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

• DOI: 10.1007/s10801-014-0571-0
• Publication Date: 2014-12-10T16:06:19Z
• Authors: Kai Fong Ernest C...
• Keywords: Colored

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<br/>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 Clements-Lindstr��m 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-Lindstr��m theorem and the Frankl-F��redi-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 $(a_1, \dots, a_n) \neq (1, \dots, 1)$.<br/>