Schubert Polynomials and Degeneracy Locus Formulas

In previous work [T6], we employed the approach to Schubert polynomials by Fomin, Stanley, and Kirillov to obtain simple, uniform proofs that the double Schubert polynomials of Lascoux and Schützenberger and Ikeda, Mihalcea, and Naruse represent degeneracy loci for the classical groups in the sense of Fulton. Using this as our starting point, and purely combinatorial methods, we obtain a new proof of the general formulas of [T5], which represent the degeneracy loci coming from any isotropic partial flag variety. Along the way, we also find several new formulas and elucidate the connections between some earlier ones. 0. Introduction In the 1990s, Fulton [Fu1, Fu2] introduced a notion of degeneracy loci determined by flags of vector bundles associated to any classical Lie group G. For any two (isotropic) flags E• and F• of subbundles of a (symplectic or orthogonal) vector bundle E over a base variety M , and an element w in the Weyl group of G, there is a locus Xw ⊂ M defined by incidence relations between the flags. The degeneracy locus problem is to find a universal polynomial Pw in the Chern classes of the vector bundles involved such that [Xw] = Pw ∩ [M ]. We ask that Pw should be combinatorially explicit and manifestly respect the symmetries (that is, the descent sets) of both w and w, whenever possible. When G is the general linear group, the degeneracy locus problem was solved by Buch, Kresch, Yong, and the author [BKTY]. This answer was extended in a type uniform way to the symplectic and orthogonal Lie groups in [T5]. The formulas of [BKTY, T5] rely in part on a theory of Schubert polynomials, which express the classes of the degeneracy loci in terms of the Chern roots of the vector bundles E• and F•. The desired combinatorial theory of Schubert polynomials, together with its connection to geometry, was established in the papers [LS, L, Fu1] (for Lie type A) and [BH, T2, T3, IMN1, T5] (for Lie types B, C, and D). In previous work [T6, §7.3], we employed Fomin, Stanley, and Kirillov’s nilCoxeter algebra approach to Schubert polynomials [FS, FK] to give simple, uniform proofs that the double Schubert polynomials of Lascoux and Schützenberger [LS, L] and Ikeda, Mihalcea, and Naruse [IMN1] represent degeneracy loci of vector bundles, in the above sense. Our main goal in this paper is to begin with the same definition of Schubert polynomials from [T5, T6] and, by purely combinatorial methods, derive the splitting formulas for these polynomials found in [T5, §3 and §6]. The latter results then imply the general degeneracy locus formulas of [T5]. Date: February 18, 2016. 2010 Mathematics Subject Classification. Primary 14M15; Secondary 05E05, 14N15. The author was supported in part by NSF Grant DMS-1303352.