A Tableau Formula for Eta Polynomials

We use the Pieri and Giambelli formulas of Buch et al. (Invent Math 178:345–405, 2009; J Reine Angew, 2013) and the calculus of raising operators developed in Buch et al. (A Giambelli formula for isotropic Grassmannians, arXiv:0811.2781, 2008) and Tamvakis (J Reine Angew Math 652, 207–244, 2011) to prove a tableau formula for the eta polynomials of Buch et al. (J Reine Angew, 2013) and the Stanley symmetric functions which correspond to Grassmannian elements of the Weyl group \(\widetilde{W}_n\) of type \(\text {D}_n\). We define the skew elements of \(\widetilde{W}_n\) and exhibit a bijection between the set of reduced words for any skew \(w\in \widetilde{W}_n\) and a set of certain standard typed tableaux on a skew shape \(\lambda /\mu \) associated to \(w\).