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rgugliel / AlVin

A C++ implementation of the Vinberg's algorithm for Q, Q( sqrt(d) ) and Q( cos(2 pi / 7) )
GNU General Public License v3.0
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Initial vertex stabiliser miscalculated #7

Closed sashakolpakov closed 5 years ago

sashakolpakov commented 5 years ago

AlVin makes an error starting Vinberg's algorithm with a wrong vertex stabiliser for Bugaenko's quadratic form -2T,1,1,1,1,1,1,1,1 (here the quad form is incorrect, c.f. below) over Q[sqrt 5] in dimension 8+1. Namely, it starts with the vectors 

e1 = (0, -1, 1, 0, 0, 0, 0, 0, 0)
e2 = (0, 0, -1, 1, 0, 0, 0, 0, 0)
e3 = (0, 0, 0, -1, 1, 0, 0, 0, 0)
e4 = (0, 0, 0, 0, -1, 1, 0, 0, 0)
e5 = (0, 0, 0, 0, 0, -1, 1, 0, 0)
e6 = (0, 0, 0, 0, 0, 0, -1, 1, 0)
e7 = (0, 0, 0, 0, 0, 0, 0, -1, 1)
e8 = (0, 0, 0, 0, 0, 0, 0, 0, -1)

while the right set of normals has to start with the E8 root system

e1 = (0, -1, 1, 0, 0, 0, 0, 0, 0)
e2 = (0, 0, -1, 1, 0, 0, 0, 0, 0)
e3 = (0, 0, 0, -1, 1, 0, 0, 0, 0)
e4 = (0, 0, 0, 0, -1, 1, 0, 0, 0)
e5 = (0, 0, 0, 0, 0, -1, 1, 0, 0)
e6 = (0, 0, 0, 0, 0, 0, -1, 1, 0)
e7 = (0, 0, 0, 0, 0, 0, 0, -1, -1)
e8 = 1/2*(1, 1, 1, 1, 1, 1, 1, 1, 1)
sashakolpakov commented 5 years ago

Bugaenko's example should correspond to a non-diagonal form: AlVin is not supposed to handle it.