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Mathematics  2015 

On Postnikov-Shapiro Algebras and their generalizations

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Abstract:

In this paper we consider the original and different generalizations of Postnikov-Shapiro algebra, see~\cite{PSh}. Firstly, for a given graph $G$ and a positive integer $t$, we generalize the notion of Postnikov-Shapiro algebras counting forests in $G$ to an algebra counting $t$-labelled forests. We also prove that for large $t$ we can restore the Tutte polynomial of $G$ from the Hilbert series of such algebra. Secondly, we prove that the original Postnikov-Shapiro algebra counting forests depends only on the matroid of $G$. And conversely, we can reconstruct this matroid from the latter algebra. Similar facts hold for analogous algebras counting trees in connected graphs. Thirdly, we present a generalization of such algebras for hypergraphs. Namely, we construct a certain family of algebras for a given hypergraph, such that for almost algebras from this family, their Hilbert series is the same. Finally, we present the definition of a hypergraphical matroid, whose Tutte polynomial allows us to calculate this generic Hilbert series.

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