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We compute the linear strand of the minimal free resolution of the idealgenerated by k x k sub-permanents of an n x n generic matrix and of the idealgenerated by square-free monomials of degree k. The latter calculation givesthe full minimal free resolution by work of Biagioli-Faridi-Rosas. Ourmotivation is to lay groundwork for the use of commutative algebra in algebraiccomplexity theory. We also compute several Hilbert functions relevant forcomplexity theory.

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