The classification of local Artinian Gorenstein algebras is equivalent to the study
of orbits of a certain non-reductive group action on a polynomial ring. We give an explicit
formula for the orbits and their tangent spaces. We apply our technique to analyse when
an algebra is isomorphic to its associated graded algebra. We classify algebras with Hilbert
function (1, 3, 3, 3, 1), obtaining finitely many isomorphism types, and those with Hilbert
function (1, 2, 2, 2, 1, 1, 1). We consider fields of arbitrary, large enough, characteristic.

EB00000002163K

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• Classifying local Artinian Gorenstein algebras