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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.

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

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$R_x$ etc. are rotations about the indicated axis. See for example this article in chem.libretexts or this pdf

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  • $\begingroup$ Thanks for replying. Just it is super short and general. I extended a bit the question, hoping that this makes it easier to address more precisely the question. $\endgroup$ Commented Jul 13, 2022 at 9:39
  • $\begingroup$ 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, $$\begin{bmatrix}1&0&0\\0&0&1\\0&-1&0\end{bmatrix}$$ $\endgroup$
    – Karsten
    Commented Aug 12, 2022 at 13:54

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