Visualizing diagram factorizations in Temperley–Lieb algebras Sarah Salmon Northern Arizona University Department of Mathematics and Statistics sks254@nau.edu Arizona–Nevada Academy of Science Annual Meeting April 12, 2014 Directed by Dr. Dana C. Ernst Joint work with Michael Hastings S. Salmon Visualizing diagram factorizations in TL algebras 1 / 20

Type A Temperley–Lieb diagrams A diagram of 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 Visualizing diagram factorizations in TL algebras 2 / 20

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. = δ2 S. Salmon Visualizing diagram factorizations in TL algebras 3 / 20

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 Visualizing diagram factorizations in TL algebras 4 / 20

Type A diagrams Example For TL(A3 ), we have the following set of 4-diagrams as a basis: S. Salmon Visualizing diagram factorizations in TL algebras 5 / 20

Type A diagrams Example For TL(A3 ), we have the following set of 4-diagrams as a basis: S. Salmon Visualizing diagram factorizations in TL algebras 5 / 20

Type A generators and relations For TL(An ), we define n simple diagrams as follows: di = 1 n + 1 · · · · · · i i + 1 S. Salmon Visualizing diagram factorizations in TL algebras 6 / 20

Type A generators and relations For TL(An ), we define n simple diagrams as follows: di = 1 n + 1 · · · · · · i i + 1 Theorem TL(An ) satisfies the following: S. Salmon Visualizing diagram factorizations in TL algebras 6 / 20

Type A generators and relations For TL(An ), we define n simple diagrams as follows: di = 1 n + 1 · · · · · · i i + 1 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 Visualizing diagram factorizations in TL algebras 6 / 20

Type A generators and relations For TL(An ), we define n simple diagrams as follows: di = 1 n + 1 · · · · · · i i + 1 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 Visualizing diagram factorizations in TL algebras 6 / 20

Products of simple diagrams Example Consider the product d1 d2 d1 d3 d2 d4 d3 in TL(A4 ). = = Note that d1 d2 d1 d3 d2 d4 d3 = d1 d3 d2 d4 d3. S. Salmon Visualizing diagram factorizations in TL algebras 7 / 20

Products of simple diagrams Example Consider the product d1 d2 d1 d3 d2 d4 d3 in TL(A4 ). = = Note that d1 d2 d1 d3 d2 d4 d3 = d1 d3 d2 d4 d3. S. Salmon Visualizing diagram factorizations in TL algebras 7 / 20

Factorization of type A We have discovered an algorithm to determine the factorization given a diagram. S. Salmon Visualizing diagram factorizations in TL algebras 8 / 20

Factorization of type A We have discovered an algorithm to determine the factorization given a diagram. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 8 / 20

Factorization of type A We have discovered an algorithm to determine the factorization given a diagram. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 8 / 20

Factorization of type A We have discovered an algorithm to determine the factorization given a diagram. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 8 / 20

Factorization of type A We have discovered an algorithm to determine the factorization given a diagram. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 8 / 20

Factorization of type A We have discovered an algorithm to determine the factorization given a diagram. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 8 / 20

Factorization of type A We have discovered an algorithm to determine the factorization given a diagram. ←→ By our algorithm, the diagram equals d4 d7 d3 d5 d8 d2 d6 d1 d7 . S. Salmon Visualizing diagram factorizations in TL algebras 8 / 20

Factorization of type A Let’s verify our calculation. d4 d7 d3 d5 d8 d2 d6 d1 d7 = = S. Salmon Visualizing diagram factorizations in TL algebras 9 / 20

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. S. Salmon Visualizing diagram factorizations in TL algebras 10 / 20

Type B Temperley–Lieb diagrams In TL(Bn ), we multiply diagrams as in type A subject to the following relations: = = = 2 = 2 = 2 S. Salmon Visualizing diagram factorizations in TL algebras 11 / 20

Type B diagrams Example For TL(B3 ), we have the following set of 4-diagrams as a basis: S. Salmon Visualizing diagram factorizations in TL algebras 12 / 20

Type B diagrams Example For TL(B3 ), we have the following set of 4-diagrams as a basis: S. Salmon Visualizing diagram factorizations in TL algebras 12 / 20

Type B generators and relations For TL(Bn ), we define n simple diagrams as follows (i ≥ 2): d1 = · · · 1 2 n n + 1 , di = 1 n + 1 · · · · · · i i + 1 S. Salmon Visualizing diagram factorizations in TL algebras 13 / 20

Type B generators and relations For TL(Bn ), we define n simple diagrams as follows (i ≥ 2): d1 = · · · 1 2 n n + 1 , di = 1 n + 1 · · · · · · i i + 1 Theorem TL(Bn ) satisfies the following: S. Salmon Visualizing diagram factorizations in TL algebras 13 / 20

Type B generators and relations For TL(Bn ), we define n simple diagrams as follows (i ≥ 2): d1 = · · · 1 2 n n + 1 , di = 1 n + 1 · · · · · · i i + 1 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 Visualizing diagram factorizations in TL algebras 13 / 20

Type B generators and relations For TL(Bn ), we define n simple diagrams as follows (i ≥ 2): d1 = · · · 1 2 n n + 1 , di = 1 n + 1 · · · · · · i i + 1 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 Visualizing diagram factorizations in TL algebras 13 / 20

Type B generators and relations For TL(Bn ), we define n simple diagrams as follows (i ≥ 2): d1 = · · · 1 2 n n + 1 , di = 1 n + 1 · · · · · · i i + 1 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 Visualizing diagram factorizations in TL algebras 13 / 20

Proof of last relation Proof For i = 1 and j = 2, d1 d2 d1 d2 = · · · · · · · · · · · · S. Salmon Visualizing diagram factorizations in TL algebras 14 / 20

Proof of last relation Proof For i = 1 and j = 2, d1 d2 d1 d2 = · · · · · · · · · · · · = 2 · · · S. Salmon Visualizing diagram factorizations in TL algebras 14 / 20

Proof of last relation Proof For i = 1 and j = 2, d1 d2 d1 d2 = · · · · · · · · · · · · = 2 · · · = 2 · · · · · · S. Salmon Visualizing diagram factorizations in TL algebras 14 / 20

Proof of last relation Proof For i = 1 and j = 2, d1 d2 d1 d2 = · · · · · · · · · · · · = 2 · · · = 2 · · · · · · = 2d1 d2 S. Salmon Visualizing diagram factorizations in TL algebras 14 / 20

Product of simple diagrams in type B Example Here is an example of a product of several simple diagrams in TL(B4 ). d1 d2 d4 d1 d3 d2 = = S. Salmon Visualizing diagram factorizations in TL algebras 15 / 20

Factorization of type B Example Using a diagram, let’s work towards the factorization. S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20

Factorization of type B Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20

Factorization of type B Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20

Factorization of type B Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20

Factorization of type B Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20

Factorization of type B Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20

Factorization of type B Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20

Factorization of type B Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20

Factorization of type B Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20

Factorization of type B Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 16 / 20

Factorization of type B 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 Visualizing diagram factorizations in TL algebras 16 / 20

Factorization of type B 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 Visualizing diagram factorizations in TL algebras 17 / 20

An exception Example There is one case where we must slightly adjust our algorithm. S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20

An exception Example There is one case where we must slightly adjust our algorithm. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20

An exception Example There is one case where we must slightly adjust our algorithm. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20

An exception Example There is one case where we must slightly adjust our algorithm. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20

An exception Example There is one case where we must slightly adjust our algorithm. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20

An exception Example There is one case where we must slightly adjust our algorithm. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20

An exception Example There is one case where we must slightly adjust our algorithm. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20

An exception Example There is one case where we must slightly adjust our algorithm. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20

An exception Example There is one case where we must slightly adjust our algorithm. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20

An exception Example There is one case where we must slightly adjust our algorithm. ←→ S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20

An exception Example There is one case where we must slightly adjust our algorithm. ←→ This diagram equals d5 d4 d6 d3 d2 d1 d2 d3 S. Salmon Visualizing diagram factorizations in TL algebras 18 / 20

An exception Here is the product of simple diagrams: d5 d4 d6 d3 d2 d1 d2 d3 = = S. Salmon Visualizing diagram factorizations in TL algebras 19 / 20

An exception Here is the product of simple diagrams: d5 d4 d6 d3 d2 d1 d2 d3 = = S. Salmon Visualizing diagram factorizations in TL algebras 19 / 20

Further work and acknowledgements Further work Will our algorithm work on other types where diagrammatic representations are known to exist? For example, TL(Cn ): Acknowledgements • Northern Arizona University • Department of Mathematics and Statistics • Dr. Dana Ernst S. Salmon Visualizing diagram factorizations in TL algebras 20 / 20