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Counting generators in type B Temperley–Lieb diagrams Sarah Otis and Leal Rivanis Plymouth State University Department of Mathematics [email protected] [email protected] HRUMC 2010 April 17, 2010 S. Otis & L. Rivanis Counting generators in TL-diagrams 1 / 12

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

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

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

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

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

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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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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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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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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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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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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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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

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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

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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

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

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

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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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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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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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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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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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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

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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

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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

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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

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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

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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

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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

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

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

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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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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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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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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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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

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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

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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

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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

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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

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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

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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

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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

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Temperley–Lieb diagrams of type B S. Otis & L. Rivanis Counting generators in TL-diagrams 7 / 12

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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

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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

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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

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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

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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

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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

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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

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Generating diagrams S. Otis & L. Rivanis Counting generators in TL-diagrams 8 / 12

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Generating diagrams Consider the following Temperley–Lieb diagrams of type Bn . S. Otis & L. Rivanis Counting generators in TL-diagrams 8 / 12

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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

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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

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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

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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

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Example S. Otis & L. Rivanis Counting generators in TL-diagrams 9 / 12

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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

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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

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Current results S. Otis & L. Rivanis Counting generators in TL-diagrams 10 / 12

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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

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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

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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

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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

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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

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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

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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

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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

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Example of main result S. Otis & L. Rivanis Counting generators in TL-diagrams 11 / 12

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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Example of main result (continued) Example (continued) S. Otis & L. Rivanis Counting generators in TL-diagrams 12 / 12

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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

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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