Syzygies of the apolar ideals of the determinant and permanent

2017/09/26 by Alper, Jarod, Rowlands, Rowan
#Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1709.09286

Abstract

We investigate the space of syzygies of the apolar ideals det_n^⊥ and \rm perm_n^⊥ of the determinant det_n and permanent \rm perm_n polynomials. Shafiei had proved that these ideals are generated by quadrics and provided a minimal generating set. Extending on her work, in characteristic distinct from two, we prove that the space of relations of det_n^⊥ is generated by linear relations and we describe a minimal generating set. The linear relations of \rm perm_n^⊥ do not generate all relations, but we provide a minimal generating set of linear and quadratic relations. For both det_n^⊥ and \rm perm_n^⊥, we give formulas for the Betti numbers β_1,j, β_2,j and β_3,4 for all j as well as conjectural descriptions of other Betti numbers. Finally, we provide representation-theoretic descriptions of certain spaces of linear syzygies.

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