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.

Transcript

1. A factorization of Temperley–Lieb diagrams
   Sarah Salmon
   Northern Arizona University
   Department of Mathematics and Statistics
   sks254@nau.edu
   Nebraska Conference for Undergraduate Women in Mathematics
   February 1, 2014
   Joint work with Michael Hastings
   S. Salmon A factorization of TL-diagrams 1 / 19
   

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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
   

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

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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
   

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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
   

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

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

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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
   

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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
   

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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
    

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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
    

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12. Historical context
    Comments
    • 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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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

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

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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

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

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

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

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

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

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

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

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

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

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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