Geodesic
Hilbert functions of colored quotient rings and a generalization of the clements-lindström theorem
Journal of Algebraic Combinatorics, Tome 42 (2015) no. 1, pp. 1-23.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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-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 $(a_1,\dots,a_n)\neq(1,\dots,1)$.
Classification : 13A02, 13C05, 13D40, 13F55, 05D99, 05E40, 05E45
Keywords: Hilbert function, Macaulay-Lex rings, Kruskal-Katona theorem, Clements-Lindström theorem, $f$-vector, colored complexes 
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     author = {Chong, Kai Fong Ernest},
     title = {Hilbert functions of colored quotient rings and a generalization of the clements-lindstr\"om theorem},
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     url = {http://geodesic.mathdoc.fr/item/JAC_2015__42_1_a11/}
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Chong, Kai Fong Ernest. Hilbert functions of colored quotient rings and a generalization of the clements-lindström theorem. Journal of Algebraic Combinatorics, Tome 42 (2015) no. 1, pp. 1-23. http://geodesic.mathdoc.fr/item/JAC_2015__42_1_a11/