Let $D_{4}$ be the dihedral group of a square, $r$ and $s$ denote rotation by $90^{\circ}$ in the clockwise direction and reflection respectively. Suppose through a series of reflections and rotations you attained a reflection prove that there are odd number of reflections you have done.
It can be proved by using the relation $rs=sr^{-1}$ and that every element of the dihedral group can be written uniquely as $ r^is^j,0\leq i\leq3,0\leq j\leq1$.
I'm looking for a proof by the action of dihedral group on a polynomial say $f(x_1,x_2,x_3,x_4) \in \mathbb{Z}[x_1,x_2,x_3,x_4]$ (I label the vertices from 1 to 4 cyclically) such that sign of the polynomial changes when we apply a reflection and its unchanged by a rotation.
I tried using the polynomial $f(x_1,x_2,x_3,x_4) = \prod_{1\leq i < j \leq 4} (x_i-x_j)$ but it gives a negative sign for a rotation.
Thanks in advance
Best Answer
Might this be the kind of thing you're after? Take $$ f(x_1,x_2,x_3,x_4)=x_1x_2x_3^2+x_2x_3x_4^2+x_3x_4x_1^2+x_4x_1x_2^2 $$ For any point in $\Bbb Z^4$, cyclically permuting the coordinates clearly doesn't change the value of the function, but flipping them does: $$ f(0,1,2,3)=18\neq 6=f(3,2,1,0) $$ Or, if you want a polynomial such that cyclic permutation leaves it unchanged while flipping just changes the sign, then $f(x_1,x_2,x_3,x_4)-f(x_4,x_3,x_2,x_1)$ is the one you're looking for.