Balanced Cohen-Macaulay complexes

Stanley, Richard P.

doi:10.1090/S0002-9947-1979-0526314-6

Balanced Cohen-Macaulay complexes
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Trans. Amer. Math. Soc. 249 (1979), 139-157 Request permission

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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Decompositions, shellings, and diameters of simplicial complexes and convex polyhedra