Monomial bases related to the n! conjecture

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The purpose of this paper is to find a new way to prove the n! conjecture for particular partitions. The idea is to construct a monomial and explicit basis for the space Mμ. We succeed completely for hook-shaped partitions, i.e., μ=(K+1,1ˡ). We are able to exhibit a basis and to verify that its cardinality is indeed n!, that it is linearly independent and that it spans Mμ. We derive from this study an explicit and simple basis for Iμ, the annihilator ideal of Δμ. This method is also successful for giving directly a basis for the homogeneous subspace of Mμ consisting of elements of 0 x-degree.