I am studying the article "Symmetrization operators in polynomial rings" by A. Lascoux and M.-P. Schützenberger (MSN). Specifically, I am trying to prove the following claim involving operators defined on the ring of rational functions $\mathbb{C}(x_1, x_2, x_3)$, where the variables $x_1, x_2, x_3$ are permuted by the simple transpositions $s_i = (i, i+1)$, for $i = 1, 2$.
Define linear operators $D_i$, $i = 1, 2$, as:
$$D_i = P_i + Q_i s_i,$$where$$ P_i(x_i, x_{i+1}) = \frac{(\alpha x_i + \beta)(\gamma x_{i+1} + \delta)}{x_i - x_{i+1}}, \quad Q_i = \eta - P_i,$$and $\alpha, \beta, \gamma, \delta, \eta \in \mathbb{C}$.
Given:
- $D_1 D_2 D_1 = D_2 D_1 D_2$,
- $D_1$ is invertible, and $P_1 \neq 0$.
Claim:
We must have $\Delta := \alpha \delta - \beta \gamma \neq 0$, $\eta \neq 0$, and $\eta \neq \Delta$.
Brute-force computation quickly becomes infeasible, and the authors hint that a deeper approach using non-commutative operator calculus associated with reflection groups (E. Gutkin) might be necessary. However, this technique seems challenging to navigate without more intuition or intermediate insights.
Can anyone provide comments or suggestions on how to approach such computations? Specifically, I am seeking ideas, conceptual guidance, or references that could shed light on a more efficient approach to this problem.
