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

Conca, Aldo; Welker, Volkmar

doi:10.2140/ant.2019.13.455

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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

DOI: 10.2140/ant.2019.13.455

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.

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

Vol. 13, No. 2, 2019