Schubert Polynomials for the Classical Groups

Double eta polynomials and equivariant Giambelli formulas

“…while type(λ) > 0 if and only if |w 1 | > 1, and in this case type(λ) is equal to 1 or 2 depending on whether w 1 > 0 or w 1 < 0, respectively. Using this bijection, we attach to any typed k-strict partition λ a finite set of pairs C(λ) := {(i, j) ∈ N × N | 1 i < j and w k+i + w k+j < 0} (2) and a sequence β(λ) = {β j (λ)} j 1 defined by…”

“…(The condition w k+i + w k+j < 0 in (2) is equivalent to λ i + λ j 2k + j − i.) For example, the 3-Grassmannian element w = (−4, 6, 8, −5, −2, −1, 3, 7) of W 8 corresponds to the 3-strict partition λ = (7, 4, 3, 3, 1) of type 2, and we have C(λ) = { (1,2), (1,3), (1,4), (2,3)} and β(λ) = (−4, −1, 0, 3, 7).…”

“…Here E n denotes a maximal isotropic subbundle of the (pullback of) E to the complete flag variety, which is in the same family as F n . Note that the images of the elements b p , b k , c p in the ring of type D Billey-Haiman Schubert polynomials [2] are given in [5,Section 5] by the power series η p (x; y), η k (x; y), ϑ p (x; y), respectively, and, using this, the equations defining π n are derived in [13,Subsection 7.4]. For more information on the image of the double eta polynomials H λ (c | t) in the ring of type D double Schubert polynomials of [7], see [14,Subsection 4.5].…”

Schubert polynomials and Arakelov theory of symplectic flag varieties

“…be a third sequence of commuting variables. In their fundamental paper [2], Billey and Haiman defined a family {C w } w∈W∞ of Schubert polynomials of type C, which are a free Z-basis of the ring Γ[Z]. The expansion coefficients for a product C u C v in the basis of Schubert polynomials agree with the Schubert structure constants on symplectic flag varieties for sufficiently large n. For every w ∈ W n there is a unique expression…”

“…The search for a good theory of Schubert polynomials in types B, C, and D has received much attention in the past (for example, see [2,11,13,14,25,29,30]). The best understood theory from the combinatorial point of view appears to be that of Billey and Haiman [2], whose Schubert polynomials form a Z-basis for a polynomial ring in infinitely many variables. Unfortunately, when expressed in their most explicit form, the Billey-Haiman polynomials are not suitable for the above-mentioned applications, because the variables used are not geometrically natural.…”