expressing in terms of sum of (double) schubert polynomial

Question

It is well known that Schubert polynomials form a basis for the polynomial ring $\mathbb{Z}[x_1,x_2,x_3,...]$.

I am interested in knowing how to express a particular polynomial into sum of Schubert polynomial.

Does anyone can give me reference on program( say Macaulay2) expressing a particular polynomial in terms of Schubert polynomial? I did a google search before but could not find a satisfactory answer.

Also are there any fast ways in determining whether a polynomial is a sum of double Schubert polynomial?

Any hints or reference are appreciated.

Thank you very much!

Answer 1

First break $p$ into its homogeneous components, since Schubert polynomials are homogeneous.

Now find the lex-last term $c \prod_i x_i^{L_i}$ of $p$ -- look for the largest $n$ such that $x_n$ occurs in a term, then take the terms with the highest power of that $x_n$, then of those take the terms with the highest power of $x_{n-1}$, and so on. The resulting exponent vector $L$ is the Lehmer code of a permutation $\pi$, and its Schubert polynomial $S_\pi$ has that same lex-last exponent vector. Subtract $c S_\pi$ from $p$ and continue. This process terminates.

It's rather weird to ask whether a polynomial is a $\mathbb Z$-combination of double Schubert polynomials, insofar as they are a $\mathbb Z[y_1,y_2,\ldots]$-basis. To compute those expansions, the only modifications to the above recipe are that we look for the lex-last term of the highest degree in the $x$ variables, and the coefficient $c$ is now a polynomial in the $y$ variables. Your second question is then whether these $c$ all happen to be in $\mathbb Z$, but I can't imagine a natural situation where that would happen.