Syzygies of Determinantal Thickenings

2020/08/06 by Hang Huang, Huang, Hang
Mathematics · #Algebraic structures and combinatorial models #Advanced Combinatorial Mathematics #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.2008.02690

Abstract

Let S = ℂ[x_i,j] be the ring of polynomial functions on the space of m × n matrices, and consider the action of the group GL = GL_m × GL_n via row and column operations on the matrix entries. It is proven by Raicu and Weyman that for a GL-invariant ideal I ⊆ S, the linear strands of its minimal free resolution translates via the BGG correspondence to modules over the general linear Lie superalgebra \mathfrakgl(m|n). When I=I_λ is the ideal generated by the GL-orbit of a highest weight vector of weight λ, they gave a conjectural description of the classes of these \mathfrakgl(m|n)-modules in the Grothendieck group. We prove their conjecture here. We also give a algorithmic description of how to get the classes of these \mathfrakgl(m|n)-modules for any GL-invariant ideal I ⊆ S.

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