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. Then in 1987, V. Jones showed that the Temperley--Lieb Algebra occurs naturally as a quotient of the Hecke algebra arising from a Coxeter group of type A (whose underlying group is the symmetric group). This realization of the Temperley--Lieb Algebra as a Hecke algebra quotient was generalized to the case of an arbitrary Coxeter group. 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 efficient algorithm for obtaining a reduced factorization for a given diagram.

This talk was given by my undergraduate research students Michael Hastings and Sarah Salmon on November 22, 2013 at the Friday Afternoon Mathematics Undergraduate Seminar (FAMUS) at Northern Arizona University.

Transcript

1. A factorization of Temperley–Lieb diagrams Michael Hastings and Sarah Salmon Northern Arizona University Department of Mathematics and Statistics mgh64@nau.edu sks254@nau.edu FAMUS November 22, 2013 M. Hastings & S. Salmon A factorization of TL-diagrams 1 / 23
2. Admissible type A Temperley–Lieb diagrams An admissible 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. M. Hastings & S. Salmon A factorization of TL-diagrams 2 / 23
3. Type A Temperley–Lieb diagrams Example Here is an example of an admissible 5-diagram. Here is an example of an admissible 6-diagram. Here is an example that is not an admissible diagram. M. Hastings & S. Salmon A factorization of TL-diagrams 3 / 23
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 M. Hastings & S. Salmon A factorization of TL-diagrams 4 / 23
5. Type An simple diagrams We define n simple diagrams as follows: d1 = · · · 1 2 n n + 1 . . . di = · · · · · · 1 i i + 1 n + 1 . . . dn = · · · 1 2 n n + 1 M. Hastings & S. Salmon A factorization of TL-diagrams 5 / 23
6. 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 admissible diagrams in the Temperley–Lieb algebra of type An . M. Hastings & S. Salmon A factorization of TL-diagrams 6 / 23
7. Proof of one relation in type An Proof We see that for |i − j| = 1 (here j = i + 1) di dj di = · · · · · · · · · · · · · · · · · · · · · · · · i i + 1 i + 2 = · · · · · · = di M. Hastings & S. Salmon A factorization of TL-diagrams 7 / 23
8. Products of simple diagrams Example Consider the product d1 d3 d2 d4 d3 in type A4 . = M. Hastings & S. Salmon A factorization of TL-diagrams 8 / 23
9. 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 ). M. Hastings & S. Salmon A factorization of TL-diagrams 9 / 23
10. Factorization in type An We have discovered an algorithm to reconstruct the factorization given an admissible diagram. ←→ ←→ By our algorithm, the diagram equals d2 d4 d1 d3 d2 . M. Hastings & S. Salmon A factorization of TL-diagrams 10 / 23
11. Factorization in type An Let’s verify our calculation. d2 d4 d1 d3 d2 = = M. Hastings & S. Salmon A factorization of TL-diagrams 11 / 23
12. Admissible type B Temperley–Lieb diagrams An admissible diagram 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; • 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 M. Hastings & S. Salmon A factorization of TL-diagrams 12 / 23
13. Type B Temperley–Lieb diagrams In type Bn , there are slightly different simple diagrams which generate the admissible diagrams. We define n simple diagrams as follows: d1 = · · · 1 2 n n + 1 . . . di = 1 n + 1 · · · · · · i i + 1 . . . dn = · · · n n + 1 1 2 M. Hastings & S. Salmon A factorization of TL-diagrams 13 / 23
14. Type B Temperley–Lieb diagrams 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 admissible diagrams in the Temperley–Lieb algebra of type Bn . M. Hastings & S. Salmon A factorization of TL-diagrams 14 / 23
15. Proof of one relation in type Bn Proof For i = 1 and j = 2, d1 d2 d1 d2 = · · · · · · · · · · · · = 2 · · · = 2 · · · · · · = 2d1 d2 M. Hastings & S. Salmon A factorization of TL-diagrams 15 / 23
16. Admissible 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 = = M. Hastings & S. Salmon A factorization of TL-diagrams 16 / 23
17. Factorization of type Bn Example Let’s take the same diagram and work towards the factorization. ←→ ←→ Therefore the original diagram equals d1 d4 d2 d1 d3 d2 . M. Hastings & S. Salmon A factorization of TL-diagrams 17 / 23
18. Factorization of type Bn That matches our previous calculation (up to commutation). d1 d4 d2 d1 d3 d2 = = M. Hastings & S. Salmon A factorization of TL-diagrams 18 / 23
19. An exception Unlike type A, there is one exception to our algorithm. d2 d1 d2 = · · · · · · · · · = · · · M. Hastings & S. Salmon A factorization of TL-diagrams 19 / 23
20. 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 . M. Hastings & S. Salmon A factorization of TL-diagrams 20 / 23
21. Factorization of type Bn Example Let’s try a larger diagram. ←→ ←→ 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 . M. Hastings & S. Salmon A factorization of TL-diagrams 21 / 23
22. 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 = = M. Hastings & S. Salmon A factorization of TL-diagrams 22 / 23
23. Open Questions • There are other types to look at such as C and D. • Here is an example of C. • Will our algorithm work on these other types? • What type of special cases will we run into when checking if our algorithm works? M. Hastings & S. Salmon A factorization of TL-diagrams 23 / 23