Minimal free resolutions of ideals of minors associated to pairs of matrices

András Cristian Lőrincz

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Submission date: 25. Jan. 2020
Pages: 15
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Abstract:
Consider the affine space consisting of pairs of matrices (A,B) of fixed size, and its closed subvariety given by the rank conditions rank A ≤ a, rank B ≤ b and rank(AB) ≤ c, for three non-negative integers a,b,c. These varieties are precisely the orbit closures of representations for the equioriented A₃ quiver. In this paper we construct the (equivariant) minimal free resolutions of the defining ideals of such varieties. We show how this problem is equivalent to determining the cohomology groups of the tensor product of two Schur functors of tautological bundles on a 2-step flag variety. We provide several techniques for the determination of these groups, which is of independent interest.