WolframInstitute/Hypergraph | Paclet Repository

WolframInstitute`Hypergraph`

HypergraphInsertionBracket

HypergraphInsertionBracket [hg1,hg2,…]
	gives the graded antisymmetric insertion bracket of the hypergraphs hg1, hg2, …, as an association from resulting hypergraphs to integer coefficients.

HypergraphInsertionBracket [assoc1,hg2,…]
	extends the bracket linearly whenever an argument is given as an association <\|hg -> c, …\|>, representing a formal sum of hypergraphs.

Details and Options

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Inserting a hypergraph hg' at a vertex v of a hypergraph hg identifies v with the root of hg', which by convention is the first vertex in

VertexList

[hg'].

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For two hypergraphs, the bracket sums the insertions of hg2 into hg1 at every vertex of hg1, minus the insertions of hg1 into hg2 at every vertex of hg2, each term carrying a sign from the Koszul convention applied to the (graded) vertex being replaced.

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For n hypergraphs, the bracket sums consecutive insertions of all n arguments over every permutation of their order, each term signed by the permutation's parity together with the Koszul sign of the insertion.

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Terms are collected up to isomorphism: results that agree after canonicalization are merged by adding their coefficients, and terms whose coefficients cancel to 0 are dropped from the resulting association.

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Insertion at a vertex only happens when that vertex's "Degree" annotation (see

HypergraphInsertionBracketDegree

) matches the degree of the root of the inserted hypergraph; every vertex defaults to degree 0, so plain hypergraphs (with no degree annotations) insert at every vertex.

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For n = 2, the bracket satisfies the antisymmetry of a graded Lie bracket: swapping the two arguments negates the result.

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The following option can be given: