Dana Ernst
November 22, 2013
140

# 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.

## Dana Ernst

November 22, 2013

## Transcript

1. A factorization of Temperley–Lieb diagrams
Michael Hastings and Sarah Salmon
Northern Arizona University
Department of Mathematics and Statistics
[email protected]
[email protected]
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
• 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