Transition equations for double Schubert polynomials

Question

For $w$ a permutation, the associated (ordinary) Schubert polynomial $\mathfrak{S}_w(\textbf{x})$ is a multivariate polynomial that represents for the cohomology class of the Schubert variety $X_w$ in the flag variety. There are many references for this fact. See, e.g., this text by Manivel. One key feature of Schubert polynomials is that they satisfy a recurrence often referred to as the Lascoux-Schutzenberger transition equation. Let $r$ be the largest value such that $w(r)>w(r+1)$ and $s$ be the maximum value so that $w(s) < w(r)$. We define $$ I(w,r) = \{i < r: w \lessdot w t_{ir}\}, $$ where $t_{rs}$ is the transposition $(r,s)$ and $\lessdot$ indicates a cover relation in Bruhat order. Then $$ x_r \mathfrak{S}_w(\textbf{x}) = \mathfrak{S}_{w t_{rs}}(\textbf{x}) - \sum_{i \in I(w,r)} \mathfrak{S}_{w t_{ir}}(\textbf{x}). $$ This result dates back to work of Lascoux and Schutzenberger in the 80's and can be derived from Monk's formula. There is a simple extension allowing one to work with arbitrary descents in $w$.

The double Schubert polynomial $\mathfrak{S}(\textbf{x},\textbf{y})$ is a polynomial in two alphabets of variables that represents the equivariant cohomology class of the Schubert variety $X_w$ in the flag variety. These polynomials satisfy an analogous recurrence $$ (x_r - y_s) \mathfrak{S}_w(\textbf{x},\textbf{y}) = \mathfrak{S}_{w t_{rs}}(\textbf{x},\textbf{y}) - \sum_{i \in I(w,r)} \mathfrak{S}_{w t_{ir}}(\textbf{x},\textbf{y}). $$ I've seen this identity proved in a class and can rederive it myself (as well as the more general transition equations), but can't seem to find an explicit reference in the literature. I've asked around a bit, but haven't turned anything up. Can anyone point me to a concrete reference of the transition equations for double Schubert polynomials (or something equivalent) in the literature?

P.S. There is an unpublished preprint by Lascoux called "Chern and Yang Through Ice" that describes transition for double Grothendieck polynomials (a stronger result), but I'm wary of giving it as a reference for two reasons. First, it claims a stronger result that requires a bit of work to translate to my setting. Second, I find the whole document a great challenge to parse, and am skeptical that a complete proof is presented.

Answer 1

On pg. 5 of https://arxiv.org/abs/1909.13777, Allen Knutson suggests that Lascoux proves a transition formula for double Schubert polynomials in the paper "Transition on Grothendieck polynomials", Physics and Combinatorics 2000, pp. 164-179, available online (if you pay for it) at https://www.worldscientific.com/doi/abs/10.1142/9789812810007_0007. But in that sentence maybe Knutson just means, the double version of what Lascoux does.

See also these slides of Alex Yong (which discuss joint work with Knutson which I think Knutson explains in the above arXiv preprint was ultimately unpublished) explaining a Gröbner basis approach: https://faculty.math.illinois.edu/~ayong/slides_AMS.04.10.18.pdf.

EDIT:

Okay, so having obtained the Physics and Combinatorics 2000 article "via undisclosed means", in the introduction Lascoux says

The cohomological computations can easily be extended to two sets of indeterminates, due to the extension of Monk's rule (cf. 10), or translated into planar combinatorial constructions. The same is true for Grothendieck polynomials , but for simplicity, we shall restrict here to only one set of indeterminates.

So I'm not sure how much this is a reference for double anything. The reference 10 for the double Monk's rule is: A. Konhert, S. Veigneau, Using Schubert basis to compute with multivariate polynomials, Adv. Appl. Math. 19, 45-60 (1997). (https://www.sciencedirect.com/science/article/pii/S0196885897905261)

EDIT 2:

So doing a bit more googling, I can, oddly enough, find a very specific reference for the transition formula for double Schubert polynomials of Types B,C,D (which is of course not what you asked for) in Proposition 6.12 of "Double Schubert polynomials for the classical groups" by Ikeda, Mihalcea, Naruse, Advances in Mathematics 226 (2011) 840–886. (https://core.ac.uk/download/pdf/81995975.pdf)