irreducible representation of quaternion group

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Quaternion Group

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Irreducible representations of finite p-groups

Let $G$ be a finite $p$ -group. What are irreducible representations of $G$ over a field of characteristic $q$ , such that $(p,q)=1$ ? Can we say something in general ? In particular, if there exists some technique to find those explicit representations in case of small groups of order $p^3, p^4, p^5 ...$ ?

• gr.group-theory
• rt.representation-theory
• finite-groups

• 2 $\begingroup$ The first question is standard representation theory. The irreducible representations in coprime characteristic over algebraically closed fields can be realized directly as reductions from those in characteristic zero. Over finite fields, things are a little easier, because the Schur index is always trivial. For the second question, there are algorithms for computing irreducible representations, but the question needs a more focus. $\endgroup$ –  Derek Holt Commented Jul 22, 2021 at 7:38

For the last part of the question, they are all induced from primitive representations of subgroups, so you need to understand primitive representations of $p$ -groups. If $G$ is a $p$ -group with a faithful primitive (irreducible) representation over a finite field $F$ of coprime characteristic $q$ , then all Abelian normal subgroups of $G$ are cyclic by Clifford theory. If $p$ is odd, this implies that $G$ itself is cyclic. If $p =2$ , I think $G$ can also be dihedral, (generalized) quaternion or semidihedral, as well as cyclic.

If $p$ is odd, and $e$ is the least power of $q$ such that $p^{m}|(|F|^{e}-1)$ , then a cyclic group of order $p^{m}$ has a faithful irreducible representation of degree $e$ (but no lower dimension) over $F$ .

If $p =2$ , the situation is more delicate: for example, a semidihedral group of order $16$ has a faithful $2$ -dimensional primitive representation over $F$ when $|F| = 3$ .

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A Complex Irreducible Representation of the Quaternion Group and a Non-free Projective Module over the Polynomial Ring in Two Variables over the Real Quaternions

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• R. Sridharan 2  

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It is a classical result of Frobenius and Schur ([F], p. 20–22 or [Se], p. 121–124) that any finite dimensional complex irreducible representation of a finite group, whose character is real, either descends to a real representation or can be extended to a representation of the group over the real quaternion algebra. The simplest example where the latter phenomenon holds is the standard 2-dimensional complex representation of the group of integral quaternions. An application of some general results of Barth and Hulek shows that this representation leads to a canonical rank 2 (stable) vector bundle over the complex projective plane. It can be shown that the restriction of this bundle to the affine plane gives rise to a non-free projective module of H [ X, Y ], isomorphic to the one constructed in [OS] in another context in a different manner. (The existence of this projective module led, incidentally, to the construction in ([PI]) of non diagonalisable, (in fact indecomposable), non singular symmetric 4 × 4 matrices of determinant one over the polynomial ring in two variables over the field of real numbers, producing remarkable counter examples to the so called quadratic analogue of Serre’s conjecture and opening up a new and fruitful area of research (cf. [P3]). On the other hand, it was shown in ([KPS]) that any non-free projective module over D [ X, Y ], where D is a finite dimensional division algebra over a field, extends (and essentially uniquely) to a vector bundle over the projective plane over this field, with a D -structure.

To Parimala, who wove golden garments with the gossamer strands of my thoughts, for the jubilee year

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Pseudo-parabolic Category over Quaternionic Projective Plane

Homogeneous projective varieties with semi-continuous rank function, topological classification of complex vector bundles over 8-dimensional spin \(^{c}\) manifolds.

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Sridharan, R. (1999). A Complex Irreducible Representation of the Quaternion Group and a Non-free Projective Module over the Polynomial Ring in Two Variables over the Real Quaternions. In: Passi, I.B.S. (eds) Algebra. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9996-3_16

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Enumeration and Representation Theory of Spin Space Groups

Xiaobing chen, jun ren, yanzhou zhu, yutong yu, ao zhang, pengfei liu, jiayu li, yuntian liu, caiheng li, and qihang liu, phys. rev. x 14 , 031038 – published 28 august 2024.

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Supplemental Material

• INTRODUCTION
• ENUMERATION OF SPIN SPACE GROUPS
• REPRESENTATION OF SPIN SPACE GROUPS
• REALISTIC MATERIALS
• LIST OF SYMBOLS AND ABBREVIATIONS
• ACKNOWLEDGMENTS

Fundamental physical properties, such as phase transitions, electronic structures, and spin excitations, in all magnetic ordered materials, were ultimately believed to rely on the symmetry theory of magnetic space groups. Recently, it has come to light that a more comprehensive group, known as the spin space group (SSG), which combines separate spin and spatial operations, is necessary to fully characterize the geometry and underlying properties of magnetic ordered materials. However, the basic theory of SSG has seldom been developed. In this work, we present a systematic study of the enumeration and the representation theory of the SSG. Starting from the 230 crystallographic space groups and finite translation groups with a maximum order of eight, we establish an extensive collection of over 100 000 SSGs under a four-index nomenclature as well as international notation. We then identify inequivalent SSGs specifically applicable to collinear, coplanar, and noncoplanar magnetic configurations. To facilitate the identification of the SSG, we develop an online program that can determine the SSG symmetries of any magnetic ordered crystal. Moreover, we derive the irreducible corepresentations of the little group in momentum space within the SSG framework. Finally, we illustrate the SSG symmetries and physical effects beyond the framework of magnetic space groups through several representative material examples, including a candidate altermagnet RuO 2 , spiral spin polarization in the coplanar antiferromagnet CeAuAl 3 , and geometric Hall effect in the noncoplanar antiferromagnet CoNb 3 S 6 . Our work advances the field of group theory in describing magnetic ordered materials, opening up avenues for deeper comprehension and further exploration of emergent phenomena in magnetic materials.

• Received 31 July 2023
• Revised 18 April 2024
• Accepted 16 May 2024

DOI: https://doi.org/10.1103/PhysRevX.14.031038

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

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• Research Areas
• Physical Systems

Authors & Affiliations

• 1 Department of Physics and Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology , Shenzhen 518055, China
• 2 National Center for Applied Mathematics Shenzhen, and Department of Mathematics, Southern University of Science and Technology , Shenzhen 518055, China
• 3 Guangdong Provincial Key Laboratory of Computational Science and Material Design, Southern University of Science and Technology , Shenzhen 518055, China
• 4 Shenzhen Key Laboratory of Advanced Quantum Functional Materials and Devices, Southern University of Science and Technology , Shenzhen 518055, China
• * These authors contributed equally to this work.
• † Contact author: [email protected]

Popular Summary

Symmetry, encoded within the framework of group theory, is an exceptionally powerful tool for comprehending various phenomena. In recent decades, crystallographic and magnetic symmetry groups have greatly advanced the understanding of condensed matter physics, materials science, and physical chemistry. But to fully characterize the geometry of magnetic ordered materials, a more comprehensive type of symmetry group is necessary. Such a group is known as the spin space group, or SSG, that combines separate spin and spatial operations. Unfortunately, the fundamental theory of SSGs is still lacking. In this work, we develop the enumeration and representation theory of SSGs.

We establish an extensive collection of more than 100 000 SSGs, and we identify SSGs using the international notation for collinear, coplanar, and noncoplanar magnetic configurations. In addition, we illustrate the SSG symmetries and physical effects beyond the framework of magnetic space groups through several representative material examples. To facilitate a broad community, we further develop an online program that can determine the SSG symmetries and the related physical effects of any magnetic ordered crystals.

Our work, along with two concurrent contributions, opens avenues for deeper understanding and further exploration of emergent phenomena in magnetic materials. A particular example would be the comprehensive classification of magnetic order and the emergence of new types of unconventional magnetism.

Spin Space Groups: Full Classification and Applications

Zhenyu xiao, jianzhou zhao, yanqi li, ryuichi shindou, and zhi-da song, phys. rev. x 14 , 031037 (2024), enumeration of spin-space groups: toward a complete description of symmetries of magnetic orders, yi jiang, ziyin song, tiannian zhu, zhong fang, hongming weng, zheng-xin liu, jian yang, and chen fang, phys. rev. x 14 , 031039 (2024), article text.

Vol. 14, Iss. 3 — July - September 2024

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• Condensed Matter Physics
• Materials Science

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Schematic plot of a spin group symmetry of an antiferromagnetic structure. It takes a fourfold rotation ( 4 001 ) in real space followed by a twofold rotation ( 2 001 ) in spin space, constituting a spin group symmetry { 2 001 ∥ 4 001 } . Such a symmetry operation contains separated lattice and spin rotations and is thus beyond the framework of magnetic groups.

Nontrivial SSGs generated from a t -type ( L 0 , G 0 ) subgroup pair and the corresponding magnetic configurations. In this case, the coordinate system x y z in spin space coincides with the coordinate system a b c in real space. (a) Structure with sublattice group L 0 = P 2 / c (No. 13). The group generators are indicated by green arrows. The balls with different colors denote different sublattices. (b) Structure with nonmagnetic group G 0 = P c c a (No. 54). The group generators that connect different sublattices are indicated by orange arrows. (c)–(h) Magnetic configurations and the corresponding international notations for different nontrivial SSGs. Green and orange arrows indicate the generators connecting the same and the different sublattices, respectively. The double star denotes SSGs for collinear magnetic configurations. Note that when considering the spin-only group, the SSGs generated by G s = 2 and m [e.g., panels (d) and (e); panels (g) and (h)] are equivalent.

Summary of inequivalent SSGs for collinear, coplanar, and noncoplanar magnetic configurations when considering their spin-only groups. The blue fonts denote the corresponding G s . “Polyhedral” indicates polyhedral PGs, i.e., T , T d , T h , O , and O h . “Others” indicates other G s except 1, 1 ¯ , 2, m , 2 / m , and polyhedral PGs. “ ⇀ ” means that when considering collinear spin-only group G SO l , G s = 1 ¯ and 2 yield equivalent collinear SSGs. “ ⊊ ” means that when considering coplanar spin-only group G SO p , the 86 951 nontrivial SSGs (supporting both coplanar and noncoplanar magnetic configurations) is reduced to 15 192 inequivalent coplanar SSGs.

Spin space group of altermagnet RuO 2 . (a) Crystal structure. The symmetry connecting different sublattices is indicated. The coordinate system x y z in spin space coincides with the coordinate system a b c in real space. (b) Brillouin zone in momentum space. (c) SOC-free band structure with the projection of the spin components. (d) Little groups and the corresponding projective coirreps for different k -points within the regime of SSG. Brown fonts denote the k -points manifesting spin splitting; green fonts denote the k -points manifesting extra degeneracies compared with the band structure with SOC. (e) Same as (c) but with SOC. (f) Same as (d) but within the regime of MSG.

Spin space group of spiral antiferromagnet CeAuAl 3 . (a) Crystal structure. The symmetry connecting different sublattices is indicated. The coordinate system x y z in spin space coincides with the coordinate system a b c in real space. (b) SOC-free electronic band structure. (c) Brillouin zone in momentum space. (d) Spin polarization S z at the k z = π / 2 plane for the band marked in panel (b). (e) Little groups and the corresponding projective coirreps for different k -points.

Spin space group of noncoplanar antiferromagnet CoNb 3 S 6 . (a) Crystal structure. The symmetry connecting different sublattices is indicated. The cubic structure in spin space is illustrated by the two tetrahedrons. The relationship between the x y z coordinate system in spin space and the a b c coordinate system in real space can be expressed as follows: a = − x − y , b = y + z , and c = − x + y − z . (b) Electronic band structure. (c) Brillouin zone in momentum space. (d) Geometric Hall conductivity as a function of energy. SOC is excluded in the calculations.

Magnetic primitive cell of P 1 − 6 3 001 1 ( 2 / 3     1 / 3     0 ) (174.174.3.1) under L 0 basis (black line) and under G 0 basis (orange line).

The overall procedure of reducing left regular projective reps M k ( g i ( a ) ) . (a) The left regular projective reps are N k × N k rep matrices. (b) The transformation matrices U 1 formed by common eigenvectors of ( C , C ( s ) ) can partly diagonalize M k ( g i ( a ) ) but fail to distinguish the same irreps. (c) the transformation matrices U formed by common eigenvectors of ( C , C ( s ) , C ¯ ( s ) ) can be used to completely lift the remaining degeneracy and find all inequivalent irreps.

Procedure of the identification of SSG from a given magnetic structure.

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Representation of $Q_8$ over $\mathbb{R}$

I'm trying to solve the following problem,

Give an example of a finite group $G$ and its irreducible representation $L$ over $\mathbb{R}$ such that the division algebra $Hom_G(L, L)$ is isomorphic to the algebra $\mathbb{H}$ of quaternions.

$Hom_G$ definition

I looked at the dimension $4$ irreducible representation of $Q_8$ over $\mathbb{R}$, which is the homomorphism from $Q_8$ to $GL_4(\mathbb{R})$ that maps elements of $Q_8$ to corresponding left multiplication. The matrix representation can be found here . (On that page search Four-dimensional irreducible representation over a non-splitting field)

I found this embedding of $\mathbb{H}$ into $M_4(\mathbb{R})$ and I've checked that these matrices commute with those matrix representation of elements of $Q_8$ (therefore in $Hom_G(L,L)$) but I haven't checked these are all of them and I'm a little lost here.

Could anyone please explain a better way of understanding/approaching this question than lucky guess+google search?

• abstract-algebra
• representation-theory
• group-homomorphism

Giving a homomorphism $\phi \colon \mathbb{H} \to \text{Hom}_{Q_8}(L,L)$ of $\mathbb{R}$-algebras is equivalent to giving a map

$$ \hat{\phi} \colon \mathbb{H} \times L \to L $$

which is $\mathbb{R}$-bilinear and additionally satisfies:

• $\hat{\phi}(hr,l) = \hat{\phi}(h,rl)$ if $r$ is a real number.
• $\hat{\phi}(h,ql) = q \hat{\phi}(h,l)$ if $q \in Q_8$.
• $\hat{\phi}(h_1h_2,l)= \hat{\phi}(h_1,\hat{\phi}(h_2,l))$ .

The correspondence is given by $\hat{\phi}(a,l) = \phi(a)(l)$.

It is easy to see that the multiplication of quaternions $ \mathbb{H} \times \mathbb{H} \to \mathbb{H}$ satisfies these properties, that is $ L = \mathbb{H}$, if we let $Q_8$ act on $ L = \mathbb{H}$ by seeing $Q_8$ as the group of units of $\mathbb{H}$ and act on $L$ by multiplication on the left.

So $L$ should be $\mathbb{H}$, which as a vector space over $\mathbb{R}$ is just $\mathbb{R}^4$. When you translate the action of $Q_8$ on $\mathbb{H}$ to the corresponding action on $\mathbb{R}^4$, you get precisely the irreducible $4$-dimensional representation you mentioned.

When you translate the multiplication of quaternions $ \mathbb{H} \times \mathbb{H} \to \mathbb{H}$ into a map $ \mathbb{H} \times \mathbb{R}^4 \to \mathbb{R}^4$ it gives you a map $\mathbb{H} \to M_4(\mathbb{R})$. This map is the same you mentioned as well, and being an embedding gives you that the corresponding $\phi$ is injective. But checking that this is an embedding is easy if you think of it as representing the multiplication of quaternions.

Finally to see that it is surjective, if you have a map $ f \colon \mathbb{H} \to \mathbb{H}$ which is $\mathbb{R}$-linear and the elements $1$, $i$, $j$ and $k$ act by multiplication on the left, then the map is left multiplication by $f(1)$.

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