Two Murnaghan-Nakayama rules in Schubert calculus (Q1653316)

Each integer partition \(\lambda\) has an associated Schur symmetric function \(s_{\lambda}\), and these Schur functions form a basis for the \(\mathbb Q\)-algebra of symmetric functions, which is also freely generated by the power sum symmetric functions \(p_r\). The Murnaghan-Nakayama rule is the expansion in the Schur basis of the product by a power sum, \[ p_r \cdot s_{\lambda} = \sum_{\mu} (-1)^{\operatorname{ht}(\frac{\mu}{\lambda})+ 1}s_{\mu}, \] the sum over all partitions \(\mu\) such that \(\frac{\mu}{\lambda}\) is a rim hook of size \(r\) and \(\operatorname{ht}(\frac{\mu}{\lambda}))\) is the height (number of rows) of \(\frac{\mu}{\lambda}\). Products \(p_{\lambda} = p_{\lambda_1}p_{\lambda_k}\) of power sums form another basis for symmetric functions, and the change of basis matrix between these two is the character table for the symmetric group. In this way, the Murnaghan-Nakayama rule gives a formula for the characters of the symmetric group. In this paper, the authors establish a version of the Murnaghan-Nakayama rule for Schubert polynomials and a version for the quantum cohomology ring of the Grassmannian. These rules compute all intersections of Schubert cycles with tautological classes coming from the Chern character. Like the classical rule, both rules are multiplicity-free signed sums.