Burkhardt quartic
Encyclopedia
In mathematics, the Burkhardt quartic is a quartic threefold
in 4-dimensional projective space studied by ,
with the maximum possible number of 45 nodes.
The equations defining the Burkhardt quartic become simpler if it is embedded in P5 rather than P4.
In this case it can be defined by the equations σ1 = σ4 = 0, where σi is the ith elementary symmetric function of the coordinates (x0 : x1 : x2 : x3 : x4 : x5) of P5.
The automorphism group of the Burkhardt quartic is the Burkhardt group U4(2) = PSp4(3), a simple group of order 25920, which is isomorphic to a subgroup of index 2 in the Weyl group
of E6.
Quartic threefold
In algebraic geometry, a quartic threefold is a degree 4 hypersurface of dimension 3 in 4-dimensional projective space. showed that all non-singular quartic threefolds are irrational, though some of them are unirational....
in 4-dimensional projective space studied by ,
with the maximum possible number of 45 nodes.
The equations defining the Burkhardt quartic become simpler if it is embedded in P5 rather than P4.
In this case it can be defined by the equations σ1 = σ4 = 0, where σi is the ith elementary symmetric function of the coordinates (x0 : x1 : x2 : x3 : x4 : x5) of P5.
The automorphism group of the Burkhardt quartic is the Burkhardt group U4(2) = PSp4(3), a simple group of order 25920, which is isomorphic to a subgroup of index 2 in the Weyl group
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...
of E6.