A Thom-Porteous formula for connective K-theory using algebraic cobordism

Under the assumption that the base field k has characteristic 0, we prove a formula for the push-forward class of Bott-Samelson resolutions in the algebraic cobordism ring of the flag bundle. We specialise our formula to connective K-theory providing a geometric interpretation to the double β-polynomials of Fomin and Kirillov by computing the fundamental classes of schubert varieties. As a corollary we obtain a Thom-Porteous formula generalising those of the Chow ring and of the Grothendieck ring of vector bundles.

On the generalised Todd genus of flag bundles

Let V be a complex algebraic variety. Given integers a1, …, am such thatone defines a (a1, …, am)-flag as a nested systemof subspaces of Sn, the n-dimensional complex projective space. The set of all such flags is called an incomplete flag-manifold in Sn, and is denoted by W(al, …, am). Also let E be a complex n-dimensional vector bundle over V. Then we denote by E(a1, …, am−1, n; V) an associated fibre bundle of E with fibre W(a1 − 1, …, am−1 − 1, n − 1). E(a1, …, am−1 − 1, n; V) is called an incomplete flag bundle of E over V (cf. (2), (3)). In Section 10.3 and Section 14.4 of (1), the generalised Todd genus Ty(W(0, n)) and Ty(W(0, 1, …, n)) of the n-dimensional projective space W(0, n) and the flag manifold W(0, 1, …, n) (or F(n+l)) were calculated. Here we compute Ty(W(a1, …, am)) and also Ty(E(a1, …, am−1, n; V)).