$s$-Hankel hypermatrices and $2\times 2$ determinantal ideals

Abstract

We introduce the concept of an $s$-Hankel hypermatrix, which generalizes both Hankel matrices and generic hypermatrices. We study two determinantal ideals associated to an $s$-Hankel hypermatrix: the ideal $\I {s}{t}$ generated by certain $2 \times 2$ slice minors, and the ideal $\It {s}{t}$ generated by certain $2 \times 2$ generalized minors. We describe the structure of these two ideals, with particular attention to the primary decomposition of $\I {s}{t}$, and provide the explicit list of minimal primes for large values of $s$. Finally we give some geometrical interpretations and generalize a theorem of Watanabe.