Lovász-Saks-Schrijver ideals and coordinate sections of determinantal varieties. (English) Zbl 1407.05254
Summary: Motivated by questions in algebra and combinatorics we study two ideals associated to a simple graph \(G\):
(i) the Lovász-Saks-Schrijver ideal defining the \(d\)-dimensional orthogonal representations of the graph complementary to \(G\), and
(ii) 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.
(i) the Lovász-Saks-Schrijver ideal defining the \(d\)-dimensional orthogonal representations of the graph complementary to \(G\), and
(ii) 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.
MSC:
| 05E40 | Combinatorial aspects of commutative algebra |
| 05C62 | Graph representations (geometric and intersection representations, etc.) |
| 13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |
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