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

Abstract

Given a polynomial ring $S=\mathbb{k}[x_1,\dots ,x_n]$ over a field $\mathbb{k}$, 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 Clements-Lindström 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-Lindström theorem and the Frankl-Füredi-Kalai 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 $(a_1,\dots ,a_n)\ne (1,\dots ,1)$.

17 pages; v2: notation significantly simplified, and a (new) shorter proof of Theorem 4.5 is given. To appear in the J. of Algebraic Combinatorics