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