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. Then in 1987, V. Jones showed that the Temperley--Lieb Algebra occurs naturally as a quotient of the Hecke algebra arising from a Coxeter group of type A (whose underlying group is the symmetric group). This realization of the Temperley--Lieb Algebra as a Hecke algebra quotient was generalized to the case of an arbitrary Coxeter group. 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 efficient algorithm for obtaining a reduced factorization for a given diagram.

This talk was given by my undergraduate research students Michael Hastings and Sarah Salmon on November 22, 2013 at the Friday Afternoon Mathematics Undergraduate Seminar (FAMUS) at Northern Arizona University.

Transcript

1. A factorization of Temperley–Lieb diagrams
   Michael Hastings and Sarah Salmon
   Northern Arizona University
   Department of Mathematics and Statistics
   mgh64@nau.edu
   sks254@nau.edu
   FAMUS
   November 22, 2013
   M. Hastings & S. Salmon A factorization of TL-diagrams 1 / 23
   

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2. Admissible type A Temperley–Lieb diagrams
   An admissible 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.
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3. Type A Temperley–Lieb diagrams
   Example
   Here is an example of an admissible 5-diagram.
   Here is an example of an admissible 6-diagram.
   Here is an example that is not an admissible diagram.
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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
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5. Type An simple diagrams
   We define n simple diagrams as follows:
   d1
   = · · ·
   1 2 n n + 1
   .
   .
   .
   di
   = · · · · · ·
   1 i i + 1 n + 1
   .
   .
   .
   dn
   = · · ·
   1 2 n n + 1
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6. 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 admissible diagrams in the Temperley–Lieb
   algebra of type An
   .
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7. Proof of one relation in type An
   Proof
   We see that for |i − j| = 1 (here j = i + 1)
   di
   dj
   di
   =
   · · · · · ·
   · · · · · ·
   · · · · · ·
   · · · · · ·
   i i + 1 i + 2
   = · · · · · ·
   = di
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8. Products of simple diagrams
   Example
   Consider the product d1
   d3
   d2
   d4
   d3
   in type A4
   .
   =
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9. 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
   ).
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10. Factorization in type An
    We have discovered an algorithm to reconstruct the factorization given an admissible
    diagram.
    ←→
    ←→
    By our algorithm, the diagram equals d2
    d4
    d1
    d3
    d2
    .
    M. Hastings & S. Salmon A factorization of TL-diagrams 10 / 23
    

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11. Factorization in type An
    Let’s verify our calculation.
    d2
    d4
    d1
    d3
    d2
    =
    =
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12. Admissible type B Temperley–Lieb diagrams
    An admissible diagram 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;
    • 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
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13. Type B Temperley–Lieb diagrams
    In type Bn
    , there are slightly different simple diagrams which generate the admissible
    diagrams. We define n simple diagrams as follows:
    d1
    = · · ·
    1 2 n n + 1
    .
    .
    .
    di
    =
    1 n + 1
    · · · · · ·
    i i + 1
    .
    .
    .
    dn
    = · · ·
    n n + 1
    1 2
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14. Type B Temperley–Lieb diagrams
    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 admissible diagrams in the Temperley–Lieb
    algebra of type Bn
    .
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15. Proof of one relation in type Bn
    Proof
    For i = 1 and j = 2,
    d1
    d2
    d1
    d2
    =
    · · ·
    · · ·
    · · ·
    · · ·
    = 2 · · ·
    = 2
    · · ·
    · · ·
    = 2d1
    d2
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16. Admissible 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
    =
    =
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17. Factorization of type Bn
    Example
    Let’s take the same diagram and work towards the factorization.
    ←→
    ←→
    Therefore the original diagram equals d1
    d4
    d2
    d1
    d3
    d2
    .
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18. Factorization of type Bn
    That matches our previous calculation (up to commutation).
    d1
    d4
    d2
    d1
    d3
    d2
    =
    =
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19. An exception
    Unlike type A, there is one exception to our algorithm.
    d2
    d1
    d2
    =
    · · ·
    · · ·
    · · ·
    = · · ·
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20. 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
    .
    M. Hastings & S. Salmon A factorization of TL-diagrams 20 / 23
    

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21. Factorization of type Bn
    Example
    Let’s try a larger diagram.
    ←→
    ←→
    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
    .
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22. 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
    = =
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23. Open Questions
    • There are other types to look at such as C and D.
    • Here is an example of C.
    • Will our algorithm work on these other types?
    • What type of special cases will we run into when checking if our algorithm
    works?
    M. Hastings & S. Salmon A factorization of TL-diagrams 23 / 23
    

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