Grassmann-Plücker relations and matroids with coefficients (Q1814138)

Using the concept of fuzzy rings, this paper presents a conceptual framework which allows a unified treatment of the algebraic and the combinatorial aspects of the Grassmann-Plücker identities which have played an important role in invariant and syzygy theory and in combinatorics. It has been shown that a matroid with coefficients of finite rank can also be defined in terms of its associated Grassmann- Plücker map, which in a way generalizes classical determinants. More precisely, it has been shown that such matroids correspond naturally and in a one-to-one fashion to equivalence classes of appropriately defined Grassmann-Plücker maps.