On minimal free resolutions of sub-permanents and other ideals arising in complexity theory

We compute the linear strand of the minimal free resolution of the ideal generated by k×k sub-permanents of an n×n generic matrix and of the ideal generated by square-free monomials of degree k. The latter calculation gives the full minimal free resolution by [1]. Our motivation is to lay groundwork for the use of commutative algebra in algebraic complexity theory. We also compute several Hilbert functions relevant for complexity theory.