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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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Let n be an arbitrary integer, let p be a prime factor of n . Denote by ω 1 the p th primitive unity root,

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\omega _1 : = e^{\tfrac{{2\pi i}} {p}}$$ \end{document}
.Define ω iω 1 i for 0 ≦ ip − 1 and B ≔ {1, ω 1 , …, ω p −1 } n ⊆ ℂ n .Denote by K ( n; p ) the minimum k for which there exist vectors ν 1 , …, ν kB such that for any vector wB , there is an i , 1 ≦ ik , such that ν i · w = 0, where ν · w is the usual scalar product of ν and w .Gröbner basis methods and linear algebra proof gives the lower bound K ( n; p ) ≧ n ( p − 1).Galvin posed the following problem: Let m = m ( n ) denote the minimal integer such that there exists subsets A 1 , …, A m of {1, …, 4 n } with | A i | = 2 n for each 1 ≦ in , such that for any subset B ⊆ [4 n ] with 2 n elements there is at least one i , 1 ≦ im , with A iB having n elements. We obtain here the result m ( p ) ≧ p in the case of p > 3 primes.

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