A factorization of Temperley–Lieb diagrams Sarah Salmon Northern Arizona University Department of Mathematics and Statistics sks254@nau.edu Nebraska Conference for Undergraduate Women in Mathematics February 1, 2014 Joint work with Michael Hastings S. Salmon A factorization of TL-diagrams 1 / 19

Type A Temperley–Lieb diagrams An diagram in type An must satisfy the following requirements: • The diagram starts with a box with n + 1 nodes along the north face and n + 1 nodes along the south face. • Every node must be connected to exactly one other node by a single edge. • The edges cannot cross. • The edges cannot leave the box. Example Here is a 6-diagram. This is not a diagram. S. Salmon A factorization of TL-diagrams 2 / 19

The type A Temperley–Lieb diagram algebra TL(An ) is the Z[δ]-algebra having (n + 1)-diagrams as a basis. When multiplying diagrams, it is possible to obtain a loop. In this case, we replace each loop with a coefficient δ. S. Salmon A factorization of TL-diagrams 3 / 19

The type A Temperley–Lieb diagram algebra TL(An ) is the Z[δ]-algebra having (n + 1)-diagrams as a basis. When multiplying diagrams, it is possible to obtain a loop. In this case, we replace each loop with a coefficient δ. = δ S. Salmon A factorization of TL-diagrams 3 / 19

Type An simple diagrams We define n simple diagrams as follows: d1 = · · · 1 2 n n + 1 di = 1 n + 1 · · · · · · i i + 1 S. Salmon A factorization of TL-diagrams 4 / 19

Important relations in type An Theorem TL(An ) satisfies the following: S. Salmon A factorization of TL-diagrams 5 / 19

Important relations in type An Theorem TL(An ) satisfies the following: • d2 i = δdi ; S. Salmon A factorization of TL-diagrams 5 / 19

Important relations in type An Theorem TL(An ) satisfies the following: • d2 i = δdi ; • di dj = dj di when |i − j| > 1; S. Salmon A factorization of TL-diagrams 5 / 19

Important relations in type An Theorem TL(An ) satisfies the following: • d2 i = δdi ; • di dj = dj di when |i − j| > 1; • di dj di = di when |i − j| = 1. S. Salmon A factorization of TL-diagrams 5 / 19

Important relations in type An Theorem TL(An ) satisfies the following: • d2 i = δdi ; • di dj = dj di when |i − j| > 1; • di dj di = di when |i − j| = 1. Theorem The set of simple diagrams generate all diagrams in the Temperley–Lieb algebra of type An . S. Salmon A factorization of TL-diagrams 5 / 19

Products of simple diagrams Example Consider the product d1 d2 d1 d3 d2 d4 d3 in type A4 . = = S. Salmon A factorization of TL-diagrams 6 / 19

Historical context Comments • TL(An ) was discovered in 1971 by Temperley and Lieb as an algebra with abstract generators and a presentation with the relations above. • It first arose in the context of integrable Potts models in statistical mechanics. • As well as having applications in physics, TL(An ) appears in the framework of knot theory, braid groups, Coxeter groups and their corresponding Hecke algebras, and subfactors of von Neumann algebras. • Penrose/Kauffman used a diagram algebra to model TL(An ) in 1971. • In 1987, Vaughan Jones (awarded Fields Medal in 1990) recognized that TL(An ) is isomorphic to a particular quotient of the Hecke algebra of type An (the Coxeter group of type An is the symmetric group, Sn+1 ). S. Salmon A factorization of TL-diagrams 7 / 19

Factorization of type An We have discovered an algorithm to reconstruct the factorization given a diagram. S. Salmon A factorization of TL-diagrams 8 / 19

Factorization of type An We have discovered an algorithm to reconstruct the factorization given a diagram. ←→ S. Salmon A factorization of TL-diagrams 8 / 19

Factorization of type An We have discovered an algorithm to reconstruct the factorization given a diagram. ←→ S. Salmon A factorization of TL-diagrams 8 / 19

Factorization of type An We have discovered an algorithm to reconstruct the factorization given a diagram. ←→ S. Salmon A factorization of TL-diagrams 8 / 19

Factorization of type An We have discovered an algorithm to reconstruct the factorization given a diagram. ←→ S. Salmon A factorization of TL-diagrams 8 / 19

Factorization of type An We have discovered an algorithm to reconstruct the factorization given a diagram. ←→ S. Salmon A factorization of TL-diagrams 8 / 19

Factorization of type An We have discovered an algorithm to reconstruct the factorization given a diagram. ←→ By our algorithm, the diagram equals d4 d7 d3 d5 d8 d2 d6 d1 d7 . S. Salmon A factorization of TL-diagrams 8 / 19

Factorization of type An Let’s verify our calculation. d4 d7 d3 d5 d8 d2 d6 d1 d7 = = S. Salmon A factorization of TL-diagrams 9 / 19

Type B Temperley–Lieb diagrams A diagram must satisfy the restrictions for type A and the following: • All decorations must be exposed to the west face; • There are a few technical restrictions on what decorations can occur where; • All loops (decorated or not) are replaced with a coefficient, δ; • Decorations are restricted by the relations below. = = = 2 S. Salmon A factorization of TL-diagrams 10 / 19

Type Bn simple diagrams In type Bn , there are slightly different simple diagrams which generate the diagrams. We define n simple diagrams as follows: d1 = · · · 1 2 n n + 1 di = 1 n + 1 · · · · · · i i + 1 , (i ≥ 2) S. Salmon A factorization of TL-diagrams 11 / 19

Important relations in type Bn Theorem TL(Bn ) satisfies the following: S. Salmon A factorization of TL-diagrams 12 / 19

Important relations in type Bn Theorem TL(Bn ) satisfies the following: • d2 i = δdi ; S. Salmon A factorization of TL-diagrams 12 / 19

Important relations in type Bn Theorem TL(Bn ) satisfies the following: • d2 i = δdi ; • di dj = dj di when |i − j| > 1; S. Salmon A factorization of TL-diagrams 12 / 19

Important relations in type Bn Theorem TL(Bn ) satisfies the following: • d2 i = δdi ; • di dj = dj di when |i − j| > 1; • di dj di = di when |i − j| = 1 and i, j = 1; S. Salmon A factorization of TL-diagrams 12 / 19

Important relations in type Bn Theorem TL(Bn ) satisfies the following: • d2 i = δdi ; • di dj = dj di when |i − j| > 1; • di dj di = di when |i − j| = 1 and i, j = 1; • di dj di dj = 2di dj if {i, j} = {1, 2}. S. Salmon A factorization of TL-diagrams 12 / 19

Important relations in type Bn Theorem TL(Bn ) satisfies the following: • d2 i = δdi ; • di dj = dj di when |i − j| > 1; • di dj di = di when |i − j| = 1 and i, j = 1; • di dj di dj = 2di dj if {i, j} = {1, 2}. Theorem The set of simple diagrams generate all diagrams in the Temperley–Lieb algebra of type Bn . S. Salmon A factorization of TL-diagrams 12 / 19

Proof of one relation in type Bn Proof For i = 1 and j = 2, d1 d2 d1 d2 = · · · · · · · · · · · · = 2 · · · = 2 · · · · · · = 2d1 d2 S. Salmon A factorization of TL-diagrams 13 / 19

Product of simple diagrams in type Bn Example Here is an example of a product of several simple diagrams in type B4 . d1 d2 d4 d1 d3 d2 = = S. Salmon A factorization of TL-diagrams 14 / 19

Factorization of type Bn Example Using a diagram, let’s work towards the factorization. S. Salmon A factorization of TL-diagrams 15 / 19

Factorization of type Bn Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon A factorization of TL-diagrams 15 / 19

Factorization of type Bn Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon A factorization of TL-diagrams 15 / 19

Factorization of type Bn Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon A factorization of TL-diagrams 15 / 19

Factorization of type Bn Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon A factorization of TL-diagrams 15 / 19

Factorization of type Bn Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon A factorization of TL-diagrams 15 / 19

Factorization of type Bn Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon A factorization of TL-diagrams 15 / 19

Factorization of type Bn Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon A factorization of TL-diagrams 15 / 19

Factorization of type Bn Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon A factorization of TL-diagrams 15 / 19

Factorization of type Bn Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon A factorization of TL-diagrams 15 / 19

Factorization of type Bn Example Using a diagram, let’s work towards the factorization. ←→ Therefore, the original diagram equals d1 d4 d8 d10 d3 d5 d9 d2 d4 d6 d1 d3 d5 d7 d2 d4 d6 d8 d1 d3 d5 d2 d4 . S. Salmon A factorization of TL-diagrams 15 / 19

Factorization of type Bn Let’s check our calculation: d1 d4 d8 d10 d3 d5 d9 d2 d4 d6 d1 d3 d5 d7 d2 d4 d6 d8 d1 d3 d5 d2 d4 = = S. Salmon A factorization of TL-diagrams 16 / 19

An exception Unlike type A, there is one exception to our algorithm. d2 d1 d2 = · · · · · · · · · = · · · S. Salmon A factorization of TL-diagrams 17 / 19

An exception Example There is one case where we must slightly adjust how we factor the diagram. S. Salmon A factorization of TL-diagrams 18 / 19

An exception Example There is one case where we must slightly adjust how we factor the diagram. ←→ S. Salmon A factorization of TL-diagrams 18 / 19

An exception Example There is one case where we must slightly adjust how we factor the diagram. ←→ S. Salmon A factorization of TL-diagrams 18 / 19

An exception Example There is one case where we must slightly adjust how we factor the diagram. ←→ S. Salmon A factorization of TL-diagrams 18 / 19

An exception Example There is one case where we must slightly adjust how we factor the diagram. ←→ S. Salmon A factorization of TL-diagrams 18 / 19

An exception Example There is one case where we must slightly adjust how we factor the diagram. ←→ S. Salmon A factorization of TL-diagrams 18 / 19

An exception Example There is one case where we must slightly adjust how we factor the diagram. ←→ This diagram equals d2 d5 d1 d4 d2 . S. Salmon A factorization of TL-diagrams 18 / 19

Open Questions Will our algorithm work on other types where diagrammatic representations are known to exist? For example, TL(Cn ): S. Salmon A factorization of TL-diagrams 19 / 19