Hilbert functions of colored quotient rings and a generalization of the ClementsLindstrm theorem
Given a polynomial ring \(S = \Bbbk [x_1, \dots , x_n]\) over a field \(\Bbbk \) , and a monomial ideal \(M\) of \(S\), we say the quotient ring \(R = S/M\) is Macaulay-Lex if for every graded ideal of \(R\), there exists a lexicographic ideal of \(R\) 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 ClementsLindstrm 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 ClementsLindstrm theorem and the FranklFrediKalai theorem. We also show that the \(f\)-vectors of \((a_1, \dots , a_n)\)-colored simplicial complexes or multicomplexes are never characterized by reverse-lexicographic complexes or multicomplexes when \(n>1\) and \((a_1, \ldots , a_n) \ne (1, \ldots , 1)\).
hilbert function clementslindstrm theorem macaulay-lex rings \(f\)-vector kruskalkatona theorem colored complexes 13a02 13c05 13d40 13f55 05d99 05e40 05e45