the Masters' Thesis of S. Sundar

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where k ∈Æ∪{0} and 1 ≤ i1 < i2 < · · · ip ≤ n − 1 1 ≤ j1 < j2 < · · · jp ≤ n − 1 i1 ≥ j1,i2 ≥ j2, · · · ,ip ≥ jp Proof.The proof can be found in [Jon]. We prove this by induction on n. Clearly the result is true for n = 2. Now assume that any word in 1,e1,e2, · · · ,en−1 is of the required form. Let w be a word in 1,e1,e2, · · · ,en. If w does not contain en then we are done. So suppose that w contains en. Assertion. w = τ k w1enw2 where w1,w2 are words in 1,e1,e2, · · · ,en−1. w has the form v1envenv2 where v1,v2 are words in 1,e1,e2, · · · ,en and v is a word in 1,e1,e2, · · · ,en−1. If v does not contain en−1 then en commutes with v and hence w = v1venv2. If v contains en−1 then by induction hypothesis v = τ r u1en−1u2 where u1,u2 are words in 1,e1,e2, · · · ,en−2. Now w = τ r v1u1enen−1enu2v2 w = τ r+1 v1u1enu2v2 In any case w is τ l multiple of a word which has one en less. Repeating this process proves the assertion. Hence w = τ k w1enw2 where w1,w2 are words in 1,e1,e2, · · · ,en−1. By induction hypothesis w2 = τ l v2(en−1en−2 · · · ,ejp) where v2 is a word in 1,e1,e2, · · · ,en−2. ( The product (en−1en−2 · · · ejp) could be empty). Hence w = τ s w1v2(enen−1 · · · ejp) where w1v2 is a word in 1,e1,e2, · · · ,en−1 Hence by induction hypothesis, w = τ k (ei1 ei1−1 · · · ej1 )(ei2 ei2−1 · · · ej2 ) · · · (eipeip−1 · · · ejp) where k ∈Æ∪{0} and 1 ≤ i1 < i2 < · · · ip ≤ n − 1 i1 ≥ j1,i2 ≥ j2, · · · ,ip ≥ jp 3
Hence we have written w in the form needed with i ′ s increasing. Now consider such an expression which has the least length. Then we claim that j ′ s are also increasing. Let w = τ k (ei1 ei1−1 · · · ej1 )(ei2 ei2−1 · · · ej2 ) · · · (eipeip−1 · · · ejp) be such an expression. Suppose j1 ≥ j2. Then w = τ k (ei1 ei1−1 · · · ej1 )(ei2 ei2−1 · · · ej2 ) · · · (eipeip−1 · · · ejp) w = τ k (ei1 ei1−1 · · · ej1+1)(ei2 · · · ej1 ej1+1ej1 · · · ej2 ) · · · (eipeip−1 · · · ejp) w = τ k+1 (ei1 ei1−1 · · · ej2 )(ei2 ei2−1 · · · ej1+2) · · · (eipeip−1 · · · ejp) which has length decreased by one which is a contradiction. Hence j1 ∈Æ < j2. Similarly jr < jr+1. This completes the proof. � Now we consider the following combinatorial problem. Consider�⊂Ê. Consider paths on�. The only allowed moves are either up or right i.e. from (a,b) one can go to either (a + 1,b) or (a,b + 1). Proposition 1. The number of paths � � from (0,0) to (n,n) where n 1 2n which lie in the region y ≤ x is n+1 n . Let pn = 1 � � 2n n+1 n . Then pn satisfy the following recurrence p1 = 1 n� pn = pi−1pn−i,for n ≥ 2. i=1 For a proof,we refer to [GHJ]. � The relevance of proposition 1 in our context is as follows: Given (i1,i2, · · · ,ip) and (j1,j2, · · · ,jp) such that 1 ≤ i1 < i2 < · · · ip ≤ n−1, 1 ≤ j1 < j2 < · · · jp ≤ n−1, i1 ≥ j1,i2 ≥ j2, · · · ,ip ≥ jp one can associate the path from (0,0) to (n,n) given by (0,0) → (i1,0) → (i1,j1) → (i2,j1) → · · · (ip,jp) → (n,jp) → (n,n) This is clearly a bijection from the set of paths from (0,0) to (n,n) to the set of ordered pairs ((i1,i2, · · · ,ip),(j1,j2, · · · ,jp)) which satisfies the following condition. 1 ≤ i1 < i2 < · · · ip ≤ n−1, 1 ≤ j1 < j2 < · · · jp ≤ n−1, i1 ≥ j1,i2 ≥ j2, · · · ,ip ≥ jp 4
Page 1 and 2: THE TEMPERLEY-LIEB ALGEBRA A Thesis
Page 3 and 4: Acknowledements I thank V.S.Sunder
Page 5: Chapter 1 The Temperley-Lieb Algebr
Page 9 and 10: which is associative. Hom(m,n) × H
Page 11 and 12: a = b ⊗ 1 ⊗ c ⊗ d with b ∈
Page 13 and 14: 1.4 ⋆ structure on Dn(β) Definit
Page 15 and 16: Proposition 4. Let τ be a nonzero
Page 17 and 18: (1) 1 − fk = e1 ∨ e2 ∨ · ·
Page 19 and 20: such that j ≥ i+2. Now let u = (
Page 21 and 22: and let ti = ρ(ei). Let �t = ⎛
Page 23 and 24: We refer to [Jon] for this section.
Page 25 and 26: the invloution are that of A but th
Page 27 and 28: y proposition 10, it follows that T
Page 29 and 30: (1),(2) and (3) can be proved by in
Page 31 and 32: Then Y = ⎛ ⎞ 0 1 0 . 0 0 ⎜ 1
Page 33 and 34: For a ∈ A, define ||a|| := sup{||
Page 35 and 36: (1) Ak is a subalgebra of Tk. (2)
Page 37 and 38: ˜π(a,λ) = ˜π1(a) + λ(1 − p)

the Masters' Thesis of S. Sundar

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