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.

Transcript

1. Visualizing diagram factorizations in
   Temperley–Lieb algebras, Part 2
   Sarah Salmon
   Northern Arizona University
   Department of Mathematics and Statistics
   sks254@nau.edu
   Southwestern Undergraduate Mathematics Research Conference
   March 1, 2014
   Joint work with Michael Hastings
   S. Salmon Visualizing diagram factorizations in TL algebras, Part 2 1 / 19
   

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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
   

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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
   

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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
   

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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
   

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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
   

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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
   

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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
   

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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
   

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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