Sperner property, matroids and finite-dimensional Gorenstein algebras

We discuss a combinatorial property of the vector space lattice and some polynomials associated to matroids. Stanley developed powerful methods based on Hard Lefschetz Theorem to handle combinatorial objects. It is known that proofs of the Sperner property of typical posets can be done by showing the Lefschetz property for related Artinian commutative graded algebras. We introduce certain finite-dimensional Gorenstein algebras associated to matroids to show the Sperner property for a class of ranked posets including vector space lattices. We also discuss the Grobner fans of the defining ideal of our Gorenstein algebras and some tropical hypersurfaces. https://books.google.co.in/books?hl=en&lr=&id=AZgCAQAAQBAJ&oi=fnd&pg=PA73&dq=Sperner+property,+matroids+and+finite-dimensional+Gorenstein+algebras&ots=FGpcbOXIkf&sig=b8Ba1p7u1aXAl9RxyjAkPEuyMuE#v=onepage&q=Sperner%20property%2C%20matroids%20and%20finite-dimensional%20Gorenstein%20algebras&f=false