In

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

$$e_1(\theta_1,\dots,\theta_n)=\cdots=e_n(\theta_1,\dots,\theta_n)=0,$$

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.