Degeneracy loci classes in K-theory — determinantal and Pfaffian formula

Abstract

We prove a determinantal formula and Pfaffian formulas that respectively describe the K-theoretic degeneracy loci classes for Grassmann bundles and for symplectic Grassmann and odd orthogonal bundles. The former generalizes Damon-Kempf-Laksov's determinantal formula and the latter generalize Pragacz-Kazarian's formula for the Chow ring. As an application, we introduce the factorial GΘ/GΘ ′ -functions representing the torus equivariant K-theoretic Schubert classes of the symplectic and the odd orthogonal Grassmannians, which generalize the (double) theta polynomials of Buch-Kresch-Tamvakis and Tamvakis-Wilson.

Symplectic and odd orthogonal Pfaffian formulas for algebraic cobordism

“…where E → X is a vector bundle of rank 2n with a nowhere vanishing skewsymmetric form and for every x ∈ X the fiber SG k (E) x is the symplectic Grassmannian of (n − k)-dimensional isotropic subspaces of E x . Exactly as Kazarian did in [10], in [6] for every k-strict partition λ the K-theoretic fundamental class of the Schubert variety X C λ was obtained by computing ψ * [Y C λ ] CK , the pushforward of the fundamental class of a resolution of singularities ψ : Y C λ → X C λ ֒→ SG k E. The final formula was expressed in terms of the relative Segre classes S CK i (U − E/F j ) ∨ , some characteristic classes associated to the tautological isotropic subbundle U and to the elements of the given reference flag of subbundles of E used to define the Schubert varieties 0 = F n ⊂ F n−1 ⊂ · · · ⊂ F 1 ⊂ F 0 ⊂ F −1 ⊂ · · · ⊂ F −n = E.…”

“…The following lemma is known from [6], where it was proved for CK * . One can easily check that the proof works for an arbitrary oriented Borel-Moore homology and in particular for Ω * .…”

“…In recent years, following the trend of generalised Schubert calculus, there has been an attempt to lift classical results in CH * to other functors like connective K-theory and algebraic cobordism, respectively denoted CK * and Ω * . In [6], together with T. Ikeda and H. Naruse, we established Pfaffian formulas describing the K-theoretic Schubert classes of symplectic and odd orthogonal Grassmann bundles which generalise those that have already been mentioned. The goal of this paper is to further extend these formulas to Ω * .…”

“…A (u) is the power series defined by the equation Recently, in [1], Anderson extended the results of [6] to more general degeneracy loci including those arising from even orthogonal Grassmann bundles. His work is based on the approach he and Fulton employed in their study of the Chow ring fundamental classes of degeneracy loci for all types [2,3].…”

Identities on factorial Grothendieck polynomials

“…On the other hand, there have been determinantal formulas for factorial Grothendieck polynomials [12][13][14]24]. For the purpose of this paper, we need the following determinantal formula for G λ (x|y) due to Ikeda and Naruse [14]:…”