Corepresentations of unitary and antiunitary groups

Definition

Let G be a group with a subgroup H of index 2. A corepresentation is a homomorphism into a group of operators over a vector space over the complex numbers where for all u in H the image of u is a linear operator and for all a in the coset G-H the image of a is antilinear (where '*' means complex conjugation):

${\displaystyle {\begin{aligned}&\forall u\in H,D(u)(a{\bf {x}}+b{\bf {y}})=a\times D(u){\bf {x}}+b\times D(u){\bf {y}}\\&\forall a\in G-H,D(a)(a{\bf {x}}+b{\bf {y}})=a^{*}\times D(a){\bf {x}}+b^{*}\times D(a){\bf {y}}\end{aligned}}}$

Properties

As this is a homomorphism

${\displaystyle {\begin{aligned}&D(u_{1}u_{2})=D(u_{1})D(u_{2})\\&D(ua)=D(u)D(a)\\&D(au)=D(a)D(u)^{*}\\&D(a_{1}a_{2})=D(a_{1})D(a_{2})^{*}\end{aligned}}}$

Reducibility

Two corepresentations are equivalent if there is a matrix V

${\displaystyle {\begin{aligned}&\forall u,D'(u)=VD(u)V^{-1}\\&\forall a,D'(a)=VD(a)V^{*-1}\end{aligned}}}$

Just like representations, a corepresentation is reducible if there is a proper subspace invariant under the operations of the corepresentation. If the corepresentation is given by matrices, it is reducible if it is equivalent to a corepresentation with each matrix in block diagonal form.

If the corepresentation is not reducible, then it is irreducible.

Schur's lemma

Schur's lemma for irreducible representations over the complex numbers states that if a matrix commutes with all matrices of the representation then it is a (complex) multiple of the identity matrix, that is, the set of commuting matrices is isomorphic to the complex numbers ${\displaystyle \mathbb {C} }$. The equivalent of Schur's lemma for irreducible corepresentations is that the set of commuting matrices is isomorphic to ${\displaystyle \mathbb {R} }$, ${\displaystyle \mathbb {C} }$ or the quaternions ${\displaystyle \mathbb {Q} }$.^[2] Using the intertwining number over the real numbers, this may be expressed as an intertwining number of 1, 2 or 4.

Relation to representations of the linear subgroup

Typically, irreducible corepresentations are related to the irreducible representations of the linear subgroup H.^[1][2][3][4] Let ${\displaystyle \Delta }$ be an irreducible (ordinary) representation of the linear subgroup H. Form the sum over all the antilinear operators of the square of the character of each of these operators:

${\displaystyle S=\sum _{a}\chi _{\Delta }(a^{2})}$

and set ${\displaystyle P=D(a_{0})}$ for an arbitrary element ${\displaystyle a_{0}}$.

There are three cases, distinguished by the character test eq 7.3.51 of Cracknell and Bradley.^[5]

Type(a)

If S = |H| (the intertwining number is one) then D is an irreducible corepresentation of the same dimension as ${\displaystyle \Delta }$ with

${\displaystyle {\begin{aligned}&D(u)=\Delta (u)\\&D(a)=D(aa_{0}^{-1}a_{0})=\Delta (aa_{0}^{-1})P\end{aligned}}}$

Type(b)

S = -|H| (the intertwining number is four) then D is an irreducible representation formed from two 'copies' of ${\displaystyle \Delta }$

${\displaystyle {\begin{aligned}&D(u)={\begin{pmatrix}\Delta (u)&0\\0&\Delta (u)\end{pmatrix}}\\&D(a)={\begin{pmatrix}0&\Delta (aa_{0}^{-1})P\\-\Delta (aa_{0}^{-1})P&0\end{pmatrix}}\end{aligned}}}$

Type(c)

If S = 0 (the intertwining number is two), then D is an irreducible corepresentation formed from two inequivalent representations ${\displaystyle \Delta }$ and ${\displaystyle \Delta '}$ where ${\displaystyle \Delta '(u)=\Delta (a_{0}^{-1}ua_{0})^{*}}$

${\displaystyle {\begin{aligned}&D(u)={\begin{pmatrix}\Delta (u)&0\\0&\Delta (a_{0}^{-1}ua_{0})^{*}\end{pmatrix}}\\&D(a)={\begin{pmatrix}0&\Delta (aa_{0}^{-1})\\\Delta (a_{0}^{-1}a)^{*}&0\end{pmatrix}}\end{aligned}}}$

Cracknell and Bradley^[5] show how to use these to construct corepresentations for the magnetic point groups, while Cracknell and Wong^[6] give more explicit tables for the double magnetic groups.

Character theory of corepresentations

Standard representation theory for finite groups has a square character table with row and column orthogonality properties. With a slightly different definition of conjugacy classes and use of the intertwining number, a square character table with similar orthogonality properties also exists for the corepresentations of finite magnetic groups.^[2]

Based on this character table, a character theory mirroring that of representation theory has been developed.^[7]

See also

• Mock, A. (2016). "Characterization of parity-time symmetry in photonic lattices using Heesh-Shubnikov group theory". Optics Express. 24 (20): 22693–22707. arXiv:1606.05044. Bibcode:2016OExpr..2422693M. doi:10.1364/OE.24.022693. PMID 27828339. S2CID 24476384.
• Schweiser, J. (2005). "Time inversion in the representation analysis of magnetic structures theory". C. R. Physique. 6: 375–384. doi:10.1016/j.crhy.2005.01.009.
• Angeloval, M. N.; Boyle, L. L. (2005). "On the classification and enumeration of the irreducible co-representations of magnetic space groups". Journal of Physics A: Mathematical and General. 29 (5): 993–1010. doi:10.1088/0305-4470/29/5/014.
• Scurek, R. (2004). "Understanding the CPT group in particle physics: Standard and nonstandard representations". Am. J. Phys. 75 (5): 638–643. Bibcode:2004AmJPh..72..638S. doi:10.1119/1.1629087.

References