Extending the descent-to-peak map and its applications

The descent-to-peak map serves as a bridge between algebra and combinatorics. We use it as a tool for proving the equidistribution of peak and valley sets of standard Young tableaux with a very short argument. We also introduce a new shuffle basis of quasisymmetric functions whose elements are eigenvectors of the descent-to-peak map. Using this basis, we then extend the notion of the peak algebra and of the descent-to-peak map to shuffle, tensor, and symmetric algebras.