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OrbitDiagram ​

The orbit basis of the partition algebra. A diagram asks for points to be connected; an orbit element asks for them to be connected AND NOTHING ELSE — so the two bases differ by Möbius inversion over the partition lattice.

WikipediaGroup actionGroup actionMathWorldGroupActionWikidataQ288465Encyclopedia of MathematicsGroup_action
OrbitDiagram(blocks)an orbit basis element
InDiagramBasis(x) / InOrbitBasis(x)change of basis, either direction
PartitionMobius(finer, coarser)the partition lattice's Möbius function on that interval
DiagramCoarsenings(d)every partition coarser than this one

Domain: Diagram algebras

Details
  • , and the inverse carries
  • Coarsening is partitioning the blocks, so a diagram with blocks has coarsenings
  • The change of basis is unitriangular in the number of blocks, which is what makes the two maps inverse
  • Contrast the Hopf-algebra bases, which invert over the BOOLEAN lattice where the Möbius function is only a sign — the factorials here are the cost of merging any blocks rather than adjacent ones
  • The map onto the symmetric group's centraliser algebra kills exactly when has more blocks than — a statement with no clean form in the diagram basis

Examples

See also: Diagram, PartitionAlgebra