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322 3151 3169 Murai, S. and Hibi, T. , The depth of an ideal with a given Hilbert function, Proc. Amedr. Math. Soc. , 136 (2008

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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 V λ be the set of all vectors v=(v 1,...,v n)∈ F n such that  |{j ∈ [n] : v ji}|=λi+1  for 0≦ i ≦\ k-1. We describe the lexicographic standard monomials of the ideal of polynomials from  F[x 1,...,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].

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, A. V. , Maroscia , P. and Roberts , L. G. , The Hilbert function of a reduced k -algebra , J. London Math. Soc. ( 2 ), 28 ( 1983 ), no. 3, 443 – 452 . [11

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Studia Scientiarum Mathematicarum Hungarica
Authors: Hiram López, Eliseo Sarmiento, Maria Pinto and Rafael Villarreal

. Inform. Theory 1991 37 1567 1576 Stanley, R. , Hilbert functions of graded

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