Gröbner Bases and Determinantal Ideals

Abstract

We give an introduction to the theory of determinantal ideals and rings, their Gröbner bases, initial ideals and algebras, respectively. The approach is based on the straightening law and the Knuth-Robinson-Schensted correspondence. The article contains a section treating the basic results about the passage to initial ideals and algebras.

Lovász–Saks–Schrijver ideals and coordinate sections of determinantal varieties

“…We use the following notations and facts from Gröbner bases theory, see for example [5]. Consider the polynomial ring S = k[x 1 , .…”

“…By [9, Thm. 1.16 (5)] the family of Cartwright-Sturmfels ideals is closed under any multigraded linear section. Hence it is enough to prove the statement for the ideal (f e + m e : e ∈ E).…”

“…Let X gen G be the 6 × 6 matrix associated to the graph from Example 3.2(3). That is, in the 6 × 6 generic matrix we set to 0 the entries in positions (1, 1), (1,2), (1,3), (1,4), (2, 1), (2, 2), (3, 2), (3, 3), (4, 3), (4, 4), (5, 1), (5,4).…”

Relations between the minors of a generic matrix

“…In this case the ideal Q generated by the Plücker relations is a complete intersection ideal of height 6 and Q = J 2 (3, 4) ∩ P where P is a prime ideal generated by Q and (S 2 ) (3,3|3,3) . In fact, there is an automorphism of S 2 (3, 4) carrying J 2 (3,4) into P so that S 2 (3, 4)/P ∼ = A 2 (3,4). Furthermore for t = 2, m = 3, n = 5 the ideal of quadrics in J 2 (3, 5) generate an ideal whose codimension is smaller than that of J 2 (3, 5) itself.…”

“…a term order such that in Proof. We know that A t is Cohen-Macaulay by [4,Theorem 7.10] and has dimension mn by [8,Proposition 10.16] because we have excluded the cases listed in Remark 1.2. Therefore we have reg(A t ) = dim A t + a(A t ) = mn + a(A t ).…”