Fomin, Sergey; Kirillov, Anatol N. Quadratic algebras, Dunkl elements, and Schubert calculus. Advances in geometry, 147--182, Progr. Math., 172, Birkhäuser Boston, Boston, MA, 1999. MR1667680 (2001a:05152), link

the authors introduce a combinatorial model for modern Schubert calculs. They define an algebra which contains a commutative subalgebra isomorphic to the cohomology ring of the flag manifold. More precisely, for $n\geq2$ they define $\mathcal{E}_n$ as the algebra with generators $x_{(ij)}$, $1\leq i<j\leq n$, and relations

  • $x_{(ij)}^2 = 0$,
  • $x_{(ij)}x_{(jk)}=x_{(ik)}x_{(ij)}+x_{(jk)}x_{(ik)}$ for $i<j<k$,
  • $x_{(jk)}x_{(ij)}=x_{(ij)}x_{(ik)}+x_{(ik)}x_{(jk)}$ for $i<j<k$,
  • $x_{(ij)}x_{(kl)}=x_{(kl)}x_{(ij)}$ for $\#\{i,j,k,l\}=4$.

Inside this algebra they define the so-called Dunkl elements $\theta_1,\dots,\theta_n$ as

$$\theta_j=-\sum_{1\leq i<j}x_{(ij)}+\sum_{j<k\leq n}x_{(jk)}$$

and prove that the Dunkl elements commute pairwise. Fomin and Kirillov prove that the complete set of relations among the Dunlk elements is given by


where $e_i$ denotes the $i$-elementary symmetric polynomial. Thus the commutative subring of $\mathcal{E}_n$ generated by the Dunkl elements is isomorphic to $\mathbb{Z}[x_1,\dots,x_n]/\langle e_1,\dots,e_n\rangle$, where $\langle e_1,\dots,e_n\rangle$ is the ideal generated by the symmetric polynomials $e_1,\dots,e_n$. Using a theorem of Borel one then obtains that this quotient and the cohomology ring of the flag manifold are isomorphic.

The paper contains the at least two partial motivations for introducing the algebra $\mathcal{E}_n$. The authors give two interesting representations related to $\mathcal{E}_n$.

The first one is the so-called Bruhat representation, where $x_{[ij]}$ acts on the space of linear combinations of elements of the symmetric group as $$ x_{[ij]}w=\begin{cases} w(ij) & \text{if $l(w(ij))=l(w)+1$},\\ 0 & \text{otherwise}, \end{cases} $$

where $l(w)$ denotes the length of $w\in\mathbb{S}_n$. These operators $x_{[ij]}$ satisfy the defining-relations of $\mathcal{E}_n$ but these relations do not form a complete set of relations for our operators. Hence $\mathcal{E}_n$ is a quadratic cover of the algebra generated by the Bruhat operators $x_{[ij]}$.

The other representation is related to divided differences. The operator $\partial_{(ij)}$ acts on $\mathbb{Z}[x_1,\dots,x_n]$ as

$$\partial_{(ij)}\cdot f=\frac{f-(ij)\cdot f}{x_i-x_j},$$

where $(ij)\cdot f$ is the polynomial $f$ with the variables $x_i$ and $x_j$ interchanged. Then the operators $\partial_{(ij)}$ satisfy the defining relations of $\mathcal{E}_n$. As before, there are more relations and hence $\mathcal{E}_n$ is a quadratic cover of the algebra generated by the operators $\partial_{(ij)}$.

Despite these are nice motivations for defining $\mathcal{E}_n$, I am not completely satisfied.

I would like to ask where the algebra $\mathcal{E}_n$ comes from. More precisely: Which is the "right" motivation for the definition of $\mathcal{E}_n$? Maybe there is some kind of motivation related to the representation theory of the symmetric group and its connection with Schubert calculus.


(I know this is an old post, but maybe this'll be of interest to future searchers.)

As far as Fomin--Kirillov's purposes are concerned, the Bruhat representation you mention is all that's really important, and my impression is that they restricted to the quadratic relations for simplicity and generality since the quadratic relations were enough for their purposes.

A little more explicitly, Fomin--Kirillov's main argument is that under the Bruhat representation acting on Schubert polynomials, the Dunkl element $\theta_j$ is just multiplication by $x_j$ via Monk's formula. Now applying an element of the positive cone $\mathcal{E}_n^+$ under the Bruhat representation to a Schubert polynomial gives a manifestly positive sum of Schubert polynomials. Fomin--Kirillov wanted to get a Schubert-Schubert rule, so we try to take advantage of the Monk's rule fact that

$$ \mathfrak{S}_v(x_1, \ldots, x_n) \mathfrak{S}_w(x_1, \ldots, x_n) = \mathfrak{S}_v(\theta_1, \ldots, \theta_n) \cdot \mathfrak{S}_w(x_1, \ldots, x_n).$$

However, you really need commutativity of the $\theta_j$'s for the right-hand side to be well-defined. (You could use Billey-Jockush-Stanley to expand $\mathfrak{S}_v$ in the standard monomial basis, for instance, but commutativity is clearly desirable, and that's what FK went for.) The real work in Fomin--Kirillov is proving this commutativity statement, and for that they only ended up needing the quadratic relations from the Bruhat representation. For what it's worth Postnikov's quantum Pieri rule follow-up and the recent work of Meszaros et al using the Fomin--Kirillov approach all just uses the quadratic relations. That said, Meszaros et al's rules are quite complicated--perhaps they'd be simpler if, say, cubic relations had been allowed?

One last addendum: Sottile--Bergeron use operators quite similar to the Bruhat operators in "A monoid for the Grassmannian Bruhat order," and they're able to give a monoid with only quadratic and cubic relations for which the operators yield a faithful representation. Maybe something similar holds for the Bruhat representation.

(Again, this is all just my impression. You could always ask Fomin or Kirillov.)


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