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

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