Hypergraph LSS-ideals and coordinate sections of symmetric tensors

2022/02/21 by Shekoofeh Gharakhloo, Gharakhloo, Shekoofeh, Volkmar Welker +1 · 1 citation
Mathematics · #Tensor decomposition and applications #Commutative Algebra and Its Applications

paper · pdf · doi:10.48550/arxiv.2202.10463

Abstract

Let K be a field, [n]= 1,...,n and H=([n],E) be a hypergraph. For an integer d >= 1 the Lovasz-Saks-Schrijver ideal (LSS-ideal) L_H^K (d) in K[y_ij~:~(i,j) ∈ [n] x [d]] is the ideal generated by the polynomials f^(d)_e= ∑_j=1^d ∏_{i ∈ e} y_ij for edges e of H. In this paper for an algebraically closed field K and a k-uniform hypergraph H=([n],E) we employ a connection between LSS-ideals and coordinate sections of the closure of the set S_n,k^d of homogeneous degree k symmetric tensors in n variables of rank <= d to derive results on the irreducibility of its coordinate sections. To this end we provide results on primality and the complete intersection property of L_H^K (d). We then use the combinatorial concept of positive matching decomposition of a hypergraph H to provide bounds on when L_H^K(d) turns prime to provide results on the irreducibility of coordinate sections of S_{n, k}^d.

Cited by

• Lovász--Saks--Schrijver Ideals and the Irreducible Components of the Variety of Orthogonal Representations of a Graph

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