Dana Ernst
March 01, 2014
130

# Visualizing diagram factorizations in Temperley–Lieb algebras, Part 2

This talk is a continued exploration and discussion of new results from "Visualizing diagram factorizations in Temperley--Lieb algebras, Part 1" by Michael Hastings (NAU). The diagrammatic representation is like that of type A (introduced in Part 1) with a relation decorated diagrams. Multiplying diagram is easy to do. However, taking a given diagram and finding the corresponding reduced factorization is generally difficult. We have an efficient (and colorful) algorithm for obtaining a reduced factorization for Temperley-Lieb diagrams of type B.

This talk was given by my undergraduate research student Sarah Salmon (Northern Arizona University) on March 1, 2014 at the Southwestern Undergraduate Mathematics Research Conference (SUnMaRC) at Mesa Community College, Mesa, AZ. This joint work with Michael Hastings.

March 01, 2014

## Transcript

1. Visualizing diagram factorizations in
Temperley–Lieb algebras, Part 2
Sarah Salmon
Northern Arizona University
Department of Mathematics and Statistics
[email protected]
March 1, 2014
Joint work with Michael Hastings
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 1 / 19

2. Type A Temperley–Lieb diagrams
A 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 Visualizing diagram factorizations in TL algebras, Part 2 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.
= δ2
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 3 / 19

4. The type A Temperley–Lieb diagram algebra
Here is another example of diagram multiplication.
= δ4
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 4 / 19

5. Type A diagrams
Example
For TL(A3
), we have the following set of 4-diagrams as a basis:
Theorem
In TL(An
), there are Catalan-many diagrams.
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 5 / 19

6. Type A diagrams
Example
For TL(A3
), we have the following set of 4-diagrams as a basis:
Theorem
In TL(An
), there are Catalan-many diagrams.
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 5 / 19

7. Products of simple diagrams
Example
Consider the product d1
d2
d1
d3
d2
d4
d3
in TL(A4
).
= =
Note that d1
d2
d1
d3
d2
d4
d3
= d1
d3
d2
d4
d3.
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 6 / 19

8. Products of simple diagrams
Example
Consider the product d1
d2
d1
d3
d2
d4
d3
in TL(A4
).
= =
Note that d1
d2
d1
d3
d2
d4
d3
= d1
d3
d2
d4
d3.
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 6 / 19

9. Factorization of type A
We have discovered an algorithm to determine the factorization given a diagram.
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 7 / 19

10. Factorization of type A
We have discovered an algorithm to determine the factorization given a diagram.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 7 / 19

11. Factorization of type A
We have discovered an algorithm to determine the factorization given a diagram.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 7 / 19

12. Factorization of type A
We have discovered an algorithm to determine the factorization given a diagram.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 7 / 19

13. Factorization of type A
We have discovered an algorithm to determine the factorization given a diagram.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 7 / 19

14. Factorization of type A
We have discovered an algorithm to determine the factorization given a diagram.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 7 / 19

15. Factorization of type A
We have discovered an algorithm to determine the factorization given a diagram.
←→
By our algorithm, the diagram equals d4
d7
d3
d5
d8
d2
d6
d1
d7
.
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 7 / 19

16. Factorization of type A
Let’s verify our calculation.
d4
d7
d3
d5
d8
d2
d6
d1
d7
=
=
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 8 / 19

17. 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.
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 9 / 19

18. Type B Temperley–Lieb diagrams
In TL(Bn
), we multiply diagrams as in type A subject to the following relations:
= = = 2 = 2
= 2
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 10 / 19

19. Type B diagrams
Example
For TL(B3
), we have the following set of 4-diagrams as a basis:
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 11 / 19

20. Type B diagrams
Example
For TL(B3
), we have the following set of 4-diagrams as a basis:
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 11 / 19

21. Type B generators and relations
For TL(Bn
), we deﬁne n simple diagrams as follows (i ≥ 2):
d1
= · · ·
1 2 n n + 1
, di
=
1 n + 1
· · · · · ·
i i + 1
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 12 / 19

22. Type B generators and relations
For TL(Bn
), 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:
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 12 / 19

23. Type B generators and relations
For TL(Bn
), 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
;
• di
dj
= dj
di
when |i − j| > 1;
• di
dj
di
= di
when |i − j| = 1 and i, j = 1;
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 12 / 19

24. Type B generators and relations
For TL(Bn
), 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
;
• 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 Visualizing diagram factorizations in TL algebras, Part 2 12 / 19

25. Type B generators and relations
For TL(Bn
), 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
;
• 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 Visualizing diagram factorizations in TL algebras, Part 2 12 / 19

26. Proof of last relation
Proof
For i = 1 and j = 2,
d1
d2
d1
d2
=
· · ·
· · ·
· · ·
· · ·
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 13 / 19

27. Proof of last relation
Proof
For i = 1 and j = 2,
d1
d2
d1
d2
=
· · ·
· · ·
· · ·
· · ·
= 2 · · ·
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 13 / 19

28. Proof of last relation
Proof
For i = 1 and j = 2,
d1
d2
d1
d2
=
· · ·
· · ·
· · ·
· · ·
= 2 · · ·
= 2
· · ·
· · ·
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 13 / 19

29. Proof of last relation
Proof
For i = 1 and j = 2,
d1
d2
d1
d2
=
· · ·
· · ·
· · ·
· · ·
= 2 · · ·
= 2
· · ·
· · ·
= 2d1
d2
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 13 / 19

30. Product of simple diagrams in type B
Example
Here is an example of a product of several simple diagrams in TL(B4
).
d1
d2
d4
d1
d3
d2
=
=
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 14 / 19

31. Factorization of type B
Example
Using a diagram, let’s work towards the factorization.
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 15 / 19

32. Factorization of type B
Example
Using a diagram, let’s work towards the factorization.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 15 / 19

33. Factorization of type B
Example
Using a diagram, let’s work towards the factorization.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 15 / 19

34. Factorization of type B
Example
Using a diagram, let’s work towards the factorization.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 15 / 19

35. Factorization of type B
Example
Using a diagram, let’s work towards the factorization.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 15 / 19

36. Factorization of type B
Example
Using a diagram, let’s work towards the factorization.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 15 / 19

37. Factorization of type B
Example
Using a diagram, let’s work towards the factorization.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 15 / 19

38. Factorization of type B
Example
Using a diagram, let’s work towards the factorization.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 15 / 19

39. Factorization of type B
Example
Using a diagram, let’s work towards the factorization.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 15 / 19

40. Factorization of type B
Example
Using a diagram, let’s work towards the factorization.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 15 / 19

41. Factorization of type B
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 Visualizing diagram factorizations in TL algebras, Part 2 15 / 19

42. Factorization of type B
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 Visualizing diagram factorizations in TL algebras, Part 2 16 / 19

43. An exception
Example
There is one case where we must slightly adjust our algorithm.
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 17 / 19

44. An exception
Example
There is one case where we must slightly adjust our algorithm.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 17 / 19

45. An exception
Example
There is one case where we must slightly adjust our algorithm.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 17 / 19

46. An exception
Example
There is one case where we must slightly adjust our algorithm.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 17 / 19

47. An exception
Example
There is one case where we must slightly adjust our algorithm.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 17 / 19

48. An exception
Example
There is one case where we must slightly adjust our algorithm.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 17 / 19

49. An exception
Example
There is one case where we must slightly adjust our algorithm.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 17 / 19

50. An exception
Example
There is one case where we must slightly adjust our algorithm.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 17 / 19

51. An exception
Example
There is one case where we must slightly adjust our algorithm.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 17 / 19

52. An exception
Example
There is one case where we must slightly adjust our algorithm.
←→
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 17 / 19

53. An exception
Example
There is one case where we must slightly adjust our algorithm.
←→
This diagram equals d5
d4
d6
d3
d2
d1
d2
d3
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 17 / 19

54. An exception
Here is the product of simple diagrams:
d5
d4
d6
d3
d2
d1
d2
d3
=
=
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 18 / 19

55. An exception
Here is the product of simple diagrams:
d5
d4
d6
d3
d2
d1
d2
d3
=
=
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 18 / 19

56. Further work and acknowledgements
Further work
Will our algorithm work on other types where diagrammatic representations are
known to exist? For example, TL(Cn
):
Acknowledgements
• Northern Arizona University
• Department of Mathematics and Statistics
• Dr. Dana Ernst
S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 19 / 19