Dana Ernst
February 01, 2014
140

# Towards 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. 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 algorithm for counting the number of times each simple diagram appears in a reduced factorization for a given diagram.

This talk was given by my undergraduate research student Sarah Salmon (Northern Arizona University) on February 1, 2014 at the Nebraska Conference for Undergraduate Women in Mathematics. This joint work with Michael Hastings.

## Dana Ernst

February 01, 2014

## Transcript

1. A factorization of Temperley–Lieb diagrams
Sarah Salmon
Northern Arizona University
Department of Mathematics and Statistics
[email protected]
February 1, 2014
Joint work with Michael Hastings
S. Salmon A factorization of TL-diagrams 1 / 19

2. Type A Temperley–Lieb diagrams
An 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.
Example
Here is a 6-diagram. This is not a diagram.
S. Salmon A factorization of TL-diagrams 2 / 19

3. 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.
S. Salmon A factorization of TL-diagrams 3 / 19

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
S. Salmon A factorization of TL-diagrams 3 / 19

5. Type An simple diagrams
We deﬁne n simple diagrams as follows:
d1
= · · ·
1 2 n n + 1
di
=
1 n + 1
· · · · · ·
i i + 1
S. Salmon A factorization of TL-diagrams 4 / 19

6. Important relations in type An
Theorem
TL(An
) satisﬁes the following:
S. Salmon A factorization of TL-diagrams 5 / 19

7. Important relations in type An
Theorem
TL(An
) satisﬁes the following:
• d2
i
= δdi
;
S. Salmon A factorization of TL-diagrams 5 / 19

8. Important relations in type An
Theorem
TL(An
) satisﬁes the following:
• d2
i
= δdi
;
• di
dj
= dj
di
when |i − j| > 1;
S. Salmon A factorization of TL-diagrams 5 / 19

9. 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.
S. Salmon A factorization of TL-diagrams 5 / 19

10. 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 diagrams in the Temperley–Lieb algebra of
type An
.
S. Salmon A factorization of TL-diagrams 5 / 19

11. Products of simple diagrams
Example
Consider the product d1
d2
d1
d3
d2
d4
d3
in type A4
.
= =
S. Salmon A factorization of TL-diagrams 6 / 19

12. 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
).
S. Salmon A factorization of TL-diagrams 7 / 19

13. Factorization of type An
We have discovered an algorithm to reconstruct the factorization given a diagram.
S. Salmon A factorization of TL-diagrams 8 / 19

14. Factorization of type An
We have discovered an algorithm to reconstruct the factorization given a diagram.
←→
S. Salmon A factorization of TL-diagrams 8 / 19

15. Factorization of type An
We have discovered an algorithm to reconstruct the factorization given a diagram.
←→
S. Salmon A factorization of TL-diagrams 8 / 19

16. Factorization of type An
We have discovered an algorithm to reconstruct the factorization given a diagram.
←→
S. Salmon A factorization of TL-diagrams 8 / 19

17. Factorization of type An
We have discovered an algorithm to reconstruct the factorization given a diagram.
←→
S. Salmon A factorization of TL-diagrams 8 / 19

18. Factorization of type An
We have discovered an algorithm to reconstruct the factorization given a diagram.
←→
S. Salmon A factorization of TL-diagrams 8 / 19

19. Factorization of type An
We have discovered an algorithm to reconstruct the factorization given a diagram.
←→
By our algorithm, the diagram equals d4
d7
d3
d5
d8
d2
d6
d1
d7
.
S. Salmon A factorization of TL-diagrams 8 / 19

20. Factorization of type An
Let’s verify our calculation.
d4
d7
d3
d5
d8
d2
d6
d1
d7
=
=
S. Salmon A factorization of TL-diagrams 9 / 19

21. Type B Temperley–Lieb diagrams
A diagram must satisfy the restrictions for type A 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
S. Salmon A factorization of TL-diagrams 10 / 19

22. Type Bn simple diagrams
In type Bn
, there are slightly diﬀerent simple diagrams which generate the diagrams.
We deﬁne n simple diagrams as follows:
d1
= · · ·
1 2 n n + 1
di
=
1 n + 1
· · · · · ·
i i + 1
, (i ≥ 2)
S. Salmon A factorization of TL-diagrams 11 / 19

23. Important relations in type Bn
Theorem
TL(Bn
) satisﬁes the following:
S. Salmon A factorization of TL-diagrams 12 / 19

24. Important relations in type Bn
Theorem
TL(Bn
) satisﬁes the following:
• d2
i
= δdi
;
S. Salmon A factorization of TL-diagrams 12 / 19

25. Important relations in type Bn
Theorem
TL(Bn
) satisﬁes the following:
• d2
i
= δdi
;
• di
dj
= dj
di
when |i − j| > 1;
S. Salmon A factorization of TL-diagrams 12 / 19

26. Important relations in type Bn
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;
S. Salmon A factorization of TL-diagrams 12 / 19

27. Important relations in type Bn
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}.
S. Salmon A factorization of TL-diagrams 12 / 19

28. Important relations in type Bn
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 diagrams in the Temperley–Lieb algebra of
type Bn
.
S. Salmon A factorization of TL-diagrams 12 / 19

29. Proof of one relation in type Bn
Proof
For i = 1 and j = 2,
d1
d2
d1
d2
=
· · ·
· · ·
· · ·
· · ·
= 2 · · ·
= 2
· · ·
· · ·
= 2d1
d2
S. Salmon A factorization of TL-diagrams 13 / 19

30. Product of simple 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
=
=
S. Salmon A factorization of TL-diagrams 14 / 19

31. Factorization of type Bn
Example
Using a diagram, let’s work towards the factorization.
S. Salmon A factorization of TL-diagrams 15 / 19

32. Factorization of type Bn
Example
Using a diagram, let’s work towards the factorization.
←→
S. Salmon A factorization of TL-diagrams 15 / 19

33. Factorization of type Bn
Example
Using a diagram, let’s work towards the factorization.
←→
S. Salmon A factorization of TL-diagrams 15 / 19

34. Factorization of type Bn
Example
Using a diagram, let’s work towards the factorization.
←→
S. Salmon A factorization of TL-diagrams 15 / 19

35. Factorization of type Bn
Example
Using a diagram, let’s work towards the factorization.
←→
S. Salmon A factorization of TL-diagrams 15 / 19

36. Factorization of type Bn
Example
Using a diagram, let’s work towards the factorization.
←→
S. Salmon A factorization of TL-diagrams 15 / 19

37. Factorization of type Bn
Example
Using a diagram, let’s work towards the factorization.
←→
S. Salmon A factorization of TL-diagrams 15 / 19

38. Factorization of type Bn
Example
Using a diagram, let’s work towards the factorization.
←→
S. Salmon A factorization of TL-diagrams 15 / 19

39. Factorization of type Bn
Example
Using a diagram, let’s work towards the factorization.
←→
S. Salmon A factorization of TL-diagrams 15 / 19

40. Factorization of type Bn
Example
Using a diagram, let’s work towards the factorization.
←→
S. Salmon A factorization of TL-diagrams 15 / 19

41. Factorization of type Bn
Example
Using a diagram, let’s work towards the factorization.
←→
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
.
S. Salmon A factorization of TL-diagrams 15 / 19

42. 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
= =
S. Salmon A factorization of TL-diagrams 16 / 19

43. An exception
Unlike type A, there is one exception to our algorithm.
d2
d1
d2
=
· · ·
· · ·
· · ·
= · · ·
S. Salmon A factorization of TL-diagrams 17 / 19

44. An exception
Example
There is one case where we must slightly adjust how we factor the diagram.
S. Salmon A factorization of TL-diagrams 18 / 19

45. An exception
Example
There is one case where we must slightly adjust how we factor the diagram.
←→
S. Salmon A factorization of TL-diagrams 18 / 19

46. An exception
Example
There is one case where we must slightly adjust how we factor the diagram.
←→
S. Salmon A factorization of TL-diagrams 18 / 19

47. An exception
Example
There is one case where we must slightly adjust how we factor the diagram.
←→
S. Salmon A factorization of TL-diagrams 18 / 19

48. An exception
Example
There is one case where we must slightly adjust how we factor the diagram.
←→
S. Salmon A factorization of TL-diagrams 18 / 19

49. An exception
Example
There is one case where we must slightly adjust how we factor the diagram.
←→
S. Salmon A factorization of TL-diagrams 18 / 19

50. 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
.
S. Salmon A factorization of TL-diagrams 18 / 19

51. Open Questions
Will our algorithm work on other types where diagrammatic representations are
known to exist? For example, TL(Cn
):
S. Salmon A factorization of TL-diagrams 19 / 19