Standard monomials for partitions

Summary

Let F be a field, and α₀,...,α_k-1 be k distinct elements of F. Let λ =(λ₁,...,λ_k) be a partition of n and V_λ be the set of all vectors v=(v₁,...,v_n)∈ Fⁿ such that  |{j ∈ [n] : v_j=α_i}|=λ_i+1  for 0≦ i ≦\ k-1. We describe the lexicographic standard monomials of the ideal of polynomials from  F[x₁,...,x_n] which vanish on the set V_λ. In the proof we give a new description of the orthogonal complement (S^λ)^⊥ (with respect to the James scalar product) of the Specht module S^λ. As applications, a basis of (S^λ)^⊥ is exhibited, and we obtain a combinatorial description of the Hilbert function of V_λ..  Our approach gives also the deglex standard monomials of V_λ, and hence provides a new proof of a result of A. M. Garsia and C. Procesi [10].