Combinatorics of Quantum Schubert Polynomials

Let $S_n$ be the symmetric group. Let $s_i$ denote the adjacent transposition $(i \ i+1)$. For any permutation $w\in S_n$, an expression $w=s_{i_1}s_{i_2}\cdots s_{i_p}$ of minimal possible length is called a reduced decomposition of $w$, where $p=\ell(w)$, the length of $w$ (number of inversions). For any function $f=f(x_1,\ldots,x_n)$ and $w\in S_n$, define $wf:=f(x_{w^{-1}(1)},\ldots,x_{w^{-1}(n)})$ and define the divided difference operator $\partial_i$ via $\partial_i f:=\frac{f-s_if}{x_i-x_{i+1}}$.

The Schubert polynomials $\mathfrak{G}_w$ are defined as follows: $\mathfrak{G}_w:=\partial_{w^{-1}w_0}x^\delta$, where $w_0$ is the longest element of $S_n$, given by $w_0(i)=n+1-i$ and $x^\delta:=x_1^{n-1}x_2^{n-2}\cdots x_{n-1}$. There are correspondingly double Schubert polynomials $\mathfrak{G}(x,y)$, which are defined through $\mathfrak{G}_w(x,y)=\partial_{w^{-1}w_0}\prod_{i+j\leq n}(x_i-y_j)$.

On the other hand, quantum Schubert polynomials $\mathfrak{G}^q_w$ are defined by Fomin, Gelfand, and Postnikov in this paper (see Theorem 1.1) and by Kirillov in this paper as well. Roughly speaking, one expands the usual Schubert polynomials in terms of elementary polynomials and then replaces each elementary polynomial with a deformed quantum elementary polynomial. See (1.3) and (1.4) of the above link for further details. There are correspondingly double quantum Schubert polynomials $\mathfrak{G}(x,y)$

Here is slightly more background: In their paper on the Nil-Coxeter algebra and Schubert Polynomials, Fomin and Stanley present a beautiful construction of Schubert polynomials using operators $A_i(x)$ which are defined on the so called Nil-Coxeter Algebra. You first define the Nil-Coxeter algebra via the generators

$$u_i^2=0$$ $$u_iu_j=u_ju_i, \ \ \ |i-j|\geq 2$$ $$u_iu_{i+1}u_i=u_{i+1}u_iu_{i+1}$$

and then the operators $A_i(x):=(I+xu_{n-1})(I+xu_{n-2})\cdots (I+xu_i)$ from which one can show that $G_w(x_1,\ldots,x_n):=A_1(x_1)A_2(x_2)\cdots A_{n-1}(x_{n-1})$ correspond naturally to the usual Schubert polynomials above when one projects $G_w$ onto $w$.

FIRST QUESTION: Is there any hope for generalizing the operators $A_i(x)$ to something akin to $A_i^q(x)$ or similar in hopes of writing quantum Schubert polynomials in the form $G_w^q=A^q_1(x_1)\cdots A^q_{n-1}(x_{n-1})$? Note that in their definition, quantum Schubert polynomials actually have $n-1$ $q_i$ terms.

SECOND QUESTION: Purely in terms of reduced decompositions (and not anything related to quantum cohomology, Schubert varieties or Gromov-Witten invariants), what exactly do quantum Schubert polynomials count? It's well known that Schubert polynomials and their infinite relatives can be used for example to count the number of reduced words of a permutation $w\in S_n$. The infinite Schubert polynomial $G_w(x)=A_1(x_1)A_1(x_2)\cdots$ projected onto $w$ corresponds exactly to the Stanley Symmetric function:

$$G_w(x)=\sum_{(s_1,\cdots,s_p)\in R(w)}\sum_{\overset{\overset{b_1\cdots, b_p}{1\leq b_1\leq\cdots \leq b_p}}{s_i<s_{i+1}\Rightarrow b_i<b_{i+1}}}x_{b_1}\cdots x_{b_p}$$

where $R(w)$ is the set of all reduced decompositions of $w$. Then we have the famous result $|R(w)|=[x_1\cdots x_p]G_w(x)=\sum_{\lambda\vdash p}\alpha_{w\lambda}[x_1\cdots x_p]s_\lambda==\sum_{\lambda\vdash p}\alpha_{w\lambda}f^\lambda$, where $\lambda$ are standard Young tableau, $f^\lambda$ is the number of standard Young tableau of shape $\lambda$ and $s_\lambda$ are Schur polynomials. The $\alpha_{w\lambda}$ coefficients have a combinatorial interpretation due to Fomin and Greene: they count the number of Semistandard Young Tableau of shape $\lambda$ such that the row-read word is a reduced decomposition for $w$. It seems to me that in the quantum version, one gets quantum Schur polynomials instead, with quantum $\alpha_{w\lambda}$ but this is as far as I can see. What would the $\alpha^q_{\lambda w}$ count in the quantum case?

Here is another interpretation. In their paper on the Yang Baxter equation and Schubert Polynomials, Fomin and Kirillov show that Schubert polynomials can be constructed by looking at braid relations on wiring diagrams (see for example Fig 10, pg 134). In this way, how do quantum Schubert polynomials count such generalized configurations? More importantly, what do the weights $q_i$ contribute to the counts?

Here is an example: for $w=321$, the double Schubert polynomial can be written as

$$\mathfrak{G}_w(x,y)=(x_1-y_2)(x_1-y_1)(x_2-y_1)$$

which has the interpretation that we assign weights $x_i,y_i$ to the threads (pseudolines) of any reduced decomposition of $w$ and thus we are explicitly showing which lines cross at a given time. The quantum version is $$\mathfrak{G}^q_w(x,y)=(x_1-y_2)(x_1-y_1)(x_2-y_1)+q_1(x_1-y_2)$$.

What exactly is this $q_1$ counting?

• The first paper you link to is by Fomin, Gelfand, and Postnikov, not Kirilov, Fomin et al. – Sam Hopkins Nov 5 '13 at 2:16
• @SamHopkins: thanks, I've added another link. – Alex R. Nov 5 '13 at 19:14