Vol. 13, No. 2, 2019

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

Aldo Conca and Volkmar Welker

Vol. 13 (2019), No. 2, 455–484
Abstract

Motivated by questions in algebra and combinatorics we study two ideals associated to a simple graph G:

  • the Lovász-Saks-Schrijver ideal defining the d-dimensional orthogonal representations of the graph complementary to G, and
  • the determinantal ideal of the (d+1)-minors of a generic symmetric matrix with 0 in positions prescribed by the graph G.

In characteristic 0 these two ideals turn out to be closely related and algebraic properties such as being radical, prime or a complete intersection transfer from the Lovász–Saks–Schrijver ideal to the determinantal ideal. For Lovász–Saks–Schrijver ideals we link these properties to combinatorial properties of G and show that they always hold for d large enough. For specific classes of graphs, such a forests, we can give a complete picture and classify the radical, prime and complete intersection Lovász–Saks–Schrijver ideals.

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Keywords
determinantal rings, complete intersections, ideals associated to graphs, Gröbner bases.
Mathematical Subject Classification 2010
Primary: 05E40
Secondary: 05C62, 13P10
Milestones
Received: 8 February 2018
Revised: 3 November 2018
Accepted: 24 December 2018
Published: 2 March 2019
Authors
Aldo Conca
Dipartimento di Matematica
Università di Genova
16146 Genova
Italy
Volkmar Welker
Fachbereich Mathematik und Informatik
Philipps-Universität Marburg
35032 Marburg
Germany