Commit 2024-12-06 05:31 0ae2683a

feat: power of product of two reflections (#13270) Let $M$ be a module over a commutative ring $R$. Let $x, y \in M$ and $f, g \in M^*$ with $f(x) = g(y) = 2$. The corresponding reflections $r_1, r_2 \colon M \to M$ are given by $r_1z = z - f(z) x$ and $r_2 z = z - g(z) y$. These are linear automorphisms of $M$. We prove that for all $z \in M$ and all $m \in \mathbb{Z}$, we have: $$(r_1 r_2)^m z = z + S_{\left\lfloor \frac{m-2}{2} \right \rfloor}(t) (S_{\left\lfloor \frac{m-1}{2} \right \rfloor}(t) + S_{\left\lfloor \frac{m-3}{2} \right \rfloor}(t)) (g(x) f(z) y - g(z) y - f(z) x) + S_{\left\lfloor \frac{m-1}{2} \right \rfloor}(t) (S_{\left\lfloor \frac{m}{2} \right \rfloor}(t) + S_{\left\lfloor \frac{m-2}{2} \right \rfloor}(t)) (f(y) g(z) x - f(z) x - g(z) y)$$ $$(r_1 r_2)^m x = (S_{m}(t) + S_{m - 1}(t)) x - S_{m-1} g(x) y$$ $$r_2(r_1 r_2)^m x = (S_{m}(t) + S_{m - 1}(t)) x - S_{m} g(x) y$$ where $t = f(y) g(x) - 2$ and $S_n$ refers to a Chebyshev $S$-polynomial. These formulas may look ridiculous, but they are necessary for constructing reflection representations of a Coxeter group over an arbitrary ring. See #13291 for more details.