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2019 Lovász–Saks–Schrijver ideals and coordinate sections of determinantal varieties
Aldo Conca, Volkmar Welker
Algebra Number Theory 13(2): 455-484 (2019). 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.

Citation

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Aldo Conca. Volkmar Welker. "Lovász–Saks–Schrijver ideals and coordinate sections of determinantal varieties." Algebra Number Theory 13 (2) 455 - 484, 2019. https://doi.org/10.2140/ant.2019.13.455

Information

Received: 8 February 2018; Revised: 3 November 2018; Accepted: 24 December 2018; Published: 2019
First available in Project Euclid: 26 March 2019

zbMATH: 07042065
MathSciNet: MR3927052
Digital Object Identifier: 10.2140/ant.2019.13.455

Subjects:
Primary: 05E40
Secondary: 05C62 , 13P10

Keywords: complete intersections , determinantal rings , Gröbner bases. , ideals associated to graphs

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.13 • No. 2 • 2019
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