We investigate the space of syzygies of the apolar ideals \({\text {det}}_n^\perp \) and \({\mathrm{perm}}_n^\perp \) of the determinant \({\text {det}}_n\) and permanent \({\mathrm{perm}}_n\) 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 \({\text {det}}_n^{\perp }\) is generated by linear relations and we describe a minimal generating set. The linear relations of \({\mathrm{perm}}_n^{\perp }\) do not generate all relations, but we provide a minimal generating set of linear and quadratic relations. For both \({\text {det}}_n^\perp \) and \({\mathrm{perm}}_n^\perp \), we give formulas for the Betti numbers \(\beta _{1,j}\), \(\beta _{2,j}\) and \(\beta _{3,4}\) for all j as well as conjectural descriptions of other Betti numbers. Finally, we provide representation-theoretic descriptions of certain spaces of linear syzygies.

中文翻译：

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行列式和永久非极性理想的Syzygies

我们调查的非极性理想syzygies的空间\（{\文本{DET}} _ñ^ \ PERP \）和\（{\ mathrm {烫发}} _ñ^ \ PERP \）行列式的\（{\文本{ det}} _ n \）和永久\（{{mathrm {perm}} _ n \）多项式。Shafiei证明了这些理想是由二次生成的，并提供了最小的生成集。扩展她的工作，以不同于两个的特征，我们证明\（{{text {det}} _ n ^ {\ perp} \}的关系空间是由线性关系生成的，并且我们描述了一个最小生成集。的线性关系\（{\ mathrm {烫发}} _ N R个{\ PERP} \），不生成的所有关系，但我们提供的线性和二次关系的最小的发电机组。对彼此而言\（{\ text {det}} _ n ^ \ perp \）和\（{\ mathrm {perm}} _ n ^ \ perp \），我们给出贝蒂数\（\ beta _ {1，j} \ ），\（\测试_ {2，J} \）和\（\测试_ {3,4} \）对于所有Ĵ以及其他贝蒂数的推测说明。最后，我们提供了线性理论的某些空间的表示理论描述。