Bernstein-Gelfand-Gelfand meets geometric complexity theory: resolving the 2 x 2 permanents of a 2 x n matrix

Abstract

We describe the minimal free resolution of the ideal of $2\times 2$ subpermanents of a $2\times n$ generic matrix $M$. In contrast to the case of $2\times 2$ determinants, the $2\times 2$ permanents define an ideal which is neither prime nor Cohen-Macaulay. We combine work of Laubenbacher-Swanson on the Gröbner basis of an ideal of $2\times 2$ permanents of a generic matrix with our previous work connecting the initial ideal of $2\times 2$ permanents to a simplicial complex. The main technical tool is a spectral sequence arising from the Bernstein-Gelfand-Gelfand correspondence.

v1: 24 pages, 4 figures, v2: changes made to reflect referee input