$ on this vector, for $\pi$ the tensor product representation, and X basis elements of $\mathfrak{su}(2),$ ).

2. Show that

$$\left(\genfrac{}{}{0ex}{}{1}{0}\right)\otimes \left(\genfrac{}{}{0ex}{}{1}{0}\right),\frac{1}{\sqrt{2}}\left(\left(\genfrac{}{}{0ex}{}{1}{0}\right)\otimes \left(\genfrac{}{}{0ex}{}{0}{1}\right)+\left(\genfrac{}{}{0ex}{}{0}{1}\right)\otimes \left(\genfrac{}{}{0ex}{}{1}{0}\right)\right),\left(\genfrac{}{}{0ex}{}{0}{1}\right)\otimes \left(\genfrac{}{}{0ex}{}{0}{1}\right)$$

give a basis for the irreducible representation $V^2$ , by showing that they are eigenvectors of ${\pi }^{\prime }(S_3)$ with the right eigenvalues (weights), and computing the action of the raising and lowering operators for $\mathfrak{su}(2)$ on these vectors.

Problem 2:

Prove that the algebra $S^{\ast }(V^{\ast })$ is isomorphic to the algebra of polynomial functions on the vector space $V$

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正文：英文 · OCR 机器稿 · 待校对

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原书 PDF · 印刷页 537、538、539、540、541、542、543、544、545、546、547、548、549、550、551、552、553、554、555、556

来源版本：2025-10-20

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