$30 off During Our Annual Pro Sale. View Details »

Towards a factorization of Temperley--Lieb diagrams

Dana Ernst
February 01, 2014

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
Tweet

More Decks by Dana Ernst

Other Decks in Research

Transcript

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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

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

    View Slide

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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

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

    View Slide

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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide