Schubert Polynomials for Complex Projective Space

Question

The Borel picture of the cohomology ring of a flag variety gives a description as a coset space, without identifying any representatives for the classes. Lascoux and Schützenberger gave specific representatives in terms of the so called Schubert polynomials. Does anyone know of a presentation of this material for the simplest case of projective space.$\,\,$

Answer 1

In general for a Grassmannian $Gr_k(\mathbb C^n)$, the Schubert polynomials are Schur polynomials in $k$ variables, one for each partition $\lambda$ in a $k\times (n-k)$ rectangle. In this case $S_\lambda = \sum_{SSYT(\lambda)} \prod_{content} x_{entry}$, where the sum is over semistandard Young tableaux with entries $\leq k$.

For $k=1$, these polynomials are $(x_1^i)_{i=0,\ldots,n-1}$, which shouldn't be much of a surprise. For $k=n-1$ they're $(x_1\cdots x_i)_{i=0,\ldots,n-1}$ which is perhaps moreso. In both cases each $\lambda$ (with one row or one column, respectively) affords exactly one SSYT.

I doubt you're going to learn much about the theory from the projective space case. The Grassmannian case is a good place to start though.