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Algebras related to matroids represented in characteristic zero. (English) Zbl 0996.16027
Summary: Let \(k\) be a field of characteristic zero. We consider graded subalgebras \(A\) of \(k[x_1,\dots,x_m]/(x_1^2,\dots,x_m^2)\) generated by \(d\) linearly independent linear forms. Representations of matroids over \(k\) provide a natural description of the structure of these algebras. In return, the numerical properties of the Hilbert function of \(A\) yield some information about the Tutte polynomial of the corresponding matroid. Isomorphism classes of these algebras correspond to equivalence classes of hyperplane arrangements under the action of the general linear group.

MSC:
16W50 Graded rings and modules (associative rings and algebras)
05B35 Combinatorial aspects of matroids and geometric lattices
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
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