The punctual Hilbert scheme of 4 points in affine 3-space via computer algebra and representation theory

The n-th punctual Hilbert scheme $Hilb^n_0(A^d)$ of points of affine d-space parametrises ideals of finite co-length n of the ring of functions on d-dimensional affine space, whose radical is the maximal ideal at the origin (equivalently, subschemes of length n with support at the origin). A classical theorem of Briancon claims the irreducibility of this space for d=2 and arbitrary n. The case of a small number of points being straightforward, the first nontrivial case is the case of 4 points in 3-space. We show, answering a question of Sturmfels, that over the complex numbers $Hilb^4_0(A^3)$ is irreducible. We use a combination of arguments from computer algebra and representation theory.