Abstract
We introduce a certain class of algebras associated to matroids. We prove the Lefschetz property of the algebras for some special cases. Our result implies the Sperner property for the Boolean lattice and the vector space lattice. Nous présentons une classe d'algèbres associées aux matroïdes. Nous démontrons que dans quelques cas spécifiques, ces algèbres vérifient la propriété de Lefschetz. Notre résultat implique la propriété de Sperner pour l'algèbre de Boole et pour le poset d'espace vectoriel.
Highlights
The strong Lefschetz property for Artinian Gorenstein algebras is a ring-theoretic interpretation of the Hard Lefschetz Theorem for compact Kahler manifolds
One of the main topics of this article is an application of the Lefschetz property for a certain kind of Gorenstein algebras to the Sperner property of the ranked posets associated to some matroids
It is known that the Sperner property of the Boolean lattice is proved from the strong Lefschetz property of the algebra k[x1, . . . , xn]/(x21, . . . , x2n)
Summary
The strong Lefschetz property for Artinian Gorenstein algebras is a ring-theoretic interpretation of the Hard Lefschetz Theorem for compact Kahler manifolds. One of the main topics of this article is an application of the Lefschetz property for a certain kind of Gorenstein algebras to the Sperner property of the ranked posets associated to some matroids. The Sperner property of the vector space lattice V (q, n) consisting of the linear subspaces of the vector space Fnq can be deduced from the result on the rank of its incidence matrices due to Kantor [9]. We will give another proof of the Sperner property of V (q, n) based on the strong Lefschetz property of a Gorenstein algebra. Please see [11], the full version of this article, for more details
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