Schur--De-Rham complex and its cohomology
The author generalizes the notion of De-Rham complex \(S^{d-*} (V)\otimes\Lambda^* (V)\) associated to a finite-dimensional vector space \(V\). The new complex, called Schur-De-Rham complex is associated to a vector space \(V\) and a Young diagram \(\lambda\). This is denoted by \(S_\lambda (V)\). When the \(p\)-core of \(\lambda\) is non-trivial then the complex is exact, where \(p\) is the characteristic of the ground field. The author shows that this is not always the case. The complex has a non-trivial cohomology when the \(p\)-core of \(\lambda\) is trivial. The cohomology of \(S_\lambda(V)\) reflects deep properties of the Young diagram in positive characteristic. The author describes \(H^* (S_\lambda(V))\) for \(\lambda\) with trivial \(p\)-core and \(p\)-quotient consisting of a single diagram.
scientific article; zbMATH DE number 2153381
scientific article