Crystallography & Symmetry
patterns derived from their groups
Concept

Factor system — where it appears

The set of phases by which two symmetry operators fail to compose into the operator of their product. It is what makes a representation projective rather than ordinary, it can sometimes be removed by rephasing the operators and sometimes provably cannot, and where it cannot the little group has no one-dimensional representation at all.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as projective representation — the same set of essays touches all of them, so they are one junction rather than several.

pgg at (1/2, 0): the operators multiply up to a sign. Every product of two Bloch operators of the little group of (1/2, 0) in pgg, against the operator of the product. They agree up to a scalar, and the scalar is +1 or −1: 4 of the 16 products come back with a minus sign. No rephasing removes them, and the search that says so tries every assignment of twelfth roots of unity to the operators. A representation that multiplies only up to this sign cannot be one-dimensional, because scalars commute and these operators do not. Each entry is an exponent modulo twelve, so the table is exact.

A glide sticks two levels together

The translation attached to a glide moves no wavevector at all. It comes back as a phase factor, and at the edge of the zone the factor is minus one — after which the operators of the little group no longer multiply the way the group does, and no rephasing repairs it.

space-groups · K symmetry Where time reversal does something, and what. Every wavevector of every plane group at which time reversal changes the answer, with the square of each antiunitary operator, the unitary prediction, Herring's corrected prediction and the measured degeneracies. Case (b) is Kramers' theorem in a crystal with no spin, and it happens exactly where a glide's operator squares to −1. Case (c) is a representation being carried to a different one by the antiunitary operator, so the two become one level. Nine of these rows were open before the criterion was built — three the earlier census called unaccounted and six it could not reach at all.

The degeneracy time reversal forces

A crystal with a glide has levels that stick together at the edge of its zone for a reason no character table contains. The operation responsible is antiunitary, it squares to minus one, and Kramers' theorem then applies to a model with no spin anywhere in it — which closes nine rows an earlier census in this collection had to leave open.

space-groups · K symmetry

Named alongside it

The objects these essays reach for when they reach for this one.

DegeneracyLittle groupProjective representationAntiunitary operatorCommutantFrobenius schur indicatorGlide planeHerring criterionKramers degeneracyNon-symmorphicTime reversalZone boundary

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