Flag orbit

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A flag orbit is a set of flags which are identical under the symmetry group (or automorphism group) of a polytope.

Since only the trivial symmetry can map a flag to itself, each flag orbit of a polytope has as many flags as there are symmetries of the polytope. Thus the number of flag orbits can be obtained by dividing the total number of flags in the polytope by the size of its symmetry group.

Definition[edit | edit source]

A flag orbit is an orbit of the symmetry group acting on the flags of a polytope.

Alternatively a flag orbit is a set of flags such that the symmetry group is:

  • Transitive: For every pair of flags a  and b , there is a symmetry mapping a  to b .
  • Closed: For any flag in the orbit, a , and any symmetry s , s (a ) is also in the orbit.

Adjacency[edit | edit source]

Two flag orbits are adjacent if any of their flags are adjacent. Because the symmetry is transitive on flag orbits, if two orbits have a pair of adjacent flags than every flag in each orbit is part of an adjacent pair.

More formally two flag orbits are adjacent if there is a flag change mapping between them.

Applications[edit | edit source]

Flag orbits appear in a number of other definitions:

  • A regular polytope can be defined as a polytope with exactly 1 flag orbit.
  • A two-orbit polytope is a polytope with exactly 2 flag orbits.
  • A chiral polytope is a polytope with exactly 2 flag orbits which are not adjacent to themselves.
  • A polytope is chiral if does not have a flag orbit which is adjacent to itself.