By J.-A. Chao, W.A. Woyczynski
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Extra resources for Martingale Theory in Harmonic Analysis and Banach Spaces
The edge joining the vertices labelled p/2 and k is solid, bold or dashed when 1/l < 0, = 0, > 0. The edge joining the vertices labelled p/2 and l is solid, bold or dashed when 1/k < 0, = 0, > 0. Finally the edges joining the vertex labelled d are solid, bold or dashed when 1/p < 0, = 0, > 0. 2. Monodromy groups and equilateral triangle groups. Consider the map R1 , which is the holonomy around L23 . This is a complex reﬂection ﬁxing L23 with angle 2π/p. Similarly, R2 and A1 are the holonomies around L13 and L14 .
In this section we consider the groups where l ≤ 0 and d ≤ 0. The values of p and k, together with l and d are: p 3 k 4 −d 2 −l 12 3 3 4 5 6 3 2 2 4 30 ∞ 12 4 5 5 6 6 4 2 3 2 3 4 10 10 ∞ ∞ ∞ 5 30 6 ∞ In this case the four lines involving v5 in the complex line arrangement described above have collapsed to a point. Instead there are four zero dimensional strata z123 , z124 , z134 and z234 , where the stratum zijk = Lij ∩ Ljk ∩ Lki . We illustrate this in Figure 6. Algebraic geometers call this line arrangement the complete quadrilateral.
We can calculate the eigenvalues as above: • R1 has eigenvalues e2πi/9 , e−πi/9 , e−πi/9 , • R2 has eigenvalues e2πi/9 ,e−πi/9 , e−πi/9 , • R1 R2 has eigenvalues e−2πi/9 , e4πi/9 , e−2πi/9 , • R1 R2 R1 has eigenvalues e2πi/3 , e−πi/3 , e−πi/3 . In this case, (R1 R2 )3 = (R1 R2 R1 )2 is conjugate to a vertical Heisenberg translation and generates the centre of R1 , R2 . 6. Suppose that 1/p + 1/k − 1/2 = −1/l < 0. The stabiliser of z134 is the group A1 , R2 . This group has order 8p2 k2 /(2p + 2k − pk)2 = 2l2 .