Counting generators in type B Temperley-Lieb diagrams

In this talk, we present our results concerning Temperley–Lieb diagram algebras of types A and B, which have a basis indexed by the fully commutative elements in Coxeter groups of types A and B, respectively. In particular, we present a non-recursive method for enumerating the number of generators occurring in the fully commutative element that indexes a given diagram. One consequence of our results is a classification of the diagrams of the Temperley–Lieb algebras of types A and B indexed by cyclically fully commutative elements.

This talk was given by my undergraduate research students Sarah Otis and Leal Rivanis (Plymouth State University) on April 17, 2010 at the 2010 Hudson River Undergraduate Mathematics Conference at Keene State College.

Transcript

1. Counting generators in type B Temperley–Lieb diagrams
   Sarah Otis and Leal Rivanis
   Plymouth State University
   Department of Mathematics
   sotis1@plymouth.edu
   lsrivanis@plymouth.edu
   HRUMC 2010
   April 17, 2010
   S. Otis & L. Rivanis Counting generators in TL-diagrams 1 / 12
   

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2. Coxeter groups
   S. Otis & L. Rivanis Counting generators in TL-diagrams 2 / 12
   

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3. Coxeter groups
   Definition
   S. Otis & L. Rivanis Counting generators in TL-diagrams 2 / 12
   

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4. Coxeter groups
   Definition
   A Coxeter group consists of a group W
   S. Otis & L. Rivanis Counting generators in TL-diagrams 2 / 12
   

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5. Coxeter groups
   Definition
   A Coxeter group consists of a group W together with a generating set S consisting
   of elements of order 2
   S. Otis & L. Rivanis Counting generators in TL-diagrams 2 / 12
   

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6. Coxeter groups
   Definition
   A Coxeter group consists of a group W together with a generating set S consisting
   of elements of order 2 with presentation
   S. Otis & L. Rivanis Counting generators in TL-diagrams 2 / 12
   

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7. Coxeter groups
   Definition
   A Coxeter group consists of a group W together with a generating set S consisting
   of elements of order 2 with presentation
   W = S : s2 = e, (st)m(s,t) = e ,
   S. Otis & L. Rivanis Counting generators in TL-diagrams 2 / 12
   

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8. Coxeter groups
   Definition
   A Coxeter group consists of a group W together with a generating set S consisting
   of elements of order 2 with presentation
   W = S : s2 = e, (st)m(s,t) = e ,
   where m(s, t) ≥ 2 for s = t.
   S. Otis & L. Rivanis Counting generators in TL-diagrams 2 / 12
   

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9. Coxeter groups
   Definition
   A Coxeter group consists of a group W together with a generating set S consisting
   of elements of order 2 with presentation
   W = S : s2 = e, (st)m(s,t) = e ,
   where m(s, t) ≥ 2 for s = t.
   Comment
   S. Otis & L. Rivanis Counting generators in TL-diagrams 2 / 12
   

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10. Coxeter groups
    Definition
    A Coxeter group consists of a group W together with a generating set S consisting
    of elements of order 2 with presentation
    W = S : s2 = e, (st)m(s,t) = e ,
    where m(s, t) ≥ 2 for s = t.
    Comment
    Since s and t are elements of order 2, the relation (st)m(s,t) = e can be rewritten as
    S. Otis & L. Rivanis Counting generators in TL-diagrams 2 / 12
    

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11. Coxeter groups
    Definition
    A Coxeter group consists of a group W together with a generating set S consisting
    of elements of order 2 with presentation
    W = S : s2 = e, (st)m(s,t) = e ,
    where m(s, t) ≥ 2 for s = t.
    Comment
    Since s and t are elements of order 2, the relation (st)m(s,t) = e can be rewritten as
    m(s, t) = 2 =⇒
    S. Otis & L. Rivanis Counting generators in TL-diagrams 2 / 12
    

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12. Coxeter groups
    Definition
    A Coxeter group consists of a group W together with a generating set S consisting
    of elements of order 2 with presentation
    W = S : s2 = e, (st)m(s,t) = e ,
    where m(s, t) ≥ 2 for s = t.
    Comment
    Since s and t are elements of order 2, the relation (st)m(s,t) = e can be rewritten as
    m(s, t) = 2 =⇒ st = ts
    S. Otis & L. Rivanis Counting generators in TL-diagrams 2 / 12
    

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13. Coxeter groups
    Definition
    A Coxeter group consists of a group W together with a generating set S consisting
    of elements of order 2 with presentation
    W = S : s2 = e, (st)m(s,t) = e ,
    where m(s, t) ≥ 2 for s = t.
    Comment
    Since s and t are elements of order 2, the relation (st)m(s,t) = e can be rewritten as
    m(s, t) = 2 =⇒ st = ts short braid relations
    S. Otis & L. Rivanis Counting generators in TL-diagrams 2 / 12
    

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14. Coxeter groups
    Definition
    A Coxeter group consists of a group W together with a generating set S consisting
    of elements of order 2 with presentation
    W = S : s2 = e, (st)m(s,t) = e ,
    where m(s, t) ≥ 2 for s = t.
    Comment
    Since s and t are elements of order 2, the relation (st)m(s,t) = e can be rewritten as
    m(s, t) = 2 =⇒ st = ts short braid relations
    m(s, t) = 3 =⇒ sts = tst
    m(s, t) = 4 =⇒ stst = tsts
    .
    .
    .
    S. Otis & L. Rivanis Counting generators in TL-diagrams 2 / 12
    

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15. Coxeter groups
    Definition
    A Coxeter group consists of a group W together with a generating set S consisting
    of elements of order 2 with presentation
    W = S : s2 = e, (st)m(s,t) = e ,
    where m(s, t) ≥ 2 for s = t.
    Comment
    Since s and t are elements of order 2, the relation (st)m(s,t) = e can be rewritten as
    m(s, t) = 2 =⇒ st = ts short braid relations
    m(s, t) = 3 =⇒ sts = tst
    m(s, t) = 4 =⇒ stst = tsts
    .
    .
    .
    
    
    
    
    
    
    
    
    
    long braid relations
    S. Otis & L. Rivanis Counting generators in TL-diagrams 2 / 12
    

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16. Coxeter graphs
    S. Otis & L. Rivanis Counting generators in TL-diagrams 3 / 12
    

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17. Coxeter graphs
    Comment
    S. Otis & L. Rivanis Counting generators in TL-diagrams 3 / 12
    

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18. Coxeter graphs
    Comment
    Given a Coxeter group W , we can encode the defining relations in a Coxeter graph.
    S. Otis & L. Rivanis Counting generators in TL-diagrams 3 / 12
    

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19. Coxeter graphs
    Comment
    Given a Coxeter group W , we can encode the defining relations in a Coxeter graph.
    Example
    S. Otis & L. Rivanis Counting generators in TL-diagrams 3 / 12
    

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20. Coxeter graphs
    Comment
    Given a Coxeter group W , we can encode the defining relations in a Coxeter graph.
    Example
    The Coxeter group of type B3
    is defined to be the group generated by s1, s2, s3
    subject to:
    S. Otis & L. Rivanis Counting generators in TL-diagrams 3 / 12
    

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21. Coxeter graphs
    Comment
    Given a Coxeter group W , we can encode the defining relations in a Coxeter graph.
    Example
    The Coxeter group of type B3
    is defined to be the group generated by s1, s2, s3
    subject to:
    • s2
    i
    = e for i = 1, 2, 3
    S. Otis & L. Rivanis Counting generators in TL-diagrams 3 / 12
    

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22. Coxeter graphs
    Comment
    Given a Coxeter group W , we can encode the defining relations in a Coxeter graph.
    Example
    The Coxeter group of type B3
    is defined to be the group generated by s1, s2, s3
    subject to:
    • s2
    i
    = e for i = 1, 2, 3
    • s1
    s2
    s1
    s2
    = s2
    s1
    s2
    s1
    S. Otis & L. Rivanis Counting generators in TL-diagrams 3 / 12
    

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23. Coxeter graphs
    Comment
    Given a Coxeter group W , we can encode the defining relations in a Coxeter graph.
    Example
    The Coxeter group of type B3
    is defined to be the group generated by s1, s2, s3
    subject to:
    • s2
    i
    = e for i = 1, 2, 3
    • s1
    s2
    s1
    s2
    = s2
    s1
    s2
    s1
    • s2
    s3
    s2
    = s3
    s2
    s3
    S. Otis & L. Rivanis Counting generators in TL-diagrams 3 / 12
    

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24. Coxeter graphs
    Comment
    Given a Coxeter group W , we can encode the defining relations in a Coxeter graph.
    Example
    The Coxeter group of type B3
    is defined to be the group generated by s1, s2, s3
    subject to:
    • s2
    i
    = e for i = 1, 2, 3
    • s1
    s2
    s1
    s2
    = s2
    s1
    s2
    s1
    • s2
    s3
    s2
    = s3
    s2
    s3
    • s1
    s3
    = s3
    s1
    S. Otis & L. Rivanis Counting generators in TL-diagrams 3 / 12
    

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25. Coxeter graphs
    Comment
    Given a Coxeter group W , we can encode the defining relations in a Coxeter graph.
    Example
    The Coxeter group of type B3
    is defined to be the group generated by s1, s2, s3
    subject to:
    • s2
    i
    = e for i = 1, 2, 3
    • s1
    s2
    s1
    s2
    = s2
    s1
    s2
    s1
    • s2
    s3
    s2
    = s3
    s2
    s3
    • s1
    s3
    = s3
    s1
    The corresponding Coxeter graph is:
    S. Otis & L. Rivanis Counting generators in TL-diagrams 3 / 12
    

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26. Coxeter graphs
    Comment
    Given a Coxeter group W , we can encode the defining relations in a Coxeter graph.
    Example
    The Coxeter group of type B3
    is defined to be the group generated by s1, s2, s3
    subject to:
    • s2
    i
    = e for i = 1, 2, 3
    • s1
    s2
    s1
    s2
    = s2
    s1
    s2
    s1
    • s2
    s3
    s2
    = s3
    s2
    s3
    • s1
    s3
    = s3
    s1
    The corresponding Coxeter graph is:
    s1
    s2
    s3
    3
    4
    S. Otis & L. Rivanis Counting generators in TL-diagrams 3 / 12
    

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27. Coxeter groups of type Bn
    Comment
    S. Otis & L. Rivanis Counting generators in TL-diagrams 4 / 12
    

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28. Coxeter groups of type Bn
    Comment
    For brevity, we will no longer use si
    ,
    S. Otis & L. Rivanis Counting generators in TL-diagrams 4 / 12
    

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29. Coxeter groups of type Bn
    Comment
    For brevity, we will no longer use si
    , but instead we will just use i.
    S. Otis & L. Rivanis Counting generators in TL-diagrams 4 / 12
    

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30. Coxeter groups of type Bn
    Comment
    For brevity, we will no longer use si
    , but instead we will just use i. For example, we
    will write 1212 = 2121 in place of s1
    s2
    s1
    s2
    = s2
    s1
    s2
    s1
    .
    S. Otis & L. Rivanis Counting generators in TL-diagrams 4 / 12
    

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31. Coxeter groups of type Bn
    Comment
    For brevity, we will no longer use si
    , but instead we will just use i. For example, we
    will write 1212 = 2121 in place of s1
    s2
    s1
    s2
    = s2
    s1
    s2
    s1
    .
    Definition
    S. Otis & L. Rivanis Counting generators in TL-diagrams 4 / 12
    

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32. Coxeter groups of type Bn
    Comment
    For brevity, we will no longer use si
    , but instead we will just use i. For example, we
    will write 1212 = 2121 in place of s1
    s2
    s1
    s2
    = s2
    s1
    s2
    s1
    .
    Definition
    The Coxeter group of type Bn
    is determined by the following graph:
    S. Otis & L. Rivanis Counting generators in TL-diagrams 4 / 12
    

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33. Coxeter groups of type Bn
    Comment
    For brevity, we will no longer use si
    , but instead we will just use i. For example, we
    will write 1212 = 2121 in place of s1
    s2
    s1
    s2
    = s2
    s1
    s2
    s1
    .
    Definition
    The Coxeter group of type Bn
    is determined by the following graph:
    1 2 3 n − 1 n
    · · ·
    3 3
    Bn
    4
    S. Otis & L. Rivanis Counting generators in TL-diagrams 4 / 12
    

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34. Coxeter groups of type Bn
    Comment
    For brevity, we will no longer use si
    , but instead we will just use i. For example, we
    will write 1212 = 2121 in place of s1
    s2
    s1
    s2
    = s2
    s1
    s2
    s1
    .
    Definition
    The Coxeter group of type Bn
    is determined by the following graph:
    1 2 3 n − 1 n
    · · ·
    3 3
    Bn
    4
    Then W (Bn
    ) is subject to:
    S. Otis & L. Rivanis Counting generators in TL-diagrams 4 / 12
    

    View Slide

35. Coxeter groups of type Bn
    Comment
    For brevity, we will no longer use si
    , but instead we will just use i. For example, we
    will write 1212 = 2121 in place of s1
    s2
    s1
    s2
    = s2
    s1
    s2
    s1
    .
    Definition
    The Coxeter group of type Bn
    is determined by the following graph:
    1 2 3 n − 1 n
    · · ·
    3 3
    Bn
    4
    Then W (Bn
    ) is subject to:
    • 1212 = 2121,
    S. Otis & L. Rivanis Counting generators in TL-diagrams 4 / 12
    

    View Slide

36. Coxeter groups of type Bn
    Comment
    For brevity, we will no longer use si
    , but instead we will just use i. For example, we
    will write 1212 = 2121 in place of s1
    s2
    s1
    s2
    = s2
    s1
    s2
    s1
    .
    Definition
    The Coxeter group of type Bn
    is determined by the following graph:
    1 2 3 n − 1 n
    · · ·
    3 3
    Bn
    4
    Then W (Bn
    ) is subject to:
    • 1212 = 2121,
    • iji = jij, if |i − j| = 1 and i, j ≥ 2,
    S. Otis & L. Rivanis Counting generators in TL-diagrams 4 / 12
    

    View Slide

37. Coxeter groups of type Bn
    Comment
    For brevity, we will no longer use si
    , but instead we will just use i. For example, we
    will write 1212 = 2121 in place of s1
    s2
    s1
    s2
    = s2
    s1
    s2
    s1
    .
    Definition
    The Coxeter group of type Bn
    is determined by the following graph:
    1 2 3 n − 1 n
    · · ·
    3 3
    Bn
    4
    Then W (Bn
    ) is subject to:
    • 1212 = 2121,
    • iji = jij, if |i − j| = 1 and i, j ≥ 2,
    • ij = ji, if |i − j| ≥ 2,
    S. Otis & L. Rivanis Counting generators in TL-diagrams 4 / 12
    

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38. Coxeter groups of type Bn
    Comment
    For brevity, we will no longer use si
    , but instead we will just use i. For example, we
    will write 1212 = 2121 in place of s1
    s2
    s1
    s2
    = s2
    s1
    s2
    s1
    .
    Definition
    The Coxeter group of type Bn
    is determined by the following graph:
    1 2 3 n − 1 n
    · · ·
    3 3
    Bn
    4
    Then W (Bn
    ) is subject to:
    • 1212 = 2121,
    • iji = jij, if |i − j| = 1 and i, j ≥ 2,
    • ij = ji, if |i − j| ≥ 2,
    • i2 = e.
    S. Otis & L. Rivanis Counting generators in TL-diagrams 4 / 12
    

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39. Reduced expressions & Matsumoto’s theorem
    S. Otis & L. Rivanis Counting generators in TL-diagrams 5 / 12
    

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40. Reduced expressions & Matsumoto’s theorem
    Definition
    S. Otis & L. Rivanis Counting generators in TL-diagrams 5 / 12
    

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41. Reduced expressions & Matsumoto’s theorem
    Definition
    A “word” sx1
    sx2
    · · · sxm
    is called an expression for w ∈ W
    S. Otis & L. Rivanis Counting generators in TL-diagrams 5 / 12
    

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42. Reduced expressions & Matsumoto’s theorem
    Definition
    A “word” sx1
    sx2
    · · · sxm
    is called an expression for w ∈ W if it is equal to w when
    considered as a group element.
    S. Otis & L. Rivanis Counting generators in TL-diagrams 5 / 12
    

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43. Reduced expressions & Matsumoto’s theorem
    Definition
    A “word” sx1
    sx2
    · · · sxm
    is called an expression for w ∈ W if it is equal to w when
    considered as a group element.
    If m is minimal, it is a reduced expression.
    S. Otis & L. Rivanis Counting generators in TL-diagrams 5 / 12
    

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44. Reduced expressions & Matsumoto’s theorem
    Definition
    A “word” sx1
    sx2
    · · · sxm
    is called an expression for w ∈ W if it is equal to w when
    considered as a group element.
    If m is minimal, it is a reduced expression.
    Example
    S. Otis & L. Rivanis Counting generators in TL-diagrams 5 / 12
    

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45. Reduced expressions & Matsumoto’s theorem
    Definition
    A “word” sx1
    sx2
    · · · sxm
    is called an expression for w ∈ W if it is equal to w when
    considered as a group element.
    If m is minimal, it is a reduced expression.
    Example
    Let 132121 be an expression for w ∈ W (B3
    ).
    S. Otis & L. Rivanis Counting generators in TL-diagrams 5 / 12
    

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46. Reduced expressions & Matsumoto’s theorem
    Definition
    A “word” sx1
    sx2
    · · · sxm
    is called an expression for w ∈ W if it is equal to w when
    considered as a group element.
    If m is minimal, it is a reduced expression.
    Example
    Let 132121 be an expression for w ∈ W (B3
    ). We see that
    S. Otis & L. Rivanis Counting generators in TL-diagrams 5 / 12
    

    View Slide

47. Reduced expressions & Matsumoto’s theorem
    Definition
    A “word” sx1
    sx2
    · · · sxm
    is called an expression for w ∈ W if it is equal to w when
    considered as a group element.
    If m is minimal, it is a reduced expression.
    Example
    Let 132121 be an expression for w ∈ W (B3
    ). We see that
    132121 =
    S. Otis & L. Rivanis Counting generators in TL-diagrams 5 / 12
    

    View Slide

48. Reduced expressions & Matsumoto’s theorem
    Definition
    A “word” sx1
    sx2
    · · · sxm
    is called an expression for w ∈ W if it is equal to w when
    considered as a group element.
    If m is minimal, it is a reduced expression.
    Example
    Let 132121 be an expression for w ∈ W (B3
    ). We see that
    132121 = 131212 =
    S. Otis & L. Rivanis Counting generators in TL-diagrams 5 / 12
    

    View Slide

49. Reduced expressions & Matsumoto’s theorem
    Definition
    A “word” sx1
    sx2
    · · · sxm
    is called an expression for w ∈ W if it is equal to w when
    considered as a group element.
    If m is minimal, it is a reduced expression.
    Example
    Let 132121 be an expression for w ∈ W (B3
    ). We see that
    132121 = 131212 = 311212 =
    S. Otis & L. Rivanis Counting generators in TL-diagrams 5 / 12
    

    View Slide

50. Reduced expressions & Matsumoto’s theorem
    Definition
    A “word” sx1
    sx2
    · · · sxm
    is called an expression for w ∈ W if it is equal to w when
    considered as a group element.
    If m is minimal, it is a reduced expression.
    Example
    Let 132121 be an expression for w ∈ W (B3
    ). We see that
    132121 = 131212 = 311212 = 3212,
    S. Otis & L. Rivanis Counting generators in TL-diagrams 5 / 12
    

    View Slide

51. Reduced expressions & Matsumoto’s theorem
    Definition
    A “word” sx1
    sx2
    · · · sxm
    is called an expression for w ∈ W if it is equal to w when
    considered as a group element.
    If m is minimal, it is a reduced expression.
    Example
    Let 132121 be an expression for w ∈ W (B3
    ). We see that
    132121 = 131212 = 311212 = 3212,
    showing that the original expression is not reduced.
    S. Otis & L. Rivanis Counting generators in TL-diagrams 5 / 12
    

    View Slide

52. Reduced expressions & Matsumoto’s theorem
    Definition
    A “word” sx1
    sx2
    · · · sxm
    is called an expression for w ∈ W if it is equal to w when
    considered as a group element.
    If m is minimal, it is a reduced expression.
    Example
    Let 132121 be an expression for w ∈ W (B3
    ). We see that
    132121 = 131212 = 311212 = 3212,
    showing that the original expression is not reduced. However, the last expression on
    the right is reduced.
    S. Otis & L. Rivanis Counting generators in TL-diagrams 5 / 12
    

    View Slide

53. Reduced expressions & Matsumoto’s theorem
    Definition
    A “word” sx1
    sx2
    · · · sxm
    is called an expression for w ∈ W if it is equal to w when
    considered as a group element.
    If m is minimal, it is a reduced expression.
    Example
    Let 132121 be an expression for w ∈ W (B3
    ). We see that
    132121 = 131212 = 311212 = 3212,
    showing that the original expression is not reduced. However, the last expression on
    the right is reduced.
    Theorem (Matsumoto)
    S. Otis & L. Rivanis Counting generators in TL-diagrams 5 / 12
    

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54. Reduced expressions & Matsumoto’s theorem
    Definition
    A “word” sx1
    sx2
    · · · sxm
    is called an expression for w ∈ W if it is equal to w when
    considered as a group element.
    If m is minimal, it is a reduced expression.
    Example
    Let 132121 be an expression for w ∈ W (B3
    ). We see that
    132121 = 131212 = 311212 = 3212,
    showing that the original expression is not reduced. However, the last expression on
    the right is reduced.
    Theorem (Matsumoto)
    Any two reduced expressions for w ∈ W differ by a sequence of braid relations.
    S. Otis & L. Rivanis Counting generators in TL-diagrams 5 / 12
    

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55. Fully commutative elements
    S. Otis & L. Rivanis Counting generators in TL-diagrams 6 / 12
    

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56. Fully commutative elements
    Definition (Stembridge 1996)
    S. Otis & L. Rivanis Counting generators in TL-diagrams 6 / 12
    

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57. Fully commutative elements
    Definition (Stembridge 1996)
    Let w ∈ W . Then w is fully commutative (FC)
    S. Otis & L. Rivanis Counting generators in TL-diagrams 6 / 12
    

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58. Fully commutative elements
    Definition (Stembridge 1996)
    Let w ∈ W . Then w is fully commutative (FC) iff every reduced expression “avoids
    long braid relations.”
    S. Otis & L. Rivanis Counting generators in TL-diagrams 6 / 12
    

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59. Fully commutative elements
    Definition (Stembridge 1996)
    Let w ∈ W . Then w is fully commutative (FC) iff every reduced expression “avoids
    long braid relations.”
    Examples
    S. Otis & L. Rivanis Counting generators in TL-diagrams 6 / 12
    

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60. Fully commutative elements
    Definition (Stembridge 1996)
    Let w ∈ W . Then w is fully commutative (FC) iff every reduced expression “avoids
    long braid relations.”
    Examples
    • Let 12142 be an expression for w in B4
    .
    S. Otis & L. Rivanis Counting generators in TL-diagrams 6 / 12
    

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61. Fully commutative elements
    Definition (Stembridge 1996)
    Let w ∈ W . Then w is fully commutative (FC) iff every reduced expression “avoids
    long braid relations.”
    Examples
    • Let 12142 be an expression for w in B4
    . We see that
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62. Fully commutative elements
    Definition (Stembridge 1996)
    Let w ∈ W . Then w is fully commutative (FC) iff every reduced expression “avoids
    long braid relations.”
    Examples
    • Let 12142 be an expression for w in B4
    . We see that
    12142
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63. Fully commutative elements
    Definition (Stembridge 1996)
    Let w ∈ W . Then w is fully commutative (FC) iff every reduced expression “avoids
    long braid relations.”
    Examples
    • Let 12142 be an expression for w in B4
    . We see that
    12142 = 12124.
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64. Fully commutative elements
    Definition (Stembridge 1996)
    Let w ∈ W . Then w is fully commutative (FC) iff every reduced expression “avoids
    long braid relations.”
    Examples
    • Let 12142 be an expression for w in B4
    . We see that
    12142 = 12124.
    The element clearly contains a long braid and therefore is not FC.
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65. Fully commutative elements
    Definition (Stembridge 1996)
    Let w ∈ W . Then w is fully commutative (FC) iff every reduced expression “avoids
    long braid relations.”
    Examples
    • Let 12142 be an expression for w in B4
    . We see that
    12142 = 12124.
    The element clearly contains a long braid and therefore is not FC.
    • Let 121342 be a reduced expression for w in B4
    .
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66. Fully commutative elements
    Definition (Stembridge 1996)
    Let w ∈ W . Then w is fully commutative (FC) iff every reduced expression “avoids
    long braid relations.”
    Examples
    • Let 12142 be an expression for w in B4
    . We see that
    12142 = 12124.
    The element clearly contains a long braid and therefore is not FC.
    • Let 121342 be a reduced expression for w in B4
    . Since
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67. Fully commutative elements
    Definition (Stembridge 1996)
    Let w ∈ W . Then w is fully commutative (FC) iff every reduced expression “avoids
    long braid relations.”
    Examples
    • Let 12142 be an expression for w in B4
    . We see that
    12142 = 12124.
    The element clearly contains a long braid and therefore is not FC.
    • Let 121342 be a reduced expression for w in B4
    . Since
    121342, 121324, 123142, 123124, 123412
    is an exhaustive list of all possible reduced expressions and none of these contain
    long braids,
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68. Fully commutative elements
    Definition (Stembridge 1996)
    Let w ∈ W . Then w is fully commutative (FC) iff every reduced expression “avoids
    long braid relations.”
    Examples
    • Let 12142 be an expression for w in B4
    . We see that
    12142 = 12124.
    The element clearly contains a long braid and therefore is not FC.
    • Let 121342 be a reduced expression for w in B4
    . Since
    121342, 121324, 123142, 123124, 123412
    is an exhaustive list of all possible reduced expressions and none of these contain
    long braids, w is FC.
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69. Temperley–Lieb diagrams of type B
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70. Temperley–Lieb diagrams of type B
    We will introduce the collection of Temperley–Lieb diagrams of type Bn
    by way of
    example.
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71. Temperley–Lieb diagrams of type B
    We will introduce the collection of Temperley–Lieb diagrams of type Bn
    by way of
    example.
    Example
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72. Temperley–Lieb diagrams of type B
    We will introduce the collection of Temperley–Lieb diagrams of type Bn
    by way of
    example.
    Example
    The following figure depicts a Temperley–Lieb diagram of type B6
    .
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73. Temperley–Lieb diagrams of type B
    We will introduce the collection of Temperley–Lieb diagrams of type Bn
    by way of
    example.
    Example
    The following figure depicts a Temperley–Lieb diagram of type B6
    .
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74. Temperley–Lieb diagrams of type B
    We will introduce the collection of Temperley–Lieb diagrams of type Bn
    by way of
    example.
    Example
    The following figure depicts a Temperley–Lieb diagram of type B6
    .
    Comments
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75. Temperley–Lieb diagrams of type B
    We will introduce the collection of Temperley–Lieb diagrams of type Bn
    by way of
    example.
    Example
    The following figure depicts a Temperley–Lieb diagram of type B6
    .
    Comments
    • There is a one-to-one correspondence between the FC elements of type Bn
    and a
    certain collection of decorated (n + 1)-diagrams.
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76. Temperley–Lieb diagrams of type B
    We will introduce the collection of Temperley–Lieb diagrams of type Bn
    by way of
    example.
    Example
    The following figure depicts a Temperley–Lieb diagram of type B6
    .
    Comments
    • There is a one-to-one correspondence between the FC elements of type Bn
    and a
    certain collection of decorated (n + 1)-diagrams.
    • There is a combinatorial description of this collection of diagrams which we will
    not elaborate on.
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77. Generating diagrams
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78. Generating diagrams
    Consider the following Temperley–Lieb diagrams of type Bn
    .
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79. Generating diagrams
    Consider the following Temperley–Lieb diagrams of type Bn
    .
    1 2 3 4 n − 1 n n + 1
    · · · −→ s1
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80. Generating diagrams
    Consider the following Temperley–Lieb diagrams of type Bn
    .
    1 2 3 4 n − 1 n n + 1
    · · · −→ s1
    1 2 3 4 n − 1 n n + 1
    · · · −→ s2
    .
    .
    .
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81. Generating diagrams
    Consider the following Temperley–Lieb diagrams of type Bn
    .
    1 2 3 4 n − 1 n n + 1
    · · · −→ s1
    1 2 3 4 n − 1 n n + 1
    · · · −→ s2
    .
    .
    .
    1 2 3 4 n − 1 n n + 1
    · · · −→ sn
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82. Generating diagrams
    Consider the following Temperley–Lieb diagrams of type Bn
    .
    1 2 3 4 n − 1 n n + 1
    · · · −→ s1
    1 2 3 4 n − 1 n n + 1
    · · · −→ s2
    .
    .
    .
    1 2 3 4 n − 1 n n + 1
    · · · −→ sn
    Under this correspondence, we can create the diagrams that correspond to the FC
    elements by concatenating the diagrams and “pulling the strings tight.”
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83. Example
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84. Example
    Here is an example of a product of several “generator” diagrams that corresponds to
    the FC element 124132 ∈ W (B4
    ).
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85. Example
    Here is an example of a product of several “generator” diagrams that corresponds to
    the FC element 124132 ∈ W (B4
    ).
    =
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86. Current results
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87. Current results
    One of our current results deals with the problem of counting the number of
    occurrences of a given generator in a diagram corresponding to an unknown FC
    element in a Coxeter group of type Bn
    .
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88. Current results
    One of our current results deals with the problem of counting the number of
    occurrences of a given generator in a diagram corresponding to an unknown FC
    element in a Coxeter group of type Bn
    .
    Given a diagram d, embed d in the plane so that the lower left hand corner is at the
    origin and node i in the south face sits at (i, 0).
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89. Current results
    One of our current results deals with the problem of counting the number of
    occurrences of a given generator in a diagram corresponding to an unknown FC
    element in a Coxeter group of type Bn
    .
    Given a diagram d, embed d in the plane so that the lower left hand corner is at the
    origin and node i in the south face sits at (i, 0). Also, let i
    be the vertical line
    x = i + 1
    2
    .
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90. Current results
    One of our current results deals with the problem of counting the number of
    occurrences of a given generator in a diagram corresponding to an unknown FC
    element in a Coxeter group of type Bn
    .
    Given a diagram d, embed d in the plane so that the lower left hand corner is at the
    origin and node i in the south face sits at (i, 0). Also, let i
    be the vertical line
    x = i + 1
    2
    .
    Theorem (Ernst, Otis, Rivanis)
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91. Current results
    One of our current results deals with the problem of counting the number of
    occurrences of a given generator in a diagram corresponding to an unknown FC
    element in a Coxeter group of type Bn
    .
    Given a diagram d, embed d in the plane so that the lower left hand corner is at the
    origin and node i in the south face sits at (i, 0). Also, let i
    be the vertical line
    x = i + 1
    2
    .
    Theorem (Ernst, Otis, Rivanis)
    Let d be a diagram corresponding to an unknown FC element w in a Coxeter group
    of type Bn
    .
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92. Current results
    One of our current results deals with the problem of counting the number of
    occurrences of a given generator in a diagram corresponding to an unknown FC
    element in a Coxeter group of type Bn
    .
    Given a diagram d, embed d in the plane so that the lower left hand corner is at the
    origin and node i in the south face sits at (i, 0). Also, let i
    be the vertical line
    x = i + 1
    2
    .
    Theorem (Ernst, Otis, Rivanis)
    Let d be a diagram corresponding to an unknown FC element w in a Coxeter group
    of type Bn
    . Then the number of occurrences of si
    , #(si
    ), in any reduced expression
    for w is given by
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93. Current results
    One of our current results deals with the problem of counting the number of
    occurrences of a given generator in a diagram corresponding to an unknown FC
    element in a Coxeter group of type Bn
    .
    Given a diagram d, embed d in the plane so that the lower left hand corner is at the
    origin and node i in the south face sits at (i, 0). Also, let i
    be the vertical line
    x = i + 1
    2
    .
    Theorem (Ernst, Otis, Rivanis)
    Let d be a diagram corresponding to an unknown FC element w in a Coxeter group
    of type Bn
    . Then the number of occurrences of si
    , #(si
    ), in any reduced expression
    for w is given by
    (# intersections with i
    ) + (# beads on non-intersected edges to the right of i
    )
    2
    .
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94. Current results
    One of our current results deals with the problem of counting the number of
    occurrences of a given generator in a diagram corresponding to an unknown FC
    element in a Coxeter group of type Bn
    .
    Given a diagram d, embed d in the plane so that the lower left hand corner is at the
    origin and node i in the south face sits at (i, 0). Also, let i
    be the vertical line
    x = i + 1
    2
    .
    Theorem (Ernst, Otis, Rivanis)
    Let d be a diagram corresponding to an unknown FC element w in a Coxeter group
    of type Bn
    . Then the number of occurrences of si
    , #(si
    ), in any reduced expression
    for w is given by
    (# intersections with i
    ) + (# beads on non-intersected edges to the right of i
    )
    2
    .
    Let’s see an example.
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95. Example of main result
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96. Example of main result
    Example
    Consider the following diagram that corresponds to some FC element in type B7
    .
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97. Example of main result
    Example
    Consider the following diagram that corresponds to some FC element in type B7
    .
    Consider the generator s1
    :
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98. Example of main result
    Example
    Consider the following diagram that corresponds to some FC element in type B7
    .
    Consider the generator s1
    :
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99. Example of main result
    Example
    Consider the following diagram that corresponds to some FC element in type B7
    .
    Consider the generator s1
    :
    • Number of edges intersected with 1
    :
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100. Example of main result
     Example
     Consider the following diagram that corresponds to some FC element in type B7
     .
     Consider the generator s1
     :
     • Number of edges intersected with 1
     : 2
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101. Example of main result
     Example
     Consider the following diagram that corresponds to some FC element in type B7
     .
     Consider the generator s1
     :
     • Number of edges intersected with 1
     : 2
     • Number of beads on non-intersected edges to right of 1
     :
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102. Example of main result
     Example
     Consider the following diagram that corresponds to some FC element in type B7
     .
     Consider the generator s1
     :
     • Number of edges intersected with 1
     : 2
     • Number of beads on non-intersected edges to right of 1
     : 2
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103. Example of main result
     Example
     Consider the following diagram that corresponds to some FC element in type B7
     .
     Consider the generator s1
     :
     • Number of edges intersected with 1
     : 2
     • Number of beads on non-intersected edges to right of 1
     : 2
     • Thus, #(s1
     ) =
     2 + 2
     2
     = 2
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104. Example of main result
     Example
     Consider the following diagram that corresponds to some FC element in type B7
     .
     Consider the generator s1
     :
     • Number of edges intersected with 1
     : 2
     • Number of beads on non-intersected edges to right of 1
     : 2
     • Thus, #(s1
     ) =
     2 + 2
     2
     = 2
     Similarly, we can conclude:
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105. Example of main result
     Example
     Consider the following diagram that corresponds to some FC element in type B7
     .
     Consider the generator s1
     :
     • Number of edges intersected with 1
     : 2
     • Number of beads on non-intersected edges to right of 1
     : 2
     • Thus, #(s1
     ) =
     2 + 2
     2
     = 2
     Similarly, we can conclude:
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106. Example of main result
     Example
     Consider the following diagram that corresponds to some FC element in type B7
     .
     Consider the generator s1
     :
     • Number of edges intersected with 1
     : 2
     • Number of beads on non-intersected edges to right of 1
     : 2
     • Thus, #(s1
     ) =
     2 + 2
     2
     = 2
     Similarly, we can conclude:
     #(s2
     ) = 2, #(s3
     ) = 3, #(s4
     ) = 2, #(s5
     ) = 2, #(s6
     ) = 1, #(s7
     ) = 1
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107. Example of main result (continued)
     Example (continued)
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108. Example of main result (continued)
     Example (continued)
     It turns out that the diagram on the previous slide corresponds to the FC element
     1357246135243 in a Coxeter group of type B7
     .
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109. Example of main result (continued)
     Example (continued)
     It turns out that the diagram on the previous slide corresponds to the FC element
     1357246135243 in a Coxeter group of type B7
     .
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