Radical generic initial ideals (Q2155639)

The authors study a class of \(\mathbb{Z}^n\)-graded ideals in the \(\mathbb{Z}^n\)-graded polynomial ring over a field, known as Cartwright-Sturmfels ideals. Let \(n\in \mathbb{N}\) and \(m_1,\ldots,m_n \in \mathbb{N}\). Let \(S=K[x_{ij} : 1\leq j \leq n, ~ 0 \leq i \leq m_j]\) be the polynomial ring over a field \(K\), with the standard \(\mathbb{Z}^n\)-grading defined as deg\((x_{ij})=e_j\), where \(e_j\in \mathbb{Z}^n\), the \(j^{th}\) standard basis vector. A \(\mathbb{Z}^n\)-graded ideal \(I\) of \(S\) is called a Cartwright-Sturmfels ideal if there exists a radical Borel-fixed \(\mathbb{Z}^n\)-graded ideal which has the same multigraded Hilbert series as \(I\). If \(K\) is infinite, this is equivalent to that \(I\) has a radical \(\mathbb{Z}^n\)-graded generic initial ideal. For example, the \(\mathbb{Z}\)-graded Cartwright-Sturmfels ideals are exactly those generated by linear forms. In this paper, the authors discuss several known classes of Cartwright-Sturmfels ideals and they found a new class. For example, the authors give characterizations for determinantal ideals of same-size minors of a matrix of variables (Corollary 4.10, Theorem 4.11) and Schubert determinantal ideals (Theorem 5.6), to be Cartwright-Sturmfels ideals. Also see Theorem 6.1, for a class of Cartwright-Sturmfels binomial edge ideals.