Higher Specht polynomials give a basis of the polynomial ring that respects the decomposition into irreducible representations of the symmetric group. We use these polynomials to construct matrix factorizations of the discriminant indexed by partitions. The Cohen-Macaualy modules associated to these matrix factorizations give a noncommutative resolution of the discriminant. We also discuss extensions of these results to other complex reflection groups. This is joint work with Eleonore Faber, Simon May and Marco Talarico.