It is known that the coefficients of the monomials appearing in a Schubert polynomial are always positive. My question is: Is it always true that at least one such coefficient must be $1$? If that is not always true, then is it true that the gcd of the coefficients of the monomials appearing in a Schubert polynomial must always be $1$?
It is known that the Schubert polynomials is an integral basis for the polynomials (if it was not a basis, then the problem of expanding a product of Schubert polynomials into Schubert polynomials would be kind of strange). That the basis is integral means that monomials expands into Schubert with integer coefficients.
As mentioned in the comments, there is an ordering such that the leading coefficient in this ordering is $1$ - this property implies that the basis is integral (is it perhaps equivalent to this property)?