# An Elementary Derivation of the Matrix Elements of Real Irreducible Representations of so(3)

## Abstract

**:**

## 1. Introduction

#### 1.1. Real Representations of $\mathfrak{so}\left(3\right)$

- Γ is called of first class, denoted by ${\Gamma}^{I}$, if $\Gamma {\otimes}_{\mathbb{R}}\mathbb{C}$ is a complex irreducible representation of $\mathfrak{g}$.
- Γ is called of second class, denoted by ${\Gamma}^{II}$, if $\Gamma {\otimes}_{\mathbb{R}}\mathbb{C}$ is a complex reducible representation of $\mathfrak{g}$.

- For $J=2$, $\mathfrak{s}\mathfrak{o}\left(3\right)$ is a maximal subalgebra irreducibly embedded into $\mathfrak{s}\mathfrak{p}\left(4\right)\simeq \mathfrak{s}\mathfrak{o}\left(5\right).$
- For $J=3$, $\mathfrak{s}\mathfrak{o}\left(3\right)$ is irreducibly embedded into $\mathfrak{s}\mathfrak{o}\left(7\right)$ through the chain:$$\mathfrak{s}\mathfrak{o}\left(3\right)\subset {G}_{2,-14}\subset \mathfrak{s}\mathfrak{o}\left(7\right).$$
- For any integer $J\ge 4$, $\mathfrak{s}\mathfrak{o}\left(3\right)$ is a maximal subalgebra irreducibly embedded into $\mathfrak{s}\mathfrak{o}\left(\right)open="("\; close=")">2J+1.$
- For $J=\frac{3}{2}$, $\mathfrak{s}\mathfrak{o}\left(3\right)$ is embedded into $\mathfrak{s}\mathfrak{o}\left(4\right)$ through the chain:$$\mathfrak{s}\mathfrak{o}\left(3\right)\subset \mathfrak{s}\mathfrak{p}\left(4\right)\subset \mathfrak{s}\mathfrak{u}\left(4\right)\subset \mathfrak{s}\mathfrak{o}\left(7\right)\subset \mathfrak{s}\mathfrak{o}\left(8\right)$$
- For half-integers $J\ge \frac{5}{2}$, $\mathfrak{s}\mathfrak{o}\left(3\right)$ is embedded into $\mathfrak{s}\mathfrak{o}\left(\right)open="("\; close=")">4J+2$ through the chain:$$\mathfrak{s}\mathfrak{o}\left(3\right)\subset \mathfrak{s}\mathfrak{p}\left(\right)open="("\; close=")">2J+1\subset \mathfrak{s}\mathfrak{u}\left(\right)open="("\; close=")">2J+1$$

## 2. Construction of the Matrices ${R}_{J}^{I}\left(\right)open="("\; close=")">{X}_{k}$

**Lemma 1.**Let J be a positive integer. The following conditions hold:

- The characteristic and minimal polynomials ${p}_{J}\left(z\right)$ and ${q}_{J}\left(z\right)$ of the matrices ${D}_{J}\left(\right)open="("\; close=")">{X}_{k}$ in (5) coincide and are given by:$${p}_{J}\left(z\right)={q}_{J}\left(z\right)=-z\left(\right)open="("\; close=")">{z}^{2}+1\left(\right)open="("\; close=")">{z}^{2}+4$$
- In the representation ${D}_{J}$, the Casimir operator ${C}_{2}$ of $\mathfrak{s}\mathfrak{o}\left(3\right)$ is given by:$${C}_{2}={D}_{J}{\left(\right)}^{{X}_{1}}2+{D}_{J}{\left(\right)}^{{X}_{2}}2$$

## 3. Construction of the Matrices ${R}_{J}^{II}\left(\right)open="("\; close=")">{X}_{k}$

## 4. Tensor Products of Real Irreducible Representations

- $J,{J}^{\prime}\in \mathbb{N}$ and $J\ge {J}^{\prime}:$$${R}_{J}^{I}\otimes {R}_{{J}^{\prime}}^{I}=\sum _{\alpha =0}^{2{J}^{\prime}}{R}_{J+{J}^{\prime}-\alpha}^{I}.$$
- $J\in \mathbb{N}$, ${J}^{\prime}\equiv 1\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">mod\phantom{\rule{4.pt}{0ex}}2:$$${R}_{J}^{I}\otimes {R}_{\frac{{J}^{\prime}}{2}}^{II}=\sum _{\alpha =0}^{2{J}^{\prime}}{R}_{\frac{\left(\right)}{2}2}^{}II.$$
- $J,{J}^{\prime}$$\equiv 1\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">mod\phantom{\rule{4.pt}{0ex}}2:$$${R}_{\frac{J}{2}}^{II}\otimes {R}_{\frac{{J}^{\prime}}{2}}^{II}=\sum _{\alpha =0}^{2{J}^{\prime}}4\phantom{\rule{0.166667em}{0ex}}{R}_{\frac{\left(\right)}{J}2}^{}I.$$

- If ${R}_{J}^{I}$ is a representation of first class, then ${R}_{J}^{I}\left(X\right)$ has characteristic polynomial:$${p}_{J}\left(z\right)=-z\prod _{\alpha =1}^{J}\left(\right)open="("\; close=")">{z}^{2}+\xi \phantom{\rule{0.166667em}{0ex}}{\alpha}^{2},\phantom{\rule{0.277778em}{0ex}}$$
- If ${R}_{\frac{J}{2}}^{II}$ is a representation of the second class, then ${R}_{\frac{J}{2}}^{II}\left(X\right)$ has characteristic polynomial:$${p}_{\frac{J}{2}}\left(z\right)=\frac{1}{{2}^{2J+2}}\prod _{\beta =0}^{\frac{J-1}{2}}{\left(\right)}^{4}2$$

- The multiplicity of z, given by ${m}_{0}$, indicates the number of irreducible factors of Class I.
- The multiplicity of ${R}_{{J}_{r}}^{I}$ is given by ${m}_{r}$, whereas the multiplicity of ${R}_{{J}_{k}}^{I}$ is given by ${m}_{k}-{m}_{k+1}$ for $r-1\ge k\ge 1$.
- The multiplicity of the trivial representation ${R}_{0}^{I}$ is given by ${m}_{0}-{m}_{1}$.
- The multiplicity of ${R}_{\frac{{J}_{s}^{\prime}}{2}}^{II}$ is given by $\frac{1}{2}{n}_{s}$, whereas the multiplicity of ${R}_{\frac{{J}_{l}^{\prime}}{2}}^{II}$ is given by $\frac{{n}_{l}-{m}_{l+1}}{2}$ for $s-1\ge l\ge 1$.

**Theorem 2.**Let R be an arbitrary real representation of $\mathfrak{s}\mathfrak{o}\left(3\right)$ and $X\in \mathfrak{s}\mathfrak{o}\left(3\right)$. Then, the decomposition of R as the sum of real irreducible representations is completely determined by the characteristic polynomial $p\left(z\right)$ of the matrix $R\left(X\right)$.

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

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Campoamor-Stursberg, R.
An Elementary Derivation of the Matrix Elements of Real Irreducible Representations of so(3). *Symmetry* **2015**, *7*, 1655-1669.
https://doi.org/10.3390/sym7031655

**AMA Style**

Campoamor-Stursberg R.
An Elementary Derivation of the Matrix Elements of Real Irreducible Representations of so(3). *Symmetry*. 2015; 7(3):1655-1669.
https://doi.org/10.3390/sym7031655

**Chicago/Turabian Style**

Campoamor-Stursberg, Rutwig.
2015. "An Elementary Derivation of the Matrix Elements of Real Irreducible Representations of so(3)" *Symmetry* 7, no. 3: 1655-1669.
https://doi.org/10.3390/sym7031655