Cartan divisors Sample Clauses

Cartan divisors. ‌ The Cartan divisors comprise the components of z = 0 in the resolved geometry z = ζ235ζ2 ζ3δ5δ2δ7ζ12347z1 = 0 . (4.2.18) We now identify these with negative simple roots of E6 as well as the root −α0 corre- sponding to the extended node of the affine E6 Dynkin diagram. The classes are7 Defining Section z1 = 0 δ5 = 0 δ6 = 0 ζ4 = 0 ζ346 = 0 ζ235 = 0 ζ12347 = 0 Locus in Y˜4 Class in Y˜4 Label δ y − ζ 2 235 ζ346ζ12347x3 = 0 6 6 235 346 b δ ζ −b4ζ4ζ346 + b3y7 − δ5ζ346 = 0 b3ζ12347 + δ5δ6 = 0 δ5 + b3ζ12347 = 0 δ7 = 0 S − E1 E5 E6 E4 E3 − E4 − E6 E2 − E3 − E5 E7 D−α 0 −b δ ζ ζ − b4ζ235ζ346x2 + b3y7 = 0 D−α1 3 7 12347 7 346 12347 y − ζ ζ x + δ δ y = 0 D−α2 D−α 3 D−α4 D−α5 D−α6 (4.2.19) The labeling is consistent with the standard ordering of roots of E6. The intersection of the Cartan divisors reproduces indeed the extended Cartan matrix of E6, with z1 = 0 playing the role of the affine root, and the intersections are diagramatically depicted below: [z1] [ζ12347] [δ5] [δ6] [ζ4] [ζ346] [ζ235]
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