BrauerAlgebra

$B_n(\delta)$ : the diagrams that are perfect matchings — every point paired with exactly one other. Dimension $(2n-1)!!$ .

BrauerAlgebra(n)the Brauer algebra on

n

strands

SageMathBrauerAlgebra(n, enumeratio_delta, enumeratio_ring)OscarEnumeratioDiagramAlgebra(:brauer, n)

Typed signatures

Type	Package	Applies
`(integer) -> diagram_algebra`	`diagram`	always

Domain: Diagram algebras

Details

Dimension $(2n-1)!!$ = 1, 3, 15, 105 — the perfect matchings of $2n$ points
Contains the symmetric group: a permutation diagram is a matching whose every block joins a top point to a bottom one
The generators satisfy $s_i^2 = 1$ , the braid relation, and $s_ie_i = e_i$
Restricting further to the PLANAR matchings gives TemperleyLiebAlgebra

Examples

Basic #

$5!!$

\mathrm{AlgebraDimension}(\mathrm{BrauerAlgebra}(3))

Out

15

\mathrm{AlgebraDimension}(\mathrm{BrauerAlgebra}(4))

Out

105

Properties #

a crossing is a Brauer diagram…

\mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1, -2\bigr\rbrack, \bigl\lbrack2, -1\bigr\rbrack, \bigl\lbrack3, -3\bigr\rbrack\bigr\rbrack)∈\mathrm{BrauerAlgebra}(3)

Out

\mathrm{True}

…but not a planar one

\mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1, -2\bigr\rbrack, \bigl\lbrack2, -1\bigr\rbrack, \bigl\lbrack3, -3\bigr\rbrack\bigr\rbrack)∈\mathrm{TemperleyLiebAlgebra}(3)

Out

\mathrm{False}

About

Examples	4
Checked against	SageMath 4 agreeOscar 2 agree
Crosswalk	Wikipedia WikidataSageMathOscar

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