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 \(w\).
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Notes
Note that if \(i<m<j\), then the factorization \(R_{ij}=R_{im}R_{mj}\) is not allowed.
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The author was supported in part by NSF Grants DMS-0901341 and DMS-1303352.
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Tamvakis, H. A tableau formula for eta polynomials. Math. Ann. 358, 1005–1029 (2014). https://doi.org/10.1007/s00208-013-0982-6
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DOI: https://doi.org/10.1007/s00208-013-0982-6