Visualizing diagram factorizations in Temperley-Lieb algebras, Part 1

The Temperley-Lieb Algebra of type A, 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 this algebra occurs naturally as a quotient of an algebra whose underlying structure is the symmetric group. Given an expression of a symmetric group element, it is easy to construct the corresponding diagram. However, given a diagram, it is generally difficult to reconstruct a factorization of the corresponding permutation. We have devised an efficient and visually appealing algorithm for obtaining a reduced factorization for a given diagram. This talk will be an introduction to Temperley-Lieb diagrams and an explanation of our algorithm. An extension involving diagrams of type B will be addressed in "Visualizing diagram factorizations in Temperley-Lieb algebras, Part 2” by Sarah Salmon (NAU).

This talk was given by my undergraduate research student Michael Hastings (Northern Arizona University) on March 1, 2014 at the Southwestern Undergraduate Mathematics Research Conference (SUnMaRC) at Mesa Community College, Mesa, AZ. This joint work with Sarah Salmon.

Transcript

1. A factorization of Temperley–Lieb diagrams, Part 1 Michael Hastings Northern Arizona University Department of Mathematics and Statistics mgh64@nau.edu SUnMaRC March, 2014 Joint work with Sarah Salmon M. Hastings A factorization of TL-diagrams, Part 1 1 / 14
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 A factorization of TL-diagrams, Part 1 2 / 14
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 A factorization of TL-diagrams, Part 1 3 / 14
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 A factorization of TL-diagrams, Part 1 4 / 14
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 A factorization of TL-diagrams, Part 1 5 / 14
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 A factorization of TL-diagrams, Part 1 6 / 14
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 A factorization of TL-diagrams, Part 1 7 / 14
8. Products of simple diagrams Example Consider the product d1 d3 d2 d4 d3 in type A4 . = M. Hastings A factorization of TL-diagrams, Part 1 8 / 14
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 A factorization of TL-diagrams, Part 1 9 / 14
10. Factorization in type An We have discovered an algorithm to reconstruct the factorization given an admissible diagram. ←→ 1 1 2 2 3 ←→ By our algorithm, the diagram equals d2 d4 d1 d3 d2 . M. Hastings A factorization of TL-diagrams, Part 1 10 / 14
11. Factorization in type An Let’s verify our calculation. d2 d4 d1 d3 d2 = 1 2 3 = M. Hastings A factorization of TL-diagrams, Part 1 11 / 14
12. Factorization in type An An example of our algorithm on a more difficult diagram. 1 1 1 2 2 2 2 2 3 3 3 4 4 5 5 6 6 7 By our algorithm, the diagram equals d2 d6 d10 d1 d3 d5 d9 d11 d4 d8 d10 d7 d9 d6 d8 d5 d7 d6 . M. Hastings A factorization of TL-diagrams, Part 1 12 / 14
13. Factorization in type An Let’s verify our calculation: d2 d6 d10 d1 d3 d5 d9 d11 d4 d8 d10 d7 d9 d6 d8 d5 d7 d6 = 1 2 3 4 5 6 7 = M. Hastings A factorization of TL-diagrams, Part 1 13 / 14
14. A taste of type Bn Example ←→ M. Hastings A factorization of TL-diagrams, Part 1 14 / 14