K-theoretic Chern class formulas for vexillary degeneracy loci

Abstract

Using raising operators and geometric arguments, we establish formulas for the K-theory classes of degeneracy loci in classical types. We also find new determinantal and Pfaffian expressions for classical cases considered by Giambelli: the loci where a generic matrix drops rank, and where a generic symmetric or skew-symmetric matrix drops rank.In an appendix, we construct a K-theoretic Euler class for evenrank vector bundles with quadratic form, refining the Chow-theoretic class introduced by Edidin and Graham. We also establish a relation between top Chern classes of maximal isotropic subbundles, which is used in proving the type D degeneracy locus formulas.Here p and q are the ranks of E and F , respectively, and c(F − E) is the K-theoretic Chern class. The leading term, where m = 0, is precisely Giambelli's determinantal formula in cohomology. 1 We will prove similar formulas for vexillary degeneracy loci, incorporating the stability properties of the cohomology formulas. 2 Our results generalize the formulas of [HIMN] for types A, B, and C. In general, the loci are described by conditions of the form dim(E p i ∩ F q i ) ≥ k i , and the results of [HIMN] are recovered as the special case where k i = i and p i = p for all i.

Vexillary degeneracy loci classes in K-theory and algebraic cobordism

“…Our proof is based on our joint work [14] with Ikeda and Naruse and on [15], where determinantal formulas were obtained for Grassmannian permutations. Independently from this work, Anderson [1] also proved a determinant formula for [O Xw ] in terms of Chern classes with a similar method.…”

A Tableau Formula of Double Grothendieck Polynomials for 321-Avoiding Permutations

“…(3.Example Let w = (31254). Then f (w) = (1, 4), h(w) = (3, 5), f c (w) = (2, 3, 5) and h c (w) =(1,2,4). We find that λ ′ = (3, 2), µ ′ = (2, 0), λ = (3, 1) and µ = (1, 0).…”

On the K-theoretic fundamental class of Deligne–Lusztig varieties

“…Actually, as pointed out in [1,Appendix A.2], the coefficient ring of CK * is isomorphic to Z[β]. Notice that our sign convention for β agrees with that of [17], while it is opposite to that of [1]. The class β also plays a central role in relating CK * with CH * and K 0 .…”

“…where β ∈ CK −1 (Spec k) is identified with the push-forward of the fundamental class of P 1 to the point. Actually, as pointed out in [1,Appendix A.2], the coefficient ring of CK * is isomorphic to Z[β]. Notice that our sign convention for β agrees with that of [17], while it is opposite to that of [1].…”