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<p>In this paper, we develop a theoretical framework for computing the Hilbert series of algebras associated with layered directed graphs using the cohomology of partially ordered sets (posets). Building on previous approaches, we refine and extend known results by expressing the Hilbert series in terms of chain enumerations within subgraphs induced by vertex levels. This formulation not only simplifies calculations but also provides explicit expressions for classes of layered graphs, such as complete layered graphs. Applications and illustrative examples are presented to demonstrate the effectiveness of the method.</p>

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Hilbert series of certain layered directed graphs via poset cohomology

  • 期刊论文

  • 发表日期：

    2025年12月6日

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  • Karoline Victor Fernandes,
  • José Eder Salvador de Vasconcelos

摘要

In this paper, we develop a theoretical framework for computing the Hilbert series of algebras associated with layered directed graphs using the cohomology of partially ordered sets (posets). Building on previous approaches, we refine and extend known results by expressing the Hilbert series in terms of chain enumerations within subgraphs induced by vertex levels. This formulation not only simplifies calculations but also provides explicit expressions for classes of layered graphs, such as complete layered graphs. Applications and illustrative examples are presented to demonstrate the effectiveness of the method.