Second syzygy of determinantal ideals generated by minors of generic symmetric matrices Original Research Article

We prove a characteristic-free plethysm formula of the complex SS2theta for a map of finite free modules theta. We also study a family of subcomplexes of a Schur complex, the ν-Schur complexes. Using these machineries from characteristic-free representation theory of general linear group, we give an example of determinantal ideal of generic symmetric matrix whose second Betti number depends on the characteristic of the base field. We also give a short proof of Kuranoʹs first syzygy theorem.