5
$\begingroup$

Let $W=\bigcup_{n=1}^\infty S_n$ be the union of all symmetric groups $S_n$. For an element $w\in W$, denote by $\mathfrak{S}_w$ the Schubert polynomial associated to $w$, and by $\partial_w$ the divided difference operator associated to $w$.

Question: Let $u,v\in W$. Why is $$ \partial_u(\mathfrak{S}_u\mathfrak{S}_v)-\mathfrak{S}_v $$ always a positive polynomial, in the sense that all coefficients are non-negative integers?

Preliminary thoughts:

  • It is clear that $\partial_u(\mathfrak{S}_u\mathfrak{S}_v)$ is positive.

  • One may use the twisted Leipniz rule and induction on the length of $u$ to reduce the problem to an apparently simpler one. Indeed, let $s_i$ be the rightmost factor in a reduced expression of $u$. Then it suffices to show that $\partial_{us_i}(s_i\mathfrak{S}_u\partial_i\mathfrak{S}_v)$ is positive. (Here, $W$ acts on $\mathbb{Z}[x_1,x_2,x_3,\ldots]$ by permuting the variables.)

  • I was thinking to realize the polynomials in question as characters of representations, in the spirit of the work of M. Watanabe on KP modules, but had not much success.

Any help would be greatly appreciated. I computed many examples and cannot find a counter example. In fact, I have a stronger "positivity conjecture" which implies this one, and even to this stronger conjecture, I was not able to find neither a counterexample nor a proof.

$\endgroup$
  • $\begingroup$ Is this expression perhaps positive in Demazure key polynomials, or Demazure atoms? $\endgroup$ – Per Alexandersson Jul 11 '17 at 11:44
  • $\begingroup$ Demazure polynomials / atoms might be relevant for the question - thanks - I googled them just now. It raises the question if Demazure polynomials / atoms are positive themselves. $\endgroup$ – user66288 Jul 11 '17 at 13:09
  • $\begingroup$ Both Demazure polynomials and Demazure atoms are positive - there are several characterizations of these (as sum over lattice points in polytopes, as sum over (subset of) semi-standard tableaux, sum over skyline fillings...) $\endgroup$ – Per Alexandersson Jul 11 '17 at 19:08
  • $\begingroup$ Why don't you post your stronger conjecture too? $\endgroup$ – Wolfgang Jul 12 '17 at 12:08
  • $\begingroup$ I was reluctant to post the stronger conjecture because it is less easy to state. Basically, if you expand the product $\partial_u(\mathfrak{S}_u\mathfrak{S}_v)$ via the twisted Leipniz rule and the lexicographically smallest reduced expression of $u$, you get many summands. One of them is $\mathfrak{S}_v$. I expect each of the summands to be a positive polynomial, but I am not completely sure about this, because I have only checked it heuristically for $u=s_\alpha$ - the case I am mostly interested in. $\endgroup$ – user66288 Jul 12 '17 at 15:55
2
$\begingroup$

You might want to try to prove Conjecture 1 in Kirillov's paper when the length difference between w and v is 2, starting from Liu's formula for skew divided difference operators in terms of /partial_{ij}.

$\endgroup$
1
$\begingroup$

I found the relevant literature for this question:

https://arxiv.org/abs/0705.4546

https://arxiv.org/abs/1409.7056

The keyword is "skew divided difference operator". In the first paper, Conjecture 1 states explicitely what I was trying to solve. An affirmative answer to this conjecture would immediately imply the positivity of the polynomial addressed in the original question.

As to the date of 2007, positivity of skew divided difference operators seems to be an open problem. If someone else knows other results in this direction (the second cited paper goes in the "right" direction), please comment below this answer.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.