A relative simplicial complex is a collection of sets of the form , where are simplicial complexes. Relative complexes have played key roles in recent advances in algebraic, geometric, and topological combinatorics but, in contrast to simplicial complexes, little is known about their general combinatorial structure. In this paper, we address a basic question in this direction and give a characterization of -vectors of relative (multi)complexes on a ground set of fixed size. On the algebraic side, this yields a characterization of Hilbert functions of quotients of homogeneous ideals over polynomial rings with a fixed number of indeterminates.
Moreover, we characterize -vectors of fully Cohen–Macaulay relative complexes as well as -vectors of Cohen–Macaulay relative complexes with minimal faces of given dimensions. The latter resolves a question of Björner.
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.38
Keywords: relative simplicial complex, $f$-vector, Kruskal–Katona theorem, Hilbert functions, $h$-vector, Macaulay theorem
Codenotti, Giulia 1; Katthän, Lukas 2; Sanyal, Raman 2
@article{ALCO_2019__2_3_343_0, author = {Codenotti, Giulia and Katth\"an, Lukas and Sanyal, Raman}, title = {On $f$- and $h$-vectors of relative simplicial complexes}, journal = {Algebraic Combinatorics}, pages = {343--353}, publisher = {MathOA foundation}, volume = {2}, number = {3}, year = {2019}, doi = {10.5802/alco.38}, zbl = {07066878}, mrnumber = {3968741}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.38/} }
TY - JOUR AU - Codenotti, Giulia AU - Katthän, Lukas AU - Sanyal, Raman TI - On $f$- and $h$-vectors of relative simplicial complexes JO - Algebraic Combinatorics PY - 2019 SP - 343 EP - 353 VL - 2 IS - 3 PB - MathOA foundation UR - https://alco.centre-mersenne.org/articles/10.5802/alco.38/ DO - 10.5802/alco.38 LA - en ID - ALCO_2019__2_3_343_0 ER -
%0 Journal Article %A Codenotti, Giulia %A Katthän, Lukas %A Sanyal, Raman %T On $f$- and $h$-vectors of relative simplicial complexes %J Algebraic Combinatorics %D 2019 %P 343-353 %V 2 %N 3 %I MathOA foundation %U https://alco.centre-mersenne.org/articles/10.5802/alco.38/ %R 10.5802/alco.38 %G en %F ALCO_2019__2_3_343_0
Codenotti, Giulia; Katthän, Lukas; Sanyal, Raman. On $f$- and $h$-vectors of relative simplicial complexes. Algebraic Combinatorics, Volume 2 (2019) no. 3, pp. 343-353. doi : 10.5802/alco.38. https://alco.centre-mersenne.org/articles/10.5802/alco.38/
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