Ch. 1 Lascoux POLYNOMIALS - Operators on polynomials
1.1 $A,B,C,D$
Simple Reflections
Recall in the classical Lie types, the simple reflections are the operators on $\RR^n$ given by
\[\begin{align*} vs_i &\coloneqq [\ldots, \, v_{i+1}, \, v_i, \, \ldots], \quad 1 \le i < n, \\ vs_i^B = vs_i^C &\coloneqq [\ldots, \, -v_{i}, \ldots], \quad 1 \le i \le n, \\ vs_i^D &\coloneqq [\ldots, \, -v_i, \, -v_{i-1}, \, \ldots], 2\le i \le n. \end{align*}\]In our case, we will assume that $v \in \ZZ^n$. The weyl group $W$ of type $\heartsuit$ is generated by:
\[\begin{cases} s_1, \ldots, s_{n-1} & \text{ if } \heartsuit = A_n, \\ s_1, \ldots, s_{n-1}, s_n^B & \text{ if } \heartsuit = B_n, C_n, \\ s_1, \ldots, s_{n-1}, s_n^D & \text{ if } \heartsuit = D_n. \end{cases}\]Orbit structure
The orbit of the vector $[1, 2, \ldots, n]$ under the type $A$ weyl group is the set of permutations of $1, \ldots, n$. In types $B$ and $C$, the orbit is all signed permutations. In type $D$, the orbit is all signed permutations with an even number of negative entries.
Braid relations
As the weyl group is a coxeter group, the generators satisfy the braid relations given by the Dynkin diagram interpreted as a coxeter graphs. As a reminder, the coxeter graphs of these types are as follows:
We are following the convention of Humphreys’: Reflection groups and Coxeter groups. No label on the edge $ss’$ implies that $m(s, s’) = 3$ and no edge $ss’$ implies $m(s, s’) = 2$.
1.3 Acting on polynomials with the symmetric group
In type $A$, we introduce various types of operators on the ring $\ZZ[x_1^{\pm 1}, \ldots, x_n^{\pm 1}]$. First, define
\[\begin{align*} x_i s_i = x_{i+1}. \end{align*}\]Note that $s_i$ acts on exponents of the monomials. Then we have
\[\begin{align*} \partial_i &\coloneqq (1 - s_i) \frac{1}{x_i - x_{i+1}}, \\ \pi_i &\coloneqq x_i \partial_i, \\ \widehat{\pi}_i &\coloneqq \partial_i x_{i+1}. \end{align*}\]1.10 $B,C,D$ action on polynomials
As we do for type $A$, we define operators on polynomials by acting on their exponents.
Type $B$
$s_i^B$ acts on $x_i$ by sending $x_i$ to $x_i^{-1}$. So then
\[\begin{align*} \partial_i^B &\coloneqq (1 - s_i^B)\frac{1}{x_i^{1/2} - x_i^{-1/2}}, \\ \pi_i^B &\coloneqq x_i^{1/2} \partial_i^B, \\ \widehat{\pi}_i^B &\coloneqq \partial_i^C x^{-1/2} \end{align*}\]for $i = 1, \ldots, n$.
[remark] I’m pretty sure $\partial_i^C$ is a typo here. Need to check that later. [/remark]
Type $C$
$s_i^C$ acts on $x_i$ by sending $x_i$ to $x_i^{-1}$. So then
\[\begin{align*} \partial_i^C &\coloneqq (1 - s_i^C)\frac{1}{x_i - x_i^{-1}}, \\ \pi_i^C &\coloneqq x_i \partial_i^C, \\ \widehat{\pi}_i^C &\coloneqq \partial_i^C x^{-1} \end{align*}\]for $i = 1, \ldots, n$.
Type $D$
$s_i^D$ acts on $x_{i-1}$ and $x_i$ by
\[\begin{align*} x_i &\mapsto x_{i-1}^{-1}, \\ x_{i-1} &\mapsto x_i^{-1}. \end{align*}\]So then
\[\begin{align*} \partial_i^D &\coloneqq (1 - s_i^D)\frac{1}{x_{i-1}^{-1} - x_i}, \\ \pi_i^D &\coloneqq \left(1 - s_i^D\frac{1}{x_{i-1}x_i}\right)\frac{1}{1 - \frac{1}{x_{i-1}x_i}}, \\ \widehat{\pi}_i^D &\coloneqq (1 - s_i^D)\frac{1}{x_{i-1}x_i - 1} \end{align*}\]for $i = 2, \ldots, n$.