OrbitDiagram

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

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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

Typed signatures

Type	Package	Applies
(list) -> number	diagram	always

Domain: Diagram algebras

Details

• $d_\lambda = \sum_{\mu \succeq \lambda} x_\mu$, and the inverse carries $\mu_\Pi(\lambda,\mu) = \prod_{B}(-1)^{k_B-1}(k_B-1)!$
• Coarsening is partitioning the blocks, so a diagram with $k$ blocks has $\mathrm{Bell}(k)$ 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 $x_\lambda$ exactly when $\lambda$ has more blocks than $\delta$ — a statement with no clean form in the diagram basis

Examples

Basic #

merging four points into one block: $(-1)^3 3! $

live ↗

In$\mathrm{PartitionMobius}(\mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack2\bigr\rbrack, \bigl\lbrack-1\bigr\rbrack, \bigl\lbrack-2\bigr\rbrack\bigr\rbrack), \mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1, 2, -1, -2\bigr\rbrack\bigr\rbrack))$Out$-6$

Properties #

the coarsest partition has nothing above it

live ↗

In$\mathrm{InOrbitBasis}(\mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1, -1\bigr\rbrack\bigr\rbrack))$Out$\mathrm{OrbitDiagram}(\bigl\lbrack\bigl\lbrack1, -1\bigr\rbrack\bigr\rbrack)$

the interval of length zero

live ↗

In$\mathrm{PartitionMobius}(\mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1, -1\bigr\rbrack\bigr\rbrack), \mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1, -1\bigr\rbrack\bigr\rbrack))$Out$1$

About

Examples	3
Checked against	SageMath 2 agree
Crosswalk	Wikipedia ×2MathWorldWikidataEncyclopedia of Mathematics

See also: Diagram, PartitionAlgebra