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I came across a property of monomials in a Schubert polynomial in Lascoux's book:

http://www-igm.univ-mlv.fr/~al/ARTICLES/CoursYGKM.pdf

page 62, footnote 4. The property is as follows.

Let us adopt the convention that Schubert polynomials in $\mathbb{Z}[x_1, \dots, x_n]$ are indexed by $\mathbb{N}^n$. Then for $v=(v_1, \dots, v_n)\in \mathbb{N}^n$, a monomial $x_1^{u_1}\dots x_n^{u_n}$ appears in $Y_v$ only if: $$u_n\leq v_n,~u_n+u_{n-1}\leq v_n+v_{n-1},~\dots,~u_n+u_{n-1}+\dots+u_1\leq v_n+v_{n-1}+\dots +v_1$$

Lascoux mentioned that "it is easy to prove by induction" that this holds. But I've tried and found no obvious clue for such a inductive proof.

So I would be very grateful if anyone gives some hint for this (or point out a reference for this)? Thank you.

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  • $\begingroup$ Are these Schubert polynomials really a reindexing of the classical Schubert polynomials indexed by permutations? Or are they a generalization? (I have to admit I am unable to see why Lascoux's definition is unambiguous to begin with...) $\endgroup$
    – darij grinberg
    Nov 9, 2014 at 6:45
  • $\begingroup$ Reindexing; see my answer $\endgroup$
    – Allen Knutson
    Nov 12, 2014 at 2:40

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I think I understand, though I haven't checked in great detail, nor attempted a naive proof. (So perhaps this doesn't count as an answer.)

Given a Schubert polynomial, consider the terms containing $x_m$ with the largest $m$, then of those terms the ones with $x_m$ to the largest power, then of those terms the ones with $x_{m-1}$ to the largest power, and so on; eventually you get to a unique term I'll call the leading term. The exponents on this term are the $v$ that Lascoux is using; I believe this is also the Lehmer code of the permutation.

Personally, I like to find this by making the Rothe diagram of the permutation [Lascoux, p300], with $v_i$ the number of boxes in the $i$th row.

If you shove the boxes to the left, you get the lowest pipe dream for this Schubert polynomial; see [Bergeron-Billey]. They show that all the other pipe dreams can be obtained by upward "chute" and "ladder" moves. I believe those moves will preserve the inequalities Lascoux states.

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  • $\begingroup$ But these Schubert polynomials aren't indexed by permutations! $\endgroup$
    – darij grinberg
    Nov 9, 2014 at 17:37
  • $\begingroup$ Did you read about the Lehmer code? It gives a bijection between finite permutations of $\mathbb N$ and finitely supported functions $\mathbb N \to \mathbb N$. $\endgroup$
    – Allen Knutson
    Nov 12, 2014 at 2:39
  • $\begingroup$ Oh -- you are saying that Lascoux's Schubert polynomials are essentially the usual permutation-indexed Schubert polynomials, whose variables have been stretched out? $\endgroup$
    – darij grinberg
    Nov 12, 2014 at 4:04
  • $\begingroup$ I don't understand the "stretched out". They are exactly the same as the usual ones, just reindexed, using the Lehmer code bijection. $\endgroup$
    – Allen Knutson
    Nov 13, 2014 at 11:21
  • $\begingroup$ Oh -- so the Lehmer code bijection is a bijection between $\mathfrak{S}_{(\infty)} = \bigcup\limits_{n \geq 0}\mathfrak{S}_n$ and the set of weak compositions. I was never aware of this; it is a very nice fact! (Nicer than the well-known finite versions.) $\endgroup$
    – darij grinberg
    Nov 13, 2014 at 18:36

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