Physics:Double group

In applications of group theory in theoretical chemistry and physics, double groups are used to extend the use of point groups and space groups to include spin. They were introduced by Hans Bethe in order to deal with magnetic properties of heavy elements in the solid state. Double groups are widely used in physical and inorganic chemistry to characterize the states of certain coordination complexes for which spin-orbit coupling is important. Essentially, rotation by 2π is treated as a symmetry operation separate from the identity operation. An alternative treatment of double groups is given in (Brown 1970).

Character tables of crystallographic point groups

(Bethe 1929) discusses the problem of computing the character tables of the irreducible projective representations of finite subgroups of the special rotation group SO(3). Because SU(2) is a double cover of SO(3), i.e. SU(2) / {±I} = SO(3), this is equivalent to determining the character tables of cyclic groups, the binary dihedral groups, the binary octahedral group, the binary tetrahedal group and the binary icosahedral group. These are special cases of character tables calculated by Georg Frobenius and Issai Schur, who codified the combinatorial rules of Alfred Young, with his Young diagrams, for the symmetric groups [math]\displaystyle{ {\mathfrak S}_n }[/math] and alternating groups [math]\displaystyle{ {\mathfrak A}_n }[/math], as well as its Schur multiplier or central extension; in the case of the symmetric or alternating groups, the multipliers had values {±1}. (Murnaghan 1938) has described in detail how to compute the character tables. These apply in particular to the 32 crystallographic point groups, tabulated in (Koster Dimmock) based on Koster's work on the representation theory of the 230 space groups in 1957. Accounts, sometimes with details of proofs, can be found in (Miller 1972), (Burns 1977) and (Cornwell 1997).

Double groups in physical chemistry

The need to use a double group in molecular spectroscopy arises in a very specific situation. For example, it arises in characterizing the electronic states of compounds of copper ions in the +2 oxidation state (Cu(II) in Stock notation), as follows:

(1) Six-coordinate complexes of the Cu(II) ion, with the generic formula [CuL₆]²⁺, are subject to Jahn-Teller distortion so that the symmetry is reduced from octahedral (point group O_h) to tetragonal (point group D_4h). Since d orbitals are centrosymmetric the atomic term symbols can be classified in the D₄ subgroup.

(2) To a first approximation spin-orbit coupling can be ignored and the magnetic moment can be predicted using the spin-only approximation of 1.73 Bohr magnetons. However, for a more accurate prediction spin-orbit coupling must be taken into consideration. This means that the relevant quantum number is J (J = L + S)

(3) When J is half-integer, the character for a rotation by an angle of (α + 2π) is equal to minus the character for rotation by an angle of (α). This cannot be true for an identity operation, so the point group must be extended to include rotations by (α + 2π) as separate symmetry operations. This group is known as the double group, D₄'.

The use of the double group is more important in the case of silver(II) as the extent of spin-orbit coupling is greater than in copper(II). It would also apply to 6-coordinate complexes of gold(II).

A double group is also used for compounds of titanium in the +3 oxidation state. There is a single electron in a 3d orbital, so compounds of octahedral titanium(III) with the generic formula [TiL₆]ⁿ⁺ are paramagnetic. The compounds are centrosymmetric, so the point group O', rather than O_h, can be used to classify electronic states. Another application is with cerium(III) where the single unpaired electron is in a 4f orbital.

Character tables for the double groups D'₄ and O' are given in appendix VII of (Cotton 1971), amongst others.

McKay correspondence

Around 1980, John McKay noticed a remarkable correspondence (now known as the McKay correspondence) that appears in various ADE classifications. He observed that the binary finite subgroups Γ of SU(2) yield certain Coxeter-Dynkin diagrams, with irreducible representations corresponding to nodes and edges corresponding to the rules for tensoring by the 2-dimensional vector representation V.

Removing the node corresponding to the trivial representation, a Dynkin diagram of type A, D or E is obtained and hence the data to construct a simple complex Lie algebra [math]\displaystyle{ \mathfrak g }[/math]. Conversely the extended diagram can be interpreted as the data for a central extension of the loop algebra [math]\displaystyle{ {\mathbb C}[z,z^{-1}]\otimes{\mathfrak g} }[/math]. These infinite-dimensional Lie algebras are the simply laced affine Kac-Moody Lie algebras.

The Coxeter-Dynkin diagrams also appeared in the study of the quotient surface V / Γ or equivalently its ring of invariants [math]\displaystyle{ {\mathbb C}[V]^\Gamma }[/math], described explicitly by Felix Klein in his classic "Lectures on the Icosahedron". The resolution of the Kleinian singularity is accomplished by successively blowing up points on the surface, encoded combinatorially via a Coxeter-Dynkin diagram, as first proved by Patrick du Val in the 1930s. It is now understood how these three different pictures can be related—in a highly non-trivial way—using the ideas of quivers.

The McKay correspondence, which also enters into Macdonald identities and the Weyl-Kac character formula, simultaneously led to conjectures about characters of sporadic simple groups, especially the monster group. These conjectures about "monstrous moonshine" have now been proved by Richard Borcherds using generalized Kac-Moody algebras and vertex algebras.

Much of this mathematical progress originated in theoretical physics, in particular in applications to string theory, integrable systems and instantons.

References

• Lipson, R.H.. "Spin-orbit coupling and double groups". https://instruct.uwo.ca/chemistry/734b.  (web site)
• Murnaghan, Francis D. (1938). The theory of group representations. Baltimore: Johns Hopkins Press.  Reprinted by Dover Books in 1963.
• Koster, George F.; Dimmock, John O.; Wheeler, Robert G.; Statz, Hermann (1963). Properties of the thirty-two point groups. Cambridge, Mass.: The M.I.T. Press. 
• Miller, Willard, Jr. (1972). Symmetry groups and their applications. Pure and Applied Mathematics. 50. New York-London: Academic Press. 
• Burns, Gerald (1977). Introduction to group theory with applications. Materials Science and Technology. New York-London: Academic Press. ISBN 0-12-145750-8. 
• Cornwell, J. F. (1984). Group theory in physics. Vol. I. Academic Press. ISBN 0-12-189801-6.