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# 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. January 03, 2014

## 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
.
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.
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 [[email protected]], Sarah [[email protected]] Typeset using L
A
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