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
2. Coxeter groups S. Otis & L. Rivanis Counting generators in TL-diagrams 2 / 12
3. Coxeter groups Definition S. Otis & L. Rivanis Counting generators in TL-diagrams 2 / 12
4. Coxeter groups Definition A Coxeter group consists of a group W S. Otis & L. Rivanis Counting generators in TL-diagrams 2 / 12
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
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
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
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
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
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
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
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
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
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
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
16. Coxeter graphs S. Otis & L. Rivanis Counting generators in TL-diagrams 3 / 12
17. Coxeter graphs Comment S. Otis & L. Rivanis Counting generators in TL-diagrams 3 / 12
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
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
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
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
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
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
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
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
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
27. Coxeter groups of type Bn Comment S. Otis & L. Rivanis Counting generators in TL-diagrams 4 / 12
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
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
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
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
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
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
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
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
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
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
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
39. Reduced expressions & Matsumoto’s theorem S. Otis & L. Rivanis Counting generators in TL-diagrams 5 / 12
40. Reduced expressions & Matsumoto’s theorem Definition S. Otis & L. Rivanis Counting generators in TL-diagrams 5 / 12
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
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
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
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
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
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
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
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
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
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
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
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
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
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
55. Fully commutative elements S. Otis & L. Rivanis Counting generators in TL-diagrams 6 / 12
56. Fully commutative elements Definition (Stembridge 1996) S. Otis & L. Rivanis Counting generators in TL-diagrams 6 / 12
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
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
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
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
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 S. Otis & L. Rivanis Counting generators in TL-diagrams 6 / 12
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 S. Otis & L. Rivanis Counting generators in TL-diagrams 6 / 12
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. S. Otis & L. Rivanis Counting generators in TL-diagrams 6 / 12
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. S. Otis & L. Rivanis Counting generators in TL-diagrams 6 / 12
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 . S. Otis & L. Rivanis Counting generators in TL-diagrams 6 / 12
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 S. Otis & L. Rivanis Counting generators in TL-diagrams 6 / 12
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, S. Otis & L. Rivanis Counting generators in TL-diagrams 6 / 12
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. S. Otis & L. Rivanis Counting generators in TL-diagrams 6 / 12
69. Temperley–Lieb diagrams of type B S. Otis & L. Rivanis Counting generators in TL-diagrams 7 / 12
70. Temperley–Lieb diagrams of type B We will introduce the collection of Temperley–Lieb diagrams of type Bn by way of example. S. Otis & L. Rivanis Counting generators in TL-diagrams 7 / 12
71. Temperley–Lieb diagrams of type B We will introduce the collection of Temperley–Lieb diagrams of type Bn by way of example. Example S. Otis & L. Rivanis Counting generators in TL-diagrams 7 / 12
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 . S. Otis & L. Rivanis Counting generators in TL-diagrams 7 / 12
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 . S. Otis & L. Rivanis Counting generators in TL-diagrams 7 / 12
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 S. Otis & L. Rivanis Counting generators in TL-diagrams 7 / 12
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. S. Otis & L. Rivanis Counting generators in TL-diagrams 7 / 12
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. S. Otis & L. Rivanis Counting generators in TL-diagrams 7 / 12
77. Generating diagrams S. Otis & L. Rivanis Counting generators in TL-diagrams 8 / 12
78. Generating diagrams Consider the following Temperley–Lieb diagrams of type Bn . S. Otis & L. Rivanis Counting generators in TL-diagrams 8 / 12
79. Generating diagrams Consider the following Temperley–Lieb diagrams of type Bn . 1 2 3 4 n − 1 n n + 1 · · · −→ s1 S. Otis & L. Rivanis Counting generators in TL-diagrams 8 / 12
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 . . . S. Otis & L. Rivanis Counting generators in TL-diagrams 8 / 12
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 S. Otis & L. Rivanis Counting generators in TL-diagrams 8 / 12
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.” S. Otis & L. Rivanis Counting generators in TL-diagrams 8 / 12
83. Example S. Otis & L. Rivanis Counting generators in TL-diagrams 9 / 12
84. Example Here is an example of a product of several “generator” diagrams that corresponds to the FC element 124132 ∈ W (B4 ). S. Otis & L. Rivanis Counting generators in TL-diagrams 9 / 12
85. Example Here is an example of a product of several “generator” diagrams that corresponds to the FC element 124132 ∈ W (B4 ). = S. Otis & L. Rivanis Counting generators in TL-diagrams 9 / 12
86. Current results S. Otis & L. Rivanis Counting generators in TL-diagrams 10 / 12
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 . S. Otis & L. Rivanis Counting generators in TL-diagrams 10 / 12
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). S. Otis & L. Rivanis Counting generators in TL-diagrams 10 / 12
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 . S. Otis & L. Rivanis Counting generators in TL-diagrams 10 / 12
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) S. Otis & L. Rivanis Counting generators in TL-diagrams 10 / 12
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 . S. Otis & L. Rivanis Counting generators in TL-diagrams 10 / 12
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 S. Otis & L. Rivanis Counting generators in TL-diagrams 10 / 12
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 . S. Otis & L. Rivanis Counting generators in TL-diagrams 10 / 12
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. S. Otis & L. Rivanis Counting generators in TL-diagrams 10 / 12
95. Example of main result S. Otis & L. Rivanis Counting generators in TL-diagrams 11 / 12
96. Example of main result Example Consider the following diagram that corresponds to some FC element in type B7 . S. Otis & L. Rivanis Counting generators in TL-diagrams 11 / 12
97. Example of main result Example Consider the following diagram that corresponds to some FC element in type B7 . Consider the generator s1 : S. Otis & L. Rivanis Counting generators in TL-diagrams 11 / 12
98. Example of main result Example Consider the following diagram that corresponds to some FC element in type B7 . Consider the generator s1 : S. Otis & L. Rivanis Counting generators in TL-diagrams 11 / 12
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 : S. Otis & L. Rivanis Counting generators in TL-diagrams 11 / 12
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 S. Otis & L. Rivanis Counting generators in TL-diagrams 11 / 12
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 : S. Otis & L. Rivanis Counting generators in TL-diagrams 11 / 12
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 S. Otis & L. Rivanis Counting generators in TL-diagrams 11 / 12
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 S. Otis & L. Rivanis Counting generators in TL-diagrams 11 / 12
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: S. Otis & L. Rivanis Counting generators in TL-diagrams 11 / 12
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: S. Otis & L. Rivanis Counting generators in TL-diagrams 11 / 12
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 S. Otis & L. Rivanis Counting generators in TL-diagrams 11 / 12
107. Example of main result (continued) Example (continued) S. Otis & L. Rivanis Counting generators in TL-diagrams 12 / 12
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 . S. Otis & L. Rivanis Counting generators in TL-diagrams 12 / 12
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 . S. Otis & L. Rivanis Counting generators in TL-diagrams 12 / 12