20 1. TEMPERLEY-LIEB-JONES THEORIES Ising theory is weak: the image of each braid group is finite, though bigger than the corresponding symmetric group. This may make the Ising theory easier to find in real materials. The Jones representation ρA J at r = 4 of Bn is irreducible of degree 2 n−1 2 if n is odd, and reducible to two irreducible representations of degree 2 n 2 −1 if n is even. To describe the image, we define groups Em 1 which are nearly extra-special 2-groups when m is even. Definition 1.30. The group E1 m has a presentation with generators x1,...,xm and relations xi 2 = 1, 1 i m xixj = xjxi, |i j| 2 xi+1xi = −xixi+1, 1 i m where −1 means an order two central element. For m even, Em 1 has only one irreducible representation of degree 1: an irreducible representation V1 of degree 2m/2. For m odd, E1 m has two irreducible representations W1,W2 of degree 1, both of degree 2 m−1 2 . Let PBn be the pure braid subgroup of Bn, defined by the exact sequence 1 PBn Bn Sn 1. Theorem 1.31. (1) The image ρJ A (PBn) as an abstract group is E1 n−1 , and the unitary Jones representation ρJ A of PBn factors through to V1 and W1 W2 respectively for n odd and n even. (2) ρJ A (Bn) fits into the exact sequence 1 E1 n−1 ρA(Bn) J Sn 1. (3) Projectively, we have 1 Zn−1 2 ρA proj (Bn) Sn 1 which splits only when n is even. The projective image (3) is from [Jo3]. A related result for images of ργ A (σi) is in [R1] for n even. For a proof, see [FRW]. Recall the Majorana fermions {γi} form the algebra with relations γi = γi, γiγj + γjγi = 2δij. Theorem 1.32. The Jones algebra Jn(A) for A = ±ie±πi/8 is isomorphic to the complex Clifford algebra. Proof. Let ei = 1 2 Ui as before. Then TLn(A) is generated by {e1,...,en−1} with relations e† i = ei, ei 2 = ei, eiei±1ei = 1 2 ei. The unitary Jones representation ρJ A (σi) is σi = −1 + (1 + q)ei, hence σi 2 = 1 2ei, q = −1. Note we write ρJ A (σi) simply as σi. Then σi 2 σj 2 = σj 2 σi 2 for |i j| 2 and σi 2 σi+1 2 + σi+1σi 2 2 = 0, which is equivalent to p3 = 0. Let γi = ( −1)i−1σ2 i · · · σ2. 1 Then {γi} forms the Majorana algebra. Conversely, we have σi 2 = −1γiγi+1.
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