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Transactions of the American Mathematical Society

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Balanced Cohen-Macaulay complexes


Author: Richard P. Stanley
Journal: Trans. Amer. Math. Soc. 249 (1979), 139-157
MSC: Primary 05A99; Secondary 06A10, 13H10, 52A40, 57Q05
DOI: https://doi.org/10.1090/S0002-9947-1979-0526314-6
MathSciNet review: 526314
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Abstract: A balanced complex of type $ ({a_1},\ldots,{a_m})$ is a finite pure simplicial complex $ \Delta $ together with an ordered partition $ ({V_1},\ldots,{V_m})$ of the vertices of $ \Delta $ such that card $ ({V_i}\, \cap \,F)\, = \,{a_i}$, for every maximal face F of $ \Delta $. If $ {\mathbf{b}}\, = \,({b_1},\ldots,{b_m})$, then define $ {f_\textbf{b}}(\Delta )$ to be the number of $ F\, \in \,\Delta $ satisfying card $ ({V_i}\, \cap \,F)\, = \,{b_i}$. The formal properties of the numbers $ {f_\textbf{b}}(\Delta )$ are investigated in analogy to the f-vector of an arbitrary simplicial complex. For a special class of balanced complexes known as balanced Cohen-Macaulay complexes, simple techniques from commutative algebra lead to very strong conditions on the numbers $ {f_\textbf{b}}(\Delta )$. For a certain complex $ \Delta (P)$ coming from a poset P, our results are intimately related to properties of the Möbius function of P.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0526314-6
Keywords: Simplicial complex, f-vector, h-vector, Coehen-Macaulay ring, Hilbert function, Poincaré series, poset, Möbius function
Article copyright: © Copyright 1979 American Mathematical Society

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