A tableau formula for eta polynomials

Abstract

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 $$w$$ .