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 poster was presented by my undergraduate research students Michael Hastings and Sarah Salmon (Northern Arizona University) on January 17, 2014 at the 2014 Joint Mathematics Meetings in Baltimore, MD and on April 25, 2014 at the 2014 NAU Undergraduate Symposium at Northern Arizona University in Flagstaff, AZ.

Transcript

1. A factorization of Temperley–Lieb diagrams Michael Hastings & Sarah Salmon, Directed by Dana C. Ernst Department of Mathematics & Statistics, Northern Arizona University Properties of diagrams in type An 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; There are no loops; The edges cannot cross; The edges cannot leave the box. The type A Temperely–Lieb diagram algebra TL(An ) is the Z[δ]-algebra having (n + 1)-diagrams as a basis. When multi- plying diagrams, it is possible to obtain a loop. In this case, we replace each loop with a coefficient δ. = δ3 Simple diagrams for type An We define n simple diagrams as follows: d1 = · · · 1 2 n n + 1 , di = 1 n + 1 · · · · · · i i + 1 Theorem TL(An ) satisfies the following: d2 i = δdi ; didj = djdi when |i − j| > 1; didjdi = di when |i − j| = 1. Theorem The set of simple diagrams generate all admissible diagrams in TL(An ). Theorem We have an efficient algorithm for obtaining a “canonical” factorization of any TL(An ) diagram. We will illustrate this algorithm via example. ←→ ←→ By our algorithm, the diagram equals d2d4 d1d3 d2 . Comments TL(An ) was discovered in 1971 by Temperley and Lieb as an algebra with abstract generators and a presentation with the stated relations. Penrose/Kauffman used a diagram algebra to model TL(An ) in 1971. In 1987, Vaughan Jones (Fields Medal in 1990) recognized that TL(An ) is isomorphic to a quotient of the Hecke algebra of type An−1 (the symmetric group, Sn ). In 1987, Vaughan Jones (Fields Medal in 1990) recognized that TL(An ) is isomorphic to a quotient of the Hecke algebra of type An (the symmetric group, Sn+1 ). Properties of diagrams in type Bn A diagram must satisfy the restrictions for type An 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 Simple diagrams for type Bn In type Bn , there is a slightly different set of simple diagrams which generate the admissible diagrams. 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 ; didj = djdi when |i − j| > 1; didjdi = di when |i − j| = 1 and i, j = 1; didjdidj = 2didj if {i, j} = {1, 2}. Theorem The set of simple diagrams generate all admissible diagrams in TL(Bn ). Example Here is an example of a product of several simple diagrams in type B4 . d1d4d2d1d3d2 = = Theorem We have an efficient algorithm for obtaining a “canonical” factorization of any TL(Bn )-diagram. We will illustrate this algorithm via example. ←→ ←→ Therefore the original diagram equals d1d4 d2 d1d3 d2 . This matches our previous calculation. Comments Unlike type A, there is one exception to our algorithm. d2d1d2 = · · · · · · · · · = · · · Example ←→ ←→ This diagram equals d2d5 d1d4 d2 . Example ←→ ←→ Therefore, the original diagram equals d1d4d8d10 d3d5d9 d2d4d6 d1d3d5d7 d2d4d6d8 d1d3d5 d2d4 . Open Problem Will our algorithm work on other types where diagrammatic representations are known to exist? For example, TL(Cn ): Email: Michael [mgh64@nau.edu], Sarah [sks254@nau.edu] Typeset using L A TEX, TikZ, and beamerposter