Hilbert functions of colored quotient rings and a generalization of the Clements-Lindstrom theorem

摘要

Given a polynomial ring over a field , and a monomial ideal of , we say the quotient ring is Macaulay-Lex if for every graded ideal of , there exists a lexicographic ideal of with the same Hilbert function. In this paper, we introduce a class of quotient rings with combinatorial significance, which we call colored quotient rings. This class of rings include Clements-Lindstrom rings and colored squarefree rings as special cases that are known to be Macaulay-Lex. We construct two new classes of Macaulay-Lex rings, characterize all colored quotient rings that are Macaulay-Lex, and give a simultaneous generalization of both the Clements-Lindstrom theorem and the Frankl-Furedi-Kalai theorem. We also show that the -vectors of -colored simplicial complexes or multicomplexes are never characterized by "reverse-lexicographic" complexes or multicomplexes when n > 1and (a(1),...,a(n)) not equal (1,...,1).