The Determinantal ideals are generated by a general set of minors of a matrix of indeterminates. In the case that the generators can be identified with the facets of a simplicial complex, it is called a determinantal facet ideal.
In this talk I will discuss some algebraic properties of these ideals which can be read directly from their simplicial complexes.
One approach is to classify the prime ideals among them. Then I will consider a nice class of simplicial complexes to give a combinatorial description for primary decompositions of their associated ideals.