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By A. Jaffe (Chief Editor)

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Example text

U1 , u2 become real numbers. , we can make the following change: (l1 , l2 ) → (l1 , l2 )v, (m1 , m2 ) → (m1 , m2 )v, . . , (u1 , u2 ) → (u1 , u2 )v for some v ∈ O(2) (common orthogonal matrix for all the vectors in R2 ). Then, we can assume l2 = 0 and m2 ≥ 0, thus, we obtain d =   e   β 2 −2 β 4 +4 1 0 β 2 −2 1 2 β 2 +1 β 2 (β 2 +1) 2  √ β β 2 +1 l1 l2    β 2 −2  β β 2 −2 − β 4 −4 2β 4 √     2 2 4 = m1 m2   β −1 (β −1)(β +4) 2 4   √ β 2 −1 β(β −1) β +4 √ √   2 2 β −1 2 (β 2 −2) −4 β 2 (2β 2 −1) 2 β 2 −1 β n1 n2 − (β 2 +1)(2β 2 −1)(β 4 +4) 2 √ 2 β 2 +1 β 2 +1 4 (β −1) by the orthogonality of the matrix.

She also acknowledges a financial support from the Honda Heizaemon memorial fellowship. She is very grateful to Y. Kawahigashi and M. Izumi for constant advice and encouragement. : Subfactor of hyperfinite II1 factor with Coxeter graph E6 as invariant. J. : Principal graphs of subfactors with small index. To appear in Math. Ann. [EK] D. Evans, E. : Quantum symmetries on operator algebras. E. : Orbifold subfactors from Hecke algebras. Commun. in Math. Phys. , de la Harpe, P. : Coxeter graphs and towers of algebras.

We easily know that z w u n,l t z˜ w =u n,l =u z w n,r . Note that the multiplicity n of the edges z-w is equal to that of the edges z-w. ˜ Now we begin to construct a candidate for the list of gauge transformation matrices. First, for the connections M (∗/6) and M (˜∗/6), we fix the gauges for the simple edges as u ∗ G a 6 =u 1,l ∗˜ G =u 1,r a˜ 6 =u 1,l = (1)1,1 . 1,r Here the matrices are all 1 × 1. From the next matrices, we always fix the gauges for simple edges to 1 × 1 matrices (1)1,1 , unless otherwise specified.

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