Diagram

A diagram: a set partition of $2n$ points, a top row $1…n$ and a bottom row $1'…n'$, written as blocks of signed labels. The basis element of every algebra below.

Diagram([[1,-1],[2,-2]])blocks of signed labels — positive on the top row, negative on the bottom

Typed signatures

Type	Package	Applies
(list) -> number	diagram	always

Domain: Diagram algebras·Drawn as Graphics

Details

• Multiplication is geometric: stack $a$ above $b$, glue $a$'s bottom row to $b$'s top, keep what stays connected between the outer rows and discard the middle
• A block left entirely in the discarded middle was a closed loop, and contributes one factor of the loop parameter $\delta$ — so these algebras live over $\mathbb{Z}[\delta]$
• Use the ORDERED product (NonCommutativeMultiply, infix $\otimes$): a diagram algebra is not commutative
• Diagrams normalise, so a written diagram and a computed one are the same expression
• Every point $\pm 1 … \pm n$ must appear exactly once; a diagram missing one is malformed and is left as written

Examples

Basic #

the cup-cap squared closes one loop: $e_1^2 = \delta e_1$

live ↗

In$\mathrm{NonCommutativeMultiply}(\mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack-1, -2\bigr\rbrack\bigr\rbrack), \mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack-1, -2\bigr\rbrack\bigr\rbrack))$Out$δ\mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack-1, -2\bigr\rbrack\bigr\rbrack)$

Properties #

$e_1e_2e_1 = e_1$, with no loop closed

live ↗

In$\mathrm{NonCommutativeMultiply}(\mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack-1, -2\bigr\rbrack, \bigl\lbrack3, -3\bigr\rbrack\bigr\rbrack), \mathrm{Diagram}(\bigl\lbrack\bigl\lbrack2, 3\bigr\rbrack, \bigl\lbrack-2, -3\bigr\rbrack, \bigl\lbrack1, -1\bigr\rbrack\bigr\rbrack), \mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack-1, -2\bigr\rbrack, \bigl\lbrack3, -3\bigr\rbrack\bigr\rbrack))$Out$\mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, -3\bigr\rbrack, \bigl\lbrack-1, -2\bigr\rbrack\bigr\rbrack)$

blocks come back in canonical order

live ↗

In$\mathrm{Diagram}(\bigl\lbrack\bigl\lbrack-1, -2\bigr\rbrack, \bigl\lbrack3, -3\bigr\rbrack, \bigl\lbrack1, 2\bigr\rbrack\bigr\rbrack)$Out$\mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, -3\bigr\rbrack, \bigl\lbrack-1, -2\bigr\rbrack\bigr\rbrack)$

Possible issues #

malformed — the point $-2$ is missing — so it is left alone

live ↗

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

About

Examples	4
Checked against	SageMath 3 agree1 differ
Crosswalk	SageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts

See also: TemperleyLiebAlgebra, PartitionAlgebra, NonCommutativeMultiply