Determinantal facet ideals

Krs and Determinantal Ideals

Krs and Determinantal Ideals

Quantum Determinantal Ideals

Introduction. Fix a base field k. The quantized coordinate ring of n×n matrices over k, denoted by q(Mn(k)), is a deformation of the classical coordinate ring of n×n matrices, (Mn(k)). As such, it is a k-algebra generated by n2 indeterminates Xij , for 1 ≤ i,j ≤ n, subject to relations which we state in (1.1). Here, q is a nonzero element of the field k. When q = 1, we recover (Mn(k)), which is...

Resolutions of Facet Ideals

In this paper we study the resolution of a facet ideal associated with a special class of simplicial complexes introduced by Faridi. These simplicial complexes are called trees, and are a generalization (to higher dimensions) of the concept of a tree in graph theory. We show that the Koszul homology of the facet ideal I of a tree is generated by the homology classes of monomial cycles, determin...

Prime Ideals in Certain Quantum Determinantal Rings

The ideal I 1 generated by the 2 2 quantum minors in the coordinate algebra of quantum matrices, O q (M m;n (k)), is investigated. Analogues of the First and Second Fundamental Theorems of Invariant Theory are proved. In particular, it is shown that I 1 is a completely prime ideal, that is, O q (M m;n (k))=I 1 is an integral domain, and that O q (M m;n (k))=I 1 is the ring of coinvariants of a ...