Rank Selection and Depth Conditions for Balanced Simplicial Complexes
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We prove some new rank selection theorems for balanced simplicial complexes. Specifically, we prove that if a balanced simplicial complex satisfies Serre's condition $(S_{\ell})$ then so do all of its rank selected subcomplexes. We also provide a formula for the depth of a balanced simplicial complex in terms of reduced homologies of its rank selected subcomplexes. By passing to a barycentric subdivision, our results give information about Serre's condition and the depth of any simplicial complex. Our results extend rank selection theorems for depth proved by Stanley, Munkres, and Hibi.
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2012 ◽
Vol 55
(1)
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pp. 157-163
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2016 ◽
Vol 08
(03)
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pp. 399-429
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Keyword(s):