# What does it mean that a state belongin to a given irrep transforms like $Rx$, $Ry$ or $Rz$

The present question is related to this other question I did few days ago. Given a point group and the list of the irreps (see for example here) the meaning of an irrep which transforms like $$x$$ or $$x^2$$ is clear to me.

Instead I do not understand what is meant when it is written that an irrep transforms like $$Rx$$.

Let me explain a bit better. Say that the group is the symmetry group o a crystal structure. It can be represented as a set of $$3\times3$$ matricies $$M_i$$. These matrices can be used to transform the vector $$v=(x,y,z)$$. So I can clearly see how something which transforms like x would be affected: $$v'=M_iv$$ and $$v'=(x',y',z')$$. I can also assume that any function of $$f(x,y,z)$$, like $$f(x,y,z)=x^2$$ would be transformed like $$f(x,y,z) \rightarrow f(x',y',z')$$ (is this point true?). However I do not understand how $$Rx$$ would transform. It is not even a defined function, but a group of operations by itself.

• Assign a direction to the rotation, then perform the transformation and see if the direction of rotation is the same (symm) or reversed (antisymm). Commented Jul 13, 2022 at 11:59

$$R_x$$ etc. are rotations about the indicated axis. See for example this article in chem.libretexts or this pdf
• If you want a function for Rx, try the x-component of angular momentum: $$L_x = y p_z − z p_y.$$ If you want a matrix, use the rotation matrix that rotates 90 deg about the x-axis, $$\begin{bmatrix}1&0&0\\0&0&1\\0&-1&0\end{bmatrix}$$