Towards a factorization of Temperley--Lieb diagrams

The Temperley--Lieb Algebra, invented by Temperley and Lieb in 1971, is a finite dimensional associative algebra that arose in the context of statistical mechanics. Later in 1971, R. Penrose showed that this algebra can be realized in terms of certain diagrams. In the cases when diagrammatic representations are known to exist, it turns out that every diagram can be written as a product of "simple diagrams." These factorizations correspond precisely to factorizations in the underlying group. Given a diagrammatic representation and a reduced factorization of a group element, it is easy to construct the corresponding diagram. However, given a diagram, it is generally difficult to reconstruct the factorization of the corresponding group element. In cases that include Temperley--Lieb algebras of types A and B, we have devised an algorithm for counting the number of times each simple diagram appears in a reduced factorization for a given diagram.

This talk was given by my undergraduate research student Sarah Salmon (Northern Arizona University) on February 1, 2014 at the Nebraska Conference for Undergraduate Women in Mathematics. This joint work with Michael Hastings.

Transcript

1. 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
2. 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
3. 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
4. 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
5. 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
6. Important relations in type An Theorem TL(An ) satisfies the following: S. Salmon A factorization of TL-diagrams 5 / 19
7. Important relations in type An Theorem TL(An ) satisfies the following: • d2 i = δdi ; S. Salmon A factorization of TL-diagrams 5 / 19
8. 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
9. 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
10. 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
11. 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
12. 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
13. 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
14. 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
15. 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
16. 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
17. 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
18. 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
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
20. 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
21. 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
22. 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
23. Important relations in type Bn Theorem TL(Bn ) satisfies the following: S. Salmon A factorization of TL-diagrams 12 / 19
24. Important relations in type Bn Theorem TL(Bn ) satisfies the following: • d2 i = δdi ; S. Salmon A factorization of TL-diagrams 12 / 19
25. 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
26. 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
27. 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
28. 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
29. 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
30. 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
31. Factorization of type Bn Example Using a diagram, let’s work towards the factorization. S. Salmon A factorization of TL-diagrams 15 / 19
32. Factorization of type Bn Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon A factorization of TL-diagrams 15 / 19
33. Factorization of type Bn Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon A factorization of TL-diagrams 15 / 19
34. Factorization of type Bn Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon A factorization of TL-diagrams 15 / 19
35. Factorization of type Bn Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon A factorization of TL-diagrams 15 / 19
36. Factorization of type Bn Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon A factorization of TL-diagrams 15 / 19
37. Factorization of type Bn Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon A factorization of TL-diagrams 15 / 19
38. Factorization of type Bn Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon A factorization of TL-diagrams 15 / 19
39. Factorization of type Bn Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon A factorization of TL-diagrams 15 / 19
40. Factorization of type Bn Example Using a diagram, let’s work towards the factorization. ←→ S. Salmon A factorization of TL-diagrams 15 / 19
41. 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
42. 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
43. An exception Unlike type A, there is one exception to our algorithm. d2 d1 d2 = · · · · · · · · · = · · · S. Salmon A factorization of TL-diagrams 17 / 19
44. 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
45. 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
46. 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
47. 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
48. 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
49. 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
50. 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
51. 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