Two unfortunate properties of pure f-vectors

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Conference Proceeding

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The set of -vectors of pure simplicial complexes is an important but little understood object in combinatorics and combinatorial commutative algebra. Unfortunately, its explicit characterization appears to be a virtually intractable problem, and its structure is very irregular and complicated. The purpose of this note, where we combine a few different algebraic and combinatorial techniques, is to lend some further evidence to this fact.

We first show that pure (in fact, Cohen-Macaulay) -vectors can be nonunimodal with arbitrarily many peaks, thus improving the corresponding results known for level Hilbert functions and pure -sequences. We provide both an algebraic and a combinatorial argument for this result. Then, answering negatively a question of the second author and collaborators posed in the recent AMS Memoir on pure -sequences, we show that the interval property fails for the set of pure -vectors, even in dimension 2.

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© Copyright 2014 American Mathematical Society. Publisher’s version of record: https://doi.org/10.1090/S0002-9939-2014-12338-0

Publication Title

Proceedings of the American Mathematical Society