representation of rotation group

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What is a rotation group and how do we get its unitary representation?

The rotation group is ${\rm SO(3)}$ . It is the group of $3\times 3$ orthogonal matrices $\{g(\theta)\}$ with unit determinant. So these are already defined in terms of $3\times 3$ matrices. But we use unitary representation $\{U(g(\theta))\}$ of the rotation group in quantum mechanics. What does that even mean?

How do we define rotation group if not in terms of explicit ${\rm SO(3)}$ matrices? Is this already not a representation? Isn't the definition of rotation group already in terms of this representation?

Given the elements of the rotation group (i.e., the ${\rm SO(3)}$ matrices $g(\theta)$ ) how do we get, $U(g(\theta))$ ?

• quantum-mechanics
• group-theory
• group-representations

• 3 $\begingroup$ This is way too broad. Do you know what the formal definition of a representation is? Have you at least read the wikipedia entry? $\endgroup$ –  AccidentalFourierTransform Commented May 29, 2019 at 16:20
• $\begingroup$ @AccidentalFourierTransform yes, I know the definition of representation. Group elements are represented by matrices that obey the group structure. Does it help? $\endgroup$ –  Solidification Commented May 29, 2019 at 17:53
• $\begingroup$ May or may not be helpful but I wrote some notes on representation theory of SO(3) and QM here: scholar.harvard.edu/files/noahmiller/files/… $\endgroup$ –  user1379857 Commented May 29, 2019 at 22:59
• $\begingroup$ Do you understand that representations can have various dimensions? For example there are $5 \times 5$ and $17 \times 17$ matrices representing 3D rotations, not just $3 \times 3$ ones. $\endgroup$ –  G. Smith Commented May 30, 2019 at 0:02
• $\begingroup$ Related : Given the transformation of $SU(2)$ triplet $\vec{\phi}$ how to find the transformation of ${\Phi}\equiv\vec{\phi}\cdot\vec{\tau}$? . $\endgroup$ –  Frobenius Commented Dec 26, 2019 at 0:36

2 Answers 2

The point here is that, if $\omega\in SO(3)$ , and if $\omega_1\cdot \omega_2=\omega \in SO(3)$ is the combination rule on abstract elements, then a representation (by matrices) $U$ is a map $\omega\mapsto U(\omega)$ so that the rule $$ \omega_1\cdot \omega_2=\omega \quad \Rightarrow \quad U(\omega_1)\cdot U(\omega_2)=U(\omega) \tag{1} $$ for any $\omega_1,\omega_2,\omega\in SO(3)$ is also satisfied by the matrices $U(\omega)$ representing the elements. There is a theorem stating that, for $SO(3)$ and a bunch of others, all representation are equivalent to unitary representations, so that $U(\omega^{-1})=U^{-1}(\omega)=U^\dagger(\omega)$ .

Although the so-called defining representation is in terms of a $3$ -dimensional space on which $SO(3)$ acts "naturally", there may be matrices of dimension other than $3$ that satisfy the basic composition law of $\omega_1\cdot \omega_2=\omega $ or its matrix version of Eq.(1).

You can "obtain" a representation by larger matrices by tensoring and decomposing the resulting representation. For instance, if $\{\vert {1}\rangle ,\vert {2}\rangle,\vert {3}\rangle\}$ are a basis for the $3$ -dimensional irrep of $SO(3)$ , then the set $\{\vert i\rangle\vert j\rangle\}$ spans a 9-dimensional space with $$ U(\omega)\left[\vert i\rangle\otimes\vert j\rangle\right]:= \left[U(\omega)\vert i\rangle\right]\otimes \left[U(\omega)\vert j\rangle\right] $$ will provide you with a $9$ -dimensional representation, which turns out to be reducible. Note that I'm abusing the notation here because on the left I have $U$ as a $9\times 9$ matrix but on the right the $U$ 's are $3\times 3$ matrices. In fact, the $9\times 9$ representation is reducible: it contains $L=2,1,0$ , i.e. irreducible pieces of dimensions $5,3$ and $1$ . The $L=2$ and $L=0$ irreps are spanned by symmetric combinations like $\vert 1\rangle\vert 2\rangle+ \vert 2\rangle\vert 1\rangle$ etc, while the $L=1$ contains antisymmetric pieces.

In cases other than $SO(3)$ (or $SU(2)$ ), one can also obtain inequivalent representations by taking the conjugate. The simplest example would be $SU(3)$ , where the defining representation ( $3\times 3$ ) is often denoted by $\textbf{3}$ or $(1,0)$ in the Dynkin scheme, and where its (non-equivalent) conjugate is denoted by $\textbf{3*}$ or $(0,1)$ . One can then construct any representation by tensoring a suitable number of copies of $(1,0)$ and $(0,1)$ and decomposing the result.

Note that $SO(3)$ representations (and also $SU(2)$ representations) are "self-conjugate" in the sense that taking the conjugate yields the same representation.

[about SO(3)] SO(3) is an abstract group with a lot of well-known properties (Lie, compact, topological etc). The representation of SO(3) via 3d, orthogonal, real-valued maricies is one of many possible ones. But this is, by definition, a faithful representation, i.e. every group member has a distinct matrix that corresponds to it.

[about getting unitary reps]

The rotation matricies are orthogonal and real-valued, so they are already unitary, thus, technically, the question is moot.

If your question is how to go from the general orthogonal, 3d, real-valued matrix with unity determinant to its representation as $R=\exp\left(\dots\right)$ , i would suggest diagonalizing the matrix.

You will find that this does not work over the space $\mathbb{R}^3$ , but it does work over $\mathbb{C}^3$ . Quite simple reasoning can show that any

$R\in \mathbb{R}^3\times\mathbb{R}^3$ with $\det R=1$ and $R^T=R^{-1}$ can be diagonalized in the complex space with eigenvalues:

$\lambda_{1,2,3}=\exp\left(\pm i\phi\right), 1$ for some $\phi\in\mathbb{R}$

Thus $R=V \exp\left(i\left(\begin{array}\\ \phi &&0 && 0\\ 0 && -\phi && 0\\ 0 && 0 && 0 \end{array}\right)\right) V^{\dagger}$

Where the $V$ is the unitary matrix with eigenvectors. Now simply take the $V$ matricies into the exponential and you will have your representation.

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• > Introduction to the Representation Theory of Compact and Locally Compact Groups
• > Representations of the rotation group

Book contents

• Frontmatter
• Conventional notations and terminology
• PART I REPRESENTATIONS OF COMPACT GROUPS
• 1 Compact groups and Haar measures
• 2 Representations, general constructions
• 3 A geometrical application
• 4 Finite-dimensional representations of compact groups (Peter-Weyl theorem)
• 5 Decomposition of the regular representation
• 6 Convolution, Plancherel formula & Fourier inversion
• 7 Characters and group algebras
• 8 Induced representations and Frobenius-Weil reciprocity
• 9 Tannaka duality
• 10 Representations of the rotation group
• PART II REPRESENTATIONS OF LOCALLY COMPACT GROUPS

10 - Representations of the rotation group

Published online by Cambridge University Press:  20 March 2010

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• Representations of the rotation group
• Alain Robert
• Book: Introduction to the Representation Theory of Compact and Locally Compact Groups
• Online publication: 20 March 2010
• Chapter DOI: https://doi.org/10.1017/CBO9780511661891.012

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• DOI: 10.2307/2315370
• Corpus ID: 121751062

Representations of the Rotation and Lorentz Groups and Their Applications

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• Published 1 October 1965
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Title: level one automorphic representations of an anisotropic exceptional group over $\mathbb{q}$ of type $\mathrm{f}_{4}$.

Abstract: Up to isomorphism, there is a unique connected semisimple algebraic group over $\mathbb{Q}$ of Lie type $\mathrm{F}_{4}$, with compact real points and split over $\mathbb{Q}_{p}$ for all primes $p$. Let $\mathbf{F}_{4}$ be such a group. In this paper, we study the level one automorphic representations of $\mathbf{F}_{4}$ in the spirit of the work of Chenevier, Renard, and Taïbi. First, we give an explicit formula for the number of these representations having any given archimedean component. For this, we study the automorphism group of the two definite exceptional Jordan algebras of rank $27$ over $\mathbb{Z}$ studied by Gross, as well as the dimension of the invariants of these groups in all irreducible representations of $\mathbf{F}_{4}(\mathbb{R})$. Then, assuming standard conjectures by Arthur and Langlands for $\mathbf{F}_{4}$, we refine this counting by studying the contribution of the representations whose global Arthur parameter has any possible image (or "Sato-Tate group"). This includes a detailed description of all those images, as well as precise statements for the Arthur's multiplicity formula in each case. As a consequence, we obtain a conjectural but explicit formula for the number of algebraic, cuspidal, level one automorphic representation of $\mathrm{GL}_{26}$ over $\mathbb{Q}$ with Sato-Tate group $\mathbf{F}_{4}(\mathbb{R})$ of any given weight (assumed "$\mathrm{F}_{4}$-regular"). The first example of such representations occurs in motivic weight $36$.

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Risc: Explainable Rotation-Invariant Self-Supervised Representation Learning

13 Pages Posted: 8 Jul 2024

devansh singh

affiliation not provided to SSRN

Aboli Marathe

Carnegie Mellon University

Siddharth Roy

University of Pennsylvania

Rahee Walambe

Symbiosis international (deemed university), dr. ketan kotecha.

This paper describes a method that can perform robust detection and classification in out-of-distribution rotated images in the medical domain. In real-world medical imaging tools, noise due to the rotation of the body part is frequently observed. This noise reduces the accuracy of AI-based classification and prediction models.  Hence, it is important to develop models which are rotation invariant. To that end, the proposed method - RISC (rotation invariant self-supervised vision framework) addresses this issue of rotational corruption. We present state-of-the-art rotation-invariant classification results and provide explainability for the performance in the domain. The evaluation of the proposed method is carried out on real-world adversarial examples in Medical Imagery-OrganAMNIST, RetinaMNIST and PneumoniaMNIST. It is observed that RISC outperforms the rotation-affected benchmark methods by obtaining 22\% , 17\% and 2\% accuracy boost on OrganAMNIST, PneumoniaMNIST and RetinaMNIST rotated baselines respectively. Further, explainability results are demonstrated. This methods paper describes:  a representation learning approach that can perform robust detection and classification in out-of-distribution rotated images in the medical domain.It presents a method that incorporates self-supervised rotation invariance for correcting rotational corruptions.Gradcam-based explainability for the rotational SSL pretext task and the downstream classification outcomes for the three benchmark datasets are presented

Note: Funding Information: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Declaration of Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Keywords: computervision, self-supervisedlearning, robustness

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Devansh Singh

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University of Pennsylvania ( email )

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Rahee Walambe (Contact Author)

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Group Representations I: Rotations and Spherical Harmonics

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