On Schubert's Problem of Characteristics

TL;DR

This paper provides a unified polynomial formula for expressing characteristics of flag manifolds G/P in terms of Cartan numbers, aiding the understanding of their cohomology rings.

Contribution

It introduces a new unified formula for the characteristics of flag manifolds G/P as polynomials in Cartan numbers, simplifying their cohomology analysis.

Findings

Derived a polynomial formula for flag manifold characteristics.

Applied the formula to clarify cohomology ring presentations.

Enhanced the computational approach to Schubert calculus.

Abstract

The Schubert varieties on a flag manifold G/P give rise to a cell decomposition on G/P whose Kronecker duals, known as the Schubert classes on G/P, form an additive base of the integral cohomology of G/P. The Schubert's problem of characteristics asks to express a monomial in the Schubert classes as a linear combination in the Schubert basis. We present a unified formula expressing the characteristics of a flag manifold G/P as polynomials in the Cartan numbers of the group G. As application we develop a direct approach to our recent works on the Schubert presentation of the cohomology rings of flag manifolds G/P.